# Average Rényi Entanglement Entropy in Gaussian Boson Sampling

Jason Youm<EMAIL_ADDRESS>Joint Quantum Institute, NIST/University of
Maryland, College Park, Maryland 20742, USA  Joint Center for Quantum
Information and Computer Science, NIST/University of Maryland, College Park,
Maryland 20742, USA  Montgomery Blair High School, Silver Spring, Maryland
20901  Joseph T. Iosue<EMAIL_ADDRESS>Joint Quantum Institute,
NIST/University of Maryland, College Park, Maryland 20742, USA  Joint Center
for Quantum Information and Computer Science, NIST/University of Maryland,
College Park, Maryland 20742, USA  Adam Ehrenberg  Joint Quantum Institute,
NIST/University of Maryland, College Park, Maryland 20742, USA  Joint Center
for Quantum Information and Computer Science, NIST/University of Maryland,
College Park, Maryland 20742, USA  Yu-Xin Wang  Joint Quantum Institute,
NIST/University of Maryland, College Park, Maryland 20742, USA  Joint Center
for Quantum Information and Computer Science, NIST/University of Maryland,
College Park, Maryland 20742, USA  Alexey V. Gorshkov  Joint Quantum
Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
Joint Center for Quantum Information and Computer Science, NIST/University of
Maryland, College Park, Maryland 20742, USA

(February 11, 2024)

###### Abstract

Recently, many experiments have been conducted with the goal of demonstrating
a quantum advantage over classical computation. One popular framework for
these experiments is Gaussian Boson Sampling, where quadratic photonic input
states are interfered via a linear optical unitary and subsequently measured
in the Fock basis. In this work, we study the modal entanglement of the output
states in this framework just before the measurement stage. Specifically, we
compute Page curves as measured by various Rényi-$\alpha$ entropies, where the
Page curve describes the entanglement between two partitioned groups of output
modes averaged over all linear optical unitaries. We derive these formulas for
$\alpha=1$ (_i.e._ the von Neumann entropy), and, more generally, for all
positive integer $\alpha$, in the asymptotic limit of infinite number of modes
and for input states that are composed of single-mode-squeezed-vacuum state
with equal squeezing strength. We then analyze the limiting behaviors when the
squeezing is small and large. Having determined the averages, we then
explicitly calculate the Rényi-$\alpha$ variance for integers $\alpha>1$, and
we are able to show that these entropies are weakly typical.

##  1 Introduction

Major advances in quantum computing over the past decade have opened up
potential breakthroughs in various fields of science [1, 2]. One such
advancement is Gaussian Boson Sampling (GBS), an experimental framework for
demonstrating quantum advantage that produces a sample from a distribution
that is classically hard to compute [3, 4, 5, 6]. In this framework, squeezed
photons are transformed by a Haar-random, passive Gaussian unitary consisting
of beamsplitters and phase shifters. Although GBS is not computationally
universal, it is nonetheless crucial for understanding the leverage of quantum
devices over their classical counterparts at performing specific tasks.

The preparation of squeezed states is a much easier task than that of single
photons used in Fock-state boson sampling, hence making GBS an experimentally
favorable avenue toward demonstrating quantum advantage. Indeed, such
demonstrations have already been realized in the lab [7, 8, 9].

There is strong evidence that classically sampling from the output probability
distribution of a GBS experiment is hard in the complexity-theoretic sense [7,
8, 9, 10, 11], whereas such sampling can be done efficiently on a quantum
device. Moreover, entanglement is a critical asset across quantum computing,
with additional applications in teleportation, dense coding, and quantum
communication [12, 13, 14, 15]. Therefore, quantifying the behavior of
entanglement in Gaussian bosonic systems has been of recent interest [7, 8],
especially to sharpen the connection between entanglement and computational
complexity.

In 1993, Page conjectured an explicit formula for the average entanglement
entropy over all Haar-random pure states in an $N$-dimensional Hilbert space
of the reduced density matrix on an $rN$-dimensional subsystem for
$r\in[0,1]$. This conjecture was proven soon after in the $N\to\infty$ limit,
and the average entanglement entropy as a function of $r$ is now known as the
Page curve [16, 17, 18]. The Page curve was initially studied in the context
of information dynamics in and around black holes [19, 20, 21, 22], and has
since found applications in open quantum systems [23, 24] and quantum
information theory [25, 26, 27]. As a natural follow-up, many works
investigated the deviation of entanglement entropy from its expected value. A
system exhibits typical entanglement if the probability that a random state
has entanglement bounded away from the average is small. This behavior, along
with strong and weak classes of entanglement typicality, has been studied in
black holes, thermodynamics, and other systems [28, 29, 30, 31, 32].

Leaving the realm of standard finite-dimensional systems, the Page curve for
fermionic Gaussian states has been extensively analyzed [33, 34, 35]. A
natural next step, especially given the relevance to GBS, is to study bosonic
Gaussian states. The typicality of entanglement in random bosonic Gaussian
states was studied in Refs. [36, 37, 38, 39], and the Page curve for Gaussian
bosonic systems was first studied in Ref. [39]. However, Ref. [39] only
computed the Page curve for the Rényi-2 entropy.

In this work, we derive analytic expressions for the Rényi-$\alpha$ Page
curves of bosonic Gaussian systems for all $\alpha\in\mathbb{Z}_{>0}$, with
special considerations for the von Neumann (Rényi-1) Page curve. The entropy
of a mixed state is an indicator of its entanglement with the rest of the
universe [40], and indeed the Rényi-$\alpha$ Page curve is a measure of the
entanglement of an average state in a system as measured by the Rényi-$\alpha$
entropy, for $\alpha\in\mathbb{R}^{+}$. Each Rényi-$\alpha$ entropy has its
own unique mathematical features, and notable examples are the von
Neumann/Shannon entropy ($\alpha=1$), collision entropy ($\alpha=2$), and min-
entropy ($\alpha\to\infty$). In particular, the min-entropy, the entropy
defined by $-\log\text{max}({p_{i}})$ for each $p_{i}$ a probability of
occurence of outcome $i$ when working with a discrete random variable, is a
useful expression for working with extracting randomness from variables when
lower entropies (e.g. von Neumann) do not suffice [41]. Therefore, it is
natural to analyze the Rényi-$\alpha$ entropies in bosonic systems and extract
their unique properties.

Our expressions are asymptotically exact in the number of modes $n$. Using
these expressions, we prove various results about typicality of entanglement
with respect to these Rényi-$\alpha$ entropies. Our setup is exactly that of a
GBS experiment, and is the same as the setups considered in Refs. [38, 39]. A
pure input quantum state with $n$ squeezed vacuum modes is acted on by a Haar-
random, linear unitary (passive Gaussian) circuit. The Page curve, as a
function of $r$, is the average Rényi-$\alpha$ entanglement between a group of
$rn$ modes and the remaining system of $(1-r)n$ modes, where $r\in[0,1].$ We
expound our setup in greater detail in Section 2.

In Section 3, we present our results for the von Neumann entropy. Under an
equal squeezing regime, we first present an explicit formula for the von
Neumann Page curve in Section 3.1 in terms of the partition ratio $r$ and
initial squeezing strength $s$, which is exact up to terms that decay to zero
as $n\rightarrow\infty.$ We subsequently discuss the von Neumann entropy under
squeezing limits $s\rightarrow 0$ and $s\rightarrow\infty$ in Section 3.2.

We then generalize our analyses to the Rényi-$\alpha$ entropies for integers
$\alpha\geq 2$ in Section 4. In Section 4.1, we derive the Rényi-$\alpha$ Page
curve in terms of the partition ratio $r$ and initial squeezing strength $s$,
with additional terms that vanish to zero as $n\rightarrow\infty$. We
calculate squeezing limits of the Rényi-$\alpha$ entropy, this time with the
extra dependence on $\alpha$, in Section 4.2. Finally, we evaluate the
Rényi-$\alpha$ entropy under unequal, small initial squeezing strengths in
Section 4.3 and derive typicality of entanglement results in Section 4.4.

##  2 Setup

In this section, we present our setup. We consider a bosonic system of $n$
modes. For a pedagogical introduction to the concepts discussed in the
section, we refer to Ref. [42]. For each $i\in\\{1,2,\dots,n\\}$, define
$r_{i}\coloneqq x_{i}$ to be the position operator and $r_{n+i}\coloneqq
p_{i}$ to be the momentum operator. A density matrix $\rho$ describes a
Gaussian state if there exists a $\beta>0$ and a Hamiltonian $H$ that is at
most quadratic in the quadrature operators $r_{i}$ such that $\rho$ is the
thermal state $\rho\propto\mathrm{e}^{-\beta H}$. When $\rho$ is Gaussian, it
is fully characterized by its first and second moments $\operatorname{Tr}(\rho
r_{i})$ and
$\sigma_{ij}=\frac{1}{2}\operatorname{Tr}[\rho(r_{i}r_{j}+r_{j}r_{i})]-\operatorname{Tr}(\rho
r_{i})\operatorname{Tr}(\rho r_{j})$. $\sigma$ is called the covariance matrix
of the state $\rho$.

In this work, we investigate the properties of bosonic states after the
application of a linear optical unitary. Linear optical unitaries generate
passive (boson-number conserving, or equivalently energy conserving) Gaussian
unitaries. The set of passive Gaussian unitaries is isomorphic to
$\text{Sp}(2n)\cap O(2n)\cong\mathrm{U}(n),$ where $\text{Sp}(2n)$ is the
symplectic group of $2n\times 2n$ matrices, $O(2n)$ is the orthogonal group of
$2n\times 2n$ matrices, and $\mathrm{U}(n)$ is the unitary group of $n\times
n$ matrices [43].

We now define our notion of a random, pure Gaussian state on $n$ modes. We
initialize the $i^{\text{th}}$ mode to be in a squeezed vacuum state with
fixed squeezing parameter
$s_{i}\in\mathbb{R}\,\,\forall\,i\in\left\\{1,\dots,n\right\\}$. We then
randomly sample a passive Gaussian unitary $U\in\mathrm{U}(n)$ according to
the Haar measure [43] and apply it to the $n$ modes. The output state is
characterized by the input squeezing strengths $s_{i}$ and the unitary $U$.
For squeezing parameters $s_{1},s_{2},\dots,s_{n}$, the total expected number
of bosons in the state on the $n$ modes is $\sum_{i=1}^{n}\sinh^{2}(s_{i})$.
In this work, we primarily consider the case when all the squeezing parameters
are equal, $s_{i}=s$. Then, the average total number of bosons per mode is
$\sinh^{2}(s)$.

We partition the $n$ output modes into two groups: one group of $k=rn$ modes
for some $0\leq r\leq 1$, and the other group of $n-k=(1-r)n$ modes. We
calculate the Rényi-$\alpha$ entropy of the reduced state of the $k$ modes.
Because we are considering pure states, this is equivalent to the entropy of
the reduced state of the $n-k$ modes [44].

For the density matrix $\rho(U)$ on the $k$ modes, we denote its
Rényi-$\alpha$ entropy by
$S_{\alpha}(U)\coloneqq\frac{1}{1-\alpha}\log\operatorname{Tr}(\rho(U)^{\alpha})$.
The limit as $\alpha\to 1$ yields the von Neumman entropy
$S_{1}(U)\coloneqq-\operatorname{Tr}\rho(U)\log\rho(U)$. We note that the
dependence of these quantities on $r$, $n$, and $s$ is implicit. Furthermore,
the average value of $S_{\alpha}(U)$ over $\mathrm{U}(n)$ with respect to the
Haar measure is defined as the Rényi-$\alpha$ Page curve.

We will frequently utilize $\sigma(U)$, the covariance matrix of $\rho(U)$. We
also use the symplectic eigenvalues of $\sigma(U),$ defined as the eigenvalues
of the matrix $i\Omega\sigma(U)$ where

$\Omega\coloneqq\begin{pmatrix}0_{n\times n}&\mathbb{I}_{n\times n}\\\
-\mathbb{I}_{n\times n}&0_{n\times n}\end{pmatrix}.$ (1)

##  3 Von Neumann Entropy

In this section, we calculate an explicit formula for the von Neumann Page
curve as a function of equal squeezing strength $s$ and the partition size
ratio $r$ up to terms that decay to zero as $n$ increases. Then, we compute
and interpret the Page curve in the squeezing limits $s\rightarrow 0$ and
$s\rightarrow\infty.$ Next, we discuss an approach to the case of small,
unequal squeezing strengths. Finally, we comment on difficulties in deriving
the variance and entanglement typicality of the von Neumann entropy in this
setting.

###  3.1 Explicit formula

The von Neumann entropy is a function of the squeezing strength $s$ (assuming
all modes are equally squeezed) and the partition size ratio $r=\frac{k}{n}$,
up to terms that decay to zero as $n\rightarrow\infty.$ We explicitly
calculate this function, which also leads us to the von Neumann Page curve as
$n\rightarrow\infty.$

######  Theorem 1 (von Neumann entropy).

Choose any $s\in\mathbb{R}$ and $r\in[0,1].$ Let
$C_{i}=\frac{1}{i+1}\binom{2i}{i}$ represent the $i$th Catalan number and
${}_{2}F_{1}$ the hypergeometric function [45, 46]. Then, the average von
Neumann entropy over all unitaries $U\in U(n)$ is

$\displaystyle\begin{split}\underset{U\in
U(n)}{\mathbb{E}}S_{1}(U)=\sum_{i=1}^{\infty}\left[\frac{1}{2i}-\frac{1}{3}\textup{sech}^{2}(2s)\tanh^{2i}(2s)\,_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},\textup{sech}^{2}(2s)\right)\right](nG_{i}(r)-H_{i}(r)+o(1)),\end{split}$
(2)

where

$\displaystyle G_{i}(r)$
$\displaystyle=r-r^{i+1}C_{i}\,{}_{2}F_{1}(1-i,i,2+i,r),$ (3) $\displaystyle
H_{i}(r)$ $\displaystyle=4^{i-1}(r(1-r))^{i}$ (4)

are functions independent of $n$ detailed in Ref. [39], and $o(1)$ denotes
terms that vanish as $n\rightarrow\infty.$

The derivation for this formula is detailed in Section A.1. In the derivation
we use a formula for the von Neumann entropy in terms of the covariance matrix
of the subsystem of $k$ modes. Using the decomposition of the covariance
matrix derived in Ref. [39], we Taylor expand about the identity matrix and
utilize the negative binomial theorem. Finally, we use formulas from Ref. [39]
regarding expectation values over $\mathrm{U}(n)$ of certain functions of
matrix elements to arrive at the final expression.

The von Neumann Page curve as $n\rightarrow\infty$ is derived from Theorem 1
by dividing by $n.$ The $H_{i}(r)$ and $o(1)$ terms on the RHS subsequently
vanish. One important note is that the Page curve must be symmetric under
$r\mapsto 1-r$ since the global state on the $n$ modes is pure [44]. Since
$G_{i}(r)$ and $H_{i}(r)$ are symmetric about $r=\frac{1}{2}$ [39], the von
Neumann Page curve derived from Theorem 1 is indeed symmetric about
$r=\frac{1}{2}.$ This symmetry allows one to replace $r$ with $\min(r,1-r).$

###  3.2 Squeezing limits

Theorem 1 applies for all equal squeezing strengths $s$. We find considerable
simplifications when examining the limiting behavior as a function of $s$. We
begin with the $s\to 0$ limit, which corresponds to photons in their near-
vacuum states. One might expect the von Neumann Page curve to grow with the
total number of bosons, $\sum_{i=1}^{n}\sinh s^{2}\approx ns^{2}$ for small
$s_{i}$. Indeed, this behavior occurs for the rest of the Rényi entropies
(with integer $\alpha\geq 2$) as shown in Section 4.2. However, the pertinent
limit $\lim_{s\rightarrow 0}\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{s^{2}n}S_{1}(U)$ diverges to infinity. In Section
A.2, we prove this result and demonstrate the relevant scaling to be
$\underset{U}{\mathbb{E}}S_{1}(U)\sim s^{2}\log(1/s^{2})$, as formulated in
the following theorem.

######  Theorem 2.

For the squeezing strength limit $s\rightarrow 0$,

$\displaystyle\lim_{s\rightarrow 0}\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{\left(s^{2}\log(1/s^{2})\right)n}S_{1}(U)$
$\displaystyle=r(1-r).$ (5)

While the derivation of Theorem 2 is fairly straightforward, its significance
is more interesting. The von Neumann Page curve’s growth in order
$s^{2}\log(1/s^{2})$ for small $s$ can be attributed to the unique definition
of the von Neumann entropy $S_{1}(\rho)=-\operatorname{Tr}(\rho\log\rho),$
where $\rho$ is the density matrix of the quantum state. Therefore, the
symplectic eigenvalues $\nu_{j}$ of the covariance matrix would also scale the
von Neumann entropy in order $\nu_{j}\log\nu_{j},$ warranting the
$s^{2}\log(1/s^{2})$ dependence (as $\nu_{j}\sim s^{2}$). Furthermore, the von
Neumann entropy is non-analytic unlike the rest of the Rényi entropies.
Therefore, the argument presented in Ref. [39] that the Rényi entropy can be
expanded in lowest order $s^{2}$ doesn’t apply to the von Neumann case. More
specifically, the deviation from analytic behavior arises from the von Neumman
entropy formula in terms of the covariance matrix $\sigma$ [47, 42, 48]:

$\displaystyle S_{1}$
$\displaystyle=\frac{1}{2}\log\det\\!\left(\frac{\sigma+\mathrm{i}\Omega}{2}\right)+\frac{1}{2}\operatorname{Tr}\\!\left(\coth^{-1}(\mathrm{i}\Omega\sigma)\mathrm{i}\Omega\sigma\right),$
(6)

where $\Omega$ is the symplectic form defined in Eq. 1. Since $\coth^{-1}(x)$
is not a real analytic function, the resulting Page curve is not analytic,
either.

In a similar vein, we compute the von Neumann Page curve as
$s\rightarrow\infty$, assuming an equal squeezing regime.

######  Theorem 3.

In the strong squeezing limit $s\rightarrow\infty$,

$\displaystyle\lim_{s\rightarrow\infty}\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{sn}S_{1}(U)$ $\displaystyle=2\min(r,1-r).$ (7)

Here, the von Neumann Page curve scales with $s$ as $s\rightarrow\infty.$ This
is because, for large absolute squeezing values, higher orders in $r$ dominate
the expansion of Theorem 1, which scale with $s.$ Interestingly, Theorem 3
remains the exact same for the Rényi-2 entropy [39], and indeed we will see in
Sec. 4.2 that it is the same for all the Rényi-$\alpha$ entropies for integral
$\alpha$.

###  3.3 Unequal squeezing

In Ref. [39], the expected value of the Rényi-2 entropy for small, unequal
squeezing strengths was calculated through a power-series analysis. The
analyticity of the Rényi-2 function was what allowed writing the entropy as a
power series in $s_{i},$ where $s_{i}$ was the squeezing strength of the $i$th
mode. Using this power series expansion, along with the translational
invariance of the Haar measure, Ref. [39] derived a formula for unequal
squeezing in order $s^{2},$ which reduces to the $s\rightarrow 0$ squeezing
limit for small, equal $s_{i}.$

However, due to the von Neumann entropy not being analytic around $s=0,$ we
cannot perform the same analysis to derive a formula for unequal squeezing
strengths. Furthermore, directly approaching the unequal squeezing regime via
the Taylor expansion of the von Neumann entropy runs into another problem,
since the separation between $s$ and $r$ dependence present in the Rényi-$2$
case presented in Ref. [39] no longer applies here. Specifically, the matrix
$W$ described in Section A.1 is no longer independent of $s.$ Therefore,
carrying out the Taylor expansion would yield complicated matrix expressions
in $\sigma$ that do not simplify appreciably at first glance. Due to these
difficulties, we are unable to derive the unequal squeezing case for the von
Neumann entropy. It is nonetheless possible, though, that executing the
complicated matrix Taylor expansion and simplifying terms yields a closed-form
result for the von Neumann entropy.

##  4 General Rényi-$\alpha$ Entropies

$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$\displaystyle\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{n}S_{1}(U)$$\displaystyle\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{n}S_{15}(U)$$r$$\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{n}S_{\alpha}(U)$

Figure 1: Simulated Page curves for Rényi-$\alpha$ entropies for
$\alpha\in\\{1,2,3,4,5,6,7,15\\}$ in dots (smaller Page curves indicate larger
$\alpha$), overlaid on the analytic expressions derived in Theorem 1 and
Theorem 4. The simulated values were run 250 times. The number of modes is
$n=400$ with equal squeezing strength $s=0.5$.

In this section, we extend much of our analysis with the von Neumann entropy
in Section 3 to Rényi-$\alpha$ entropies for integer $\alpha\geq 2$. We also
derive results for the typicality of entanglement and the small, unequal
squeezing case. First, we present the Rényi-$\alpha$ Page curve
$\forall\alpha\in\mathbb{Z},\alpha\geq 2.$ Next, we investigate squeezing
limits as $s\rightarrow 0$ and $s\rightarrow\infty,$ as well as consider the
Rényi-$\alpha$ Page curve for small, unequal squeezing strengths. We also
study the variance of the entanglement and prove various regimes of
entanglement typicality. Finally, we show some numerical simulations
confirming our results.

###  4.1 Explicit formula

As before, we consider a system of $n$ modes, each with equal initial
squeezing strength $s.$

######  Theorem 4 (Rényi-$\alpha$ Page curve).

For all integer $\alpha\geq 2,$ define
$a=\left\lfloor\frac{\alpha-1}{2}\right\rfloor$. Let $\zeta=1$ if $\alpha$ is
even and $\zeta=0$ if $\alpha$ is odd. Then, the Rényi-$\alpha$ entropy is
given by

$\displaystyle\begin{split}\underset{U\in
U(n)}{\mathbb{E}}S_{\alpha}(U)=\frac{\zeta}{\alpha-1}\cdot\underset{U\in
U(n)}{\mathbb{E}}S_{2}+\frac{1}{\alpha-1}\sum_{i=1}^{\infty}\left[\sum_{m=1}^{a}\frac{\sinh^{2i}(2s)}{i(\cosh^{2}(2s)+\cot^{2}\frac{\pi
m}{\alpha})^{i}}\right]\left(nG_{i}(r)-H_{i}(r)+o(1)\right),\end{split}$ (8)

where $\underset{U\in U(n)}{\mathbb{E}}S_{2}$ is the expected value of the
Rényi-2 entropy as a function of $s$ and $r$ given as

$\displaystyle\underset{U\in
U(n)}{\mathbb{E}}S_{2}=\sum_{i=1}^{\infty}\frac{\tanh^{2i}(2s)}{2i}\left(nG_{i}(r)-H_{i}(r)+o(1)\right),$
(9)

and where $G_{i}(r)$ and $H_{i}(r)$ are defined in Eq. 3.

The proof of Theorem 4 is shown in Section B.1. In the proof, we use a formula
for the Rényi-$\alpha$ entropy in terms of the symplectic eigenvalues of
$\sigma,$ the covariance matrix of the reduced state of $k$ modes (see, e.g.,
Ref. [42]). We convert this formula into a matrix expression in terms of
$\Omega$ and $\sigma$, and perform significant simplifications. Finally, to
compute the expectation value over the unitary group, we again utilize
averaged matrix quantities derived in Ref. [39].

Since $G_{i}(r)$ is symmetric under $r\to 1-r$, we see that the Page curves
are also manifeslty symmetric under $r\to 1-r$. Various analytic
Rényi-$\alpha$ Page curves are plotted in Fig. 1. We also plot numerically-
simulated values of the Page curve, where we generate Haar-random unitary
matrices and compute the entropies using the positive symplectic eigenvalues
$\nu_{j}$ of each covariance matrix $\sigma$. Then, we compute the respective
Rényi-$\alpha$ entropy as
$S_{\alpha}(U)=\sum_{j=1}^{n}\frac{1}{1-\alpha}\ln\left(\frac{2^{\alpha}}{(\nu_{j}+1)^{\alpha}-(\nu_{j}-1)^{\alpha}}\right)$
[42]. The numerical and analytical values overlap, indicating the validity of
Theorem 1 and Theorem 4.

###  4.2 Squeezing limits

We now investigate the Rényi-$\alpha$ Page curve for all integer $\alpha\geq
2$ in the squeezing limits $s\rightarrow 0$ and $s\rightarrow\infty.$ We find
that, just as in the Rényi-2 case, the Page curve scales with $s^{2}$ when $s$
approaches zero and scales with $s$ when $s\to\infty$.

######  Theorem 5.

For integer $\alpha\geq 2,$ the Page curves of the Rényi-$\alpha$ entropy as
the equal squeezing strength $s$ approaches $0$ and $\infty,$ respectively,
are given by

$\displaystyle\lim_{s\rightarrow 0}\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{ns^{2}}S_{\alpha}(U)$
$\displaystyle=\frac{\alpha}{\alpha-1}r(1-r),$ (10)
$\displaystyle\lim_{s\rightarrow\infty}\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{ns}S_{\alpha}(U)$ $\displaystyle=2\min(r,1-r).$ (11)

In general, the Rényi entropies satisfy $S_{\alpha}\geq S_{\alpha^{\prime}}$
whenever $\alpha\leq\alpha^{\prime}$. This behavior matches with our formulas
for the Rényi-$\alpha$ limits in Theorem 5, as the Page curve decreases when
$\alpha$ increases. From Theorem 5, it is furthermore evident that
$S_{2}(U)=2S_{\infty}(U)$ when $s\to 0$. This relation between two Rényi
entropies saturates the case of a discrete random variable with outcome
probabilities $p_{1},p_{2},\dots,p_{m},$ where

$\displaystyle S_{2}=-\log\sum_{i}p_{i}^{2}\leq-\log\sup p_{i}^{2}=-2\log\sup
p_{i}=2S_{\infty}.$ (12)

In the infinite squeezing case, the Page curve remains independent of $\alpha$
in order $s$ as $\alpha\rightarrow\infty$. This indicates that, as
$s\to\infty,$ the Page curve approaches the curve of maximum entanglement.

|  | $k=\Theta\\!\left(n\right)$ |  $k=o\\!\left(n\right)$ [39] |  $k=o\\!\left(n^{1/3}\right)$ [38]
---|---|---|---|---
Equal | Rényi-$\alpha$ ($\alpha\geq 2$) | weak | strong | strong
squeezing | von Neumann | ? | weak | strong
Unequal | Rényi-$\alpha$ ($\alpha\geq 2$) | ? | weak∗ | strong
squeezing | von Neumann | ? | weak∗ | strong
Table 1:  A summary of the current status of rigorous results on entanglement
typicality in Gaussian bosonic systems. This table is the same as the one
provided in Ref. [39], but the Rényi-2 case has been generalized for all
Rényi-$\alpha$ entropies where $\alpha\in\mathbb{Z}^{+}.$ Note that “weak∗”
indicates that the result is not fully proven, but is based on Conjecture 10
in Ref. [39]. Furthermore, wherever "weak" typicality is displayed, we have
not ruled out the possibilty of strong typicality. We present typicality
results when $k,$ the number of modes in the subsystem, scales with
$\Theta\\!\left(n\right),o\\!\left(n\right),$ and $o\\!\left(n^{1/3}\right),$
respectively. Note that we only take the "worst-case" typicality; if there
exists both weak and strong typicality for one scaling, we display only the
weakly typical case. We present typicality results separately for the
Rényi-$\alpha$ entropies and the von Neumann entropy. The leftmost column
($k=\Theta\\!\left(n\right)$) stems from our findings in Section 4.4, while
the middle column ($k=o\\!\left(n\right)$) follows from Ref. [39]. The
rightmost column ($k=o\\!\left(n^{1/3}\right)$) follows from the results of
Ref. [38]. Refs. [36, 37] proved strong typicality in the regime
$k=\mathcal{O}\\!\left(1\right)$.

###  4.3 Unequal squeezing

We now consider the Rényi-$\alpha$ entropy when the initial squeezing
strengths are not necessarily equal, but are close to zero. We let the
squeezing strength on the $i$th mode be $s_{i}$ for some $s_{i}\in\mathbb{R}.$
Define $s_{\text{max}}\coloneqq\max{(|s_{1}|,|s_{2}|,\dots,|s_{n}|})$.

######  Theorem 6.

For integer $\alpha\geq 2,$ the Rényi-$\alpha$ entropy for small, unequal
squeezing strengths $s_{1},s_{2},\dots,s_{n}$ is given by

$\displaystyle\operatorname*{\mathbb{E}}_{U\in\mathrm{U}(n)}S_{\alpha}(U)=\frac{\alpha}{\alpha-1}r(1-r)\sum_{i=1}^{n}s_{i}^{2}+\mathcal{O}\\!\left(rns_{\rm
max}^{4}\right).$ (13)

To prove Theorem 6, note that the Rényi-$\alpha$ entropy is real analytic for
positive integer $\alpha$. This is because the Rényi-$\alpha$ expression, when
written in terms of the covariance matrix $\sigma$, contains only analytic
expressions in the vicinity of $s=0$, and thus the final expression is also
analytic in the small-squeezing limit [42]. Thus, we can write the entropy as
a power series in $s_{i}$. Furthermore, there exists a passive, local (i.e.
acting independently on each mode) Gaussian unitary that acts on the initial
covariance matrix $\sigma_{0}$ with the transformation $s_{i}\rightarrow-
s_{i}$. Since this unitary can absorbed into the Haar random unitary $U$, the
power series can be expanded in $s_{i}^{2}$. Note that when the absolute
squeezing strengths are close to zero, the first statement of Theorem 5 is a
special case of Theorem 6.

There exists another passive Gaussian unitary that acts on $\sigma_{0}$ via
the transformation $s_{i}\to s_{\tau(i)}$ for any permutation $\tau$ of $n$
elements. Since this unitary can also be absorbed into the Haar random unitary
$U$, the power series is symmetric under $s_{i}\to s_{\tau(i)}$. Thus, the
$s_{i}^{2}$ term in the power series expansion is of the form
$g(r,n)\sum_{i=1}^{n}s_{i}^{2}$ for some function $g(r,n)$. The equal-
squeezing result in Theorem 5 then immediately implies
$g(r,n)=\frac{\alpha}{\alpha-1}r(1-r).$

###  4.4 Typicality of entanglement

Next, we investigate the variance of the Rényi-$\alpha$ Page curve.
Chebyshev’s inequality allows one to relate the variance with the typicality
of entanglement, where typicality describes how quickly the difference between
some quantity (in this case, entanglement) and its average vanishes. This
analysis is useful in statistical mechanics, where typicality is useful for
thermodynamic variables to accurately represent their mean values.

The formal definition of typicality as described in Ref. [39] is as follows: a
unitary-dependent nonnegative random variable $S(U)$ is weakly typical if, for
any constant $\epsilon>0$,

$\displaystyle\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\text{Pr}}\left[\left|\frac{S(U)}{\underset{V\in
U(n)}{\mathbb{E}}S(V)}-1\right|<\epsilon\right]=1.$ (14)

Similarly, $S(U)$ is strongly typical if, for any constant $\epsilon>0$,

$\displaystyle\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\text{Pr}}\left[\left|S(U)-\underset{V\in
U(n)}{\mathbb{E}}S(V)\right|<\epsilon\right]=1.$ (15)

Perhaps contrary to intuition, a weakly typical variable has its
multiplicative distance vanish in the thermodynamical limit, while a strongly
typical variable has its additive distance vanish. This occurs because the
entropy goes to infinity as $n\rightarrow\infty.$ In this regard, strong
typicality implies weak typicality.

We are able to prove typicality results based on a crucial invariance
described in Ref. [39]. Specifically, Ref. [39] derives that the variance of
the Rényi-2 entropy is constant as $n\to\infty$. We extend this to general
integer $\alpha\geq 2$ using a Cauchy-Schwarz argument in Section B.4. Using a
bound for the variance via Chebyshev’s inequality, the Rényi-$\alpha$ entropy
exhibits weak typicality for $k=\Theta(n)$ (again, for all integer $\alpha\geq
2$). We present our results, as well as past analyses of entanglement
typicality in Gaussian bosonic systems, in Table 1.

##  5 Conclusions and open questions

In this paper, we have carefully studied the Rényi and von Neumann entropies
of random bosonic Gaussian states. Such states are exactly the output states
of a GBS experiment. Specifically, we have found closed formulas for the
Rényi-$\alpha$ entanglement entropy in terms of the squeezing strength $s$ and
the mode partition ratio $r$ that are exact asymptotically in $n$, where
$\alpha\in\mathbb{Z}^{+}.$ Using this, we calculated the Rényi-$\alpha$ Page
curve and various results stemming from it, such as $s\rightarrow 0$ and
$s\rightarrow\infty$ squeezing limits, and, for $\alpha\geq 2$, variance and
typicality of entanglement.

We have not reached a conclusion regarding the typicality of entanglement for
the von Neumann entropy as $k$ scales with $\Theta(n).$ We have that
$\operatorname*{Var}_{U}S_{1}(U)=\operatorname*{\mathbb{E}}_{U}S_{1}(U)^{2}-\left(\operatorname*{\mathbb{E}}_{U}S_{1}(U)\right)^{2},$
which we have not been able to simplify appreciably when using the von Neumann
entropy formula from Theorem 1. Specifically, the squaring of hypergeometric
functions yields terms in the variance expansion that we have not been able to
simplify. More research should be done to determine the typicality of
entanglement for the von Neumann entropy. However, since we’ve proven in
Section B.4 that the Rényi-$\alpha$ entropies for integer $\alpha\geq 2$ are
only weakly typical, the von Neumann entropy is unlikely to be strongly
typical. It remains to be rigorously proven, perhaps through examining special
cases of the von Neumann entropy, such as when $r=\frac{1}{2}$ or $s\to 0,$
and attempting to extract the variance from these cases.

Another point of research is in generalizing the Rényi-$\alpha$ entropy in
Theorem 4 to non-integer $\alpha$. In our work, we only considered
$\alpha\in\mathbb{Z}^{+}.$ Extending to non-integer $\alpha$ would allow a
formula for the Rényi entropies that is continuous in $\alpha$, and could be
useful to examine the behavior of the Rényi entropy as $\alpha\rightarrow 1$.

We also propose other areas of further research. Thus far, we have only
considered Haar-random unitaries that describe the linear optical circuit.
However, an interesting point of research could be in examining the
entanglement associated with random linear optical circuits of finite depth,
that is, the depth dependence of entanglement entropy. Currently, the precise
relationship between entanglement and circuit depth is largely unknown [49].
There have been some numerical studies [50], but only specialized cases where
most, but not all, initial modes are vacuum are understood analytically [51].

Another related notion deals with sampling complexity, the overall complexity
of classical computers producing a sample from approximately the same
distribution as the quantum system [52]. Quantifying the relationship between
entanglement and complexity, and studying the depth dependence of entanglement
and complexity, could yield crucial new insights into the classical hardness
of simulating quantum experiments such as GBS and, more generally, into the
relation between entanglement and complexity.

Another problem to tackle deals with unequal squeezing. There are two primary
avenues of future research. First, for small, varying squeezing strengths
$s_{1},s_{2},\dots,s_{n},$ what is the average von Neumann entanglement
entropy of the subsystem? We have outlined a method in Section 3.3 of tackling
this problem, but initial attempts have proved futile due to the heavy
presence of inverse matrices. Nonetheless, learning the behavior of the von
Neumann entropy for unequal squeezing regimes is important from a practical
standpoint, since many GBS experiments are conducted with one set of modes at
zero squeezing strength and the other at some varied squeezing strength close
to zero. Second, it is interesting to consider Rényi-$\alpha$ entropies for
systems with varying squeezing strengths that are not necessarily close to
zero. The argument given in Section 4.3 revolves around the entropy expression
being analytic around $s=0$ and vanishing in higher orders of
$s_{\text{max}}.$ However, for the most general case of
$s_{\text{max}}\in\mathbb{R},$ one cannot simply extend the $s\rightarrow 0$
squeezing limit for unequal squeezing. Nonetheless, deriving such an
expression would be important for a wider range of bosonic systems.

## Acknowledgements

This work was supported in part by the DoE ASCR Accelerated Research in
Quantum Computing program (award No. DE-SC0020312), DARPA SAVaNT ADVENT, AFOSR
MURI, DoE ASCR Quantum Testbed Pathfinder program (awards No. DE-SC0019040 and
No. DE-SC0024220), NSF QLCI (award No. OMA-2120757), NSF STAQ program, and
AFOSR. Support is also acknowledged from the U.S. Department of Energy, Office
of Science, National Quantum Information Science Research Centers, Quantum
Systems Accelerator. JTI thanks the Joint Quantum Institute at the University
of Maryland for support through a JQI fellowship. Specific product citations
are for the purpose of clarification only, and are not an endorsement by the
authors or NIST.

###### Appendices

  1. 1 Introduction
  2. 2 Setup
  3. 3 Von Neumann Entropy
    1. 3.1 Explicit formula
    2. 3.2 Squeezing limits
    3. 3.3 Unequal squeezing
  4. 4 General Rényi-$\alpha$ Entropies
    1. 4.1 Explicit formula
    2. 4.2 Squeezing limits
    3. 4.3 Unequal squeezing
    4. 4.4 Typicality of entanglement
  5. 5 Conclusions and open questions
  6. A von Neumann Entropy
    1. A.1 Page Curve for the von Neumann Entropy
    2. A.2 Limit as $s\rightarrow 0$
    3. A.3 Limit as $s\rightarrow\infty$
  7. B General Rényi-$\alpha$ Entropies
    1. B.1 Page Curve for the Rényi-$\alpha$ entropy
    2. B.2 Limit as $s\rightarrow 0$
    3. B.3 Limit as $s\rightarrow\infty$
    4. B.4 Variance and typicality of entanglement

##  Appendix A von Neumann Entropy

In this section of the appendix, we show derivations of our results regarding
the von Neumann entropy. We cover the derivation of the von Neumann entropy
presented in Theorem 1, show squeezing limits as $s\rightarrow 0$ and
$s\rightarrow\infty,$ (Theorem 2 and Theorem 3), and discuss a potential
avenue for solving the unequal squeezing case.

###  A.1 Page Curve for the von Neumann Entropy

Here, we derive the von Neumann entanglement entropy between the subsystems
with $k$ and $n-k$ modes, in terms of the partition size ratio $r=\frac{k}{n}$
and the squeezing strength $s.$ We use much of the notation from Ref. [39],
and the background on bosonic Gaussian states and operations can be found in
Ref. [42].

Consider a system of $n$ bosonic modes, where each mode $1\leq i\leq n$ is
prepared in a single-mode squeezed state with squeezing strength $s_{i}.$ This
$n$-mode Gaussian state is acted on by a passive linear optical unitary $U\in
U(n).$

We divide the output modes into two groups, with one group bearing $0\leq
k\leq n$ modes. Let $\sigma$ be the $2k\times 2k$ covariance matrix of the
reduced state on these $k$ modes. As shown in Refs. [42, 47, 48], the von
Neumann entanglement entropy $S_{1}(U)$ of the whole system is given by

$\displaystyle S_{1}(U)$
$\displaystyle=\frac{1}{2}S_{1}^{(1)}+\frac{1}{2}S_{1}^{(2)}\,\,\text{ with
}\,\,\begin{cases}S_{1}^{(1)}=\log\det\\!\left(\frac{\sigma+\mathrm{i}\Omega}{2}\right)\\\
S_{1}^{(2)}=\operatorname{Tr}\\!\left(\coth^{-1}(\mathrm{i}\Omega\sigma)\mathrm{i}\Omega\sigma\right)\end{cases},$
(A1)

where $\coth^{-1}(x)$ is the inverse hyperbolic cotangent function, and the
symplectic form $\Omega$ equals

$\displaystyle\Omega:=\begin{pmatrix}0_{n\times n}&\mathbb{I}_{n\times n}\\\
-\mathbb{I}_{n\times n}&0_{n\times n}\end{pmatrix}.$ (A2)

We aim to derive a formula for $S_{1}$ solely in terms of $r,n,$ and $s$ using
Eq. A1.

Define the $k\times n$ matrix $P$ and the $n\times n$ projector $\Pi$ such
that

$\displaystyle P$ $\displaystyle:=\begin{pmatrix}\mathbb{I}_{k\times
k}&0_{k\times(n-k)}\end{pmatrix},$ (A3) $\displaystyle\Pi$
$\displaystyle:=\begin{pmatrix}\mathbb{I}_{k\times k}&0_{k\times(n-k)}\\\
0_{(n-k)\times k}&0_{(n-k)\times(n-k)}\end{pmatrix}=\mathbb{I}_{k\times
k}\oplus 0_{(n-k)\times(n-k)}.$ (A4)

From Ref. [39], when $s_{i}=s_{j}\,\,\forall
i,j\in\left\\{1,2,\dots,n\right\\},$ the covariance matrix $\sigma$ of the
reduced state after the application of $U$ can be expressed as

$\displaystyle\sigma=\cosh{(2s)}\mathbb{I}+\sinh{(2s)}M,\,\,\text{ where
}\,\,M:=\begin{pmatrix}P\,\text{Re}(\bar{U}U^{{\dagger}})\,P^{T}&P\,\text{Im}(\bar{U}U^{{\dagger}})\,P^{T}\\\
P\,\text{Im}(\bar{U}U^{{\dagger}})\,P^{T}&-P\,\text{Re}(\bar{U}U^{{\dagger}})\,P^{T}\end{pmatrix},$
(A5)

and $\bar{U},U^{T},U^{{\dagger}}$ represent the conjugate, transpose, and
conjugate transpose of $U,$ respectively. Two useful and easily checkable
properties of $M$ are that $\Omega M=-M\Omega$ and $\text{Tr}(\Omega
M^{j})=\text{Tr}(M^{2j-1})=0,\,\,\forall j\in\mathbb{Z}^{+}.$

Using Eq. A5 and the properties above, $S_{1}^{(1)}$ from Eq. A1 is expressed
as

$\displaystyle S_{1}^{(1)}$
$\displaystyle=\operatorname{Tr}\log\\!\left(\frac{\sigma+\mathrm{i}\Omega}{2}\right)$
(A6)
$\displaystyle=\sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j}\operatorname{Tr}\left(\frac{\sigma+\mathrm{i}\Omega}{2}-\mathbb{I}\right)^{j}$
$\displaystyle=\sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j2^{j}}\operatorname{Tr}\left((\cosh(2s)-2)\mathbb{I}+\sinh(2s)M+\mathrm{i}\Omega\right)^{j}$
$\displaystyle=\operatorname{Tr}\sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j2^{j}}\sum_{\ell=0}^{j}\binom{j}{\ell}\left(\sinh(2s)M+\mathrm{i}\Omega\right)^{\ell}(\cosh(2s)-2)^{j-\ell}$
$\displaystyle=\sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j2^{j}}\left(2k(\cosh(2s)-2)^{j}+\operatorname{Tr}\sum_{\ell=1}^{j}\binom{j}{\ell}\left(\sinh(2s)M+\mathrm{i}\Omega\right)^{\ell}(\cosh(2s)-2)^{j-\ell}\right)$
$\displaystyle=2k\log\left(\frac{\cosh
2s}{2}\right)-\operatorname{Tr}\sum_{\ell=1}^{\infty}\left(\sinh(2s)M+\mathrm{i}\Omega\right)^{\ell}\sum_{j=\ell}^{\infty}\frac{(-1)^{j}}{j2^{j}}\binom{j}{\ell}(\cosh(2s)-2)^{j-\ell}$
$\displaystyle=2k\log\left(\frac{\cosh
2s}{2}\right)-\operatorname{Tr}\sum_{\ell=1}^{\infty}\frac{(-1)^{\ell}}{\ell}\left(\sinh(2s)M+\mathrm{i}\Omega\right)^{\ell}\cosh^{-\ell}(2s)$
$\displaystyle=2k\log\left(\frac{\cosh
2s}{2}\right)-\operatorname{Tr}\sum_{\ell=1}^{\infty}\frac{1}{2\ell}\left(\sinh(2s)M+\mathrm{i}\Omega\right)^{2\ell}\cosh^{-2\ell}(2s)$
$\displaystyle=2k\log\left(\frac{\cosh
2s}{2}\right)-\operatorname{Tr}\sum_{\ell=1}^{\infty}\frac{1}{2\ell}\cosh^{-2\ell}(2s)\left(\sinh^{2}(2s)M^{2}+\mathbb{I}\right)^{\ell}.$
(A7)

As discussed in Ref. [39], let $W=\Pi
U\bar{U}^{{\dagger}}\Pi\bar{U}U^{{\dagger}}\Pi.$ Then,
$\text{Tr}(M^{2i})=2\,\text{Tr}(W^{i})$ $\forall i\in\mathbb{Z}^{+}.$ We
emphasize that the dependence on $U$ and $r$ is implicit in $W$. Importantly,
though, there is no $s$ dependence in $W$. We also note that, since now
everything is written in terms of traces, we are able to replace $P$ with
$\Pi$ so that all matrices are square. Incorporating this definition into Eq.
A7, we have

$\displaystyle S_{1}^{(1)}$ $\displaystyle=2k\log\left(\frac{\cosh
2s}{2}\right)-2\sum_{\ell=1}^{\infty}\frac{1}{2\ell}\cosh^{-2\ell}(2s)\sum_{j=0}^{\ell}\binom{\ell}{j}\sinh^{2j}(2s)\operatorname{Tr}W^{j}$
(A8) $\displaystyle=2k\log\left(\frac{\cosh
2s}{2}\right)-2\sum_{\ell=1}^{\infty}\frac{1}{2\ell}\cosh^{-2\ell}(2s)\left(k+\sum_{j=1}^{\ell}\binom{\ell}{j}\sinh^{2j}(2s)\operatorname{Tr}W^{j}\right)$
$\displaystyle=2k\log\left(\frac{\cosh
2s}{2}\right)+k\log\\!\left[\tanh^{2}(2s)\right]-2\sum_{j=1}^{\infty}\sinh^{2j}(2s)\operatorname{Tr}W^{j}\sum_{\ell=j}^{\infty}\frac{1}{2\ell\cosh^{2\ell}(2s)}\binom{\ell}{j}$
$\displaystyle=k\log\\!\left[\cosh^{2}(s)\sinh^{2}(s)\right]-2\sum_{j=1}^{\infty}\sinh^{2j}(2s)\operatorname{Tr}W^{j}\frac{1}{2j\sinh(2s)^{2j}}$
$\displaystyle=k\log\\!\left[\frac{\sinh^{2}(2s)}{4}\right]-\sum_{j=1}^{\infty}\frac{1}{j}\operatorname{Tr}W^{j}.$
(A9)

We use a similar approach to simplify $S_{1}^{(2)}.$ Notably, we use the
Taylor expansion

$\displaystyle\coth^{-1}(x)=\sum_{j=0}^{\infty}\frac{x^{-2j-1}}{2j+1}\,\,\text{
for }\,\,|x|\geq 1.$ (A10)

The $|x|\geq 1$ condition is met, as the eigenvalues of $i\Omega\sigma$ come
in positive-negative pairs, each with magnitude greater than (or equal to) 1.
Thus,

$\displaystyle S_{1}^{(2)}$
$\displaystyle=\operatorname{Tr}(\coth^{-1}(\mathrm{i}\Omega\sigma)\mathrm{i}\Omega\sigma)$
$\displaystyle=\sum_{j=0}^{\infty}\frac{1}{2j+1}\operatorname{Tr}\left(\left(i\Omega\sigma\right)^{-2j}\right)$
$\displaystyle=\sum_{j=0}^{\infty}\frac{(-1)^{j}}{2j+1}\operatorname{Tr}\left(\left(\Omega\sigma\right)^{-2j}\right)$
$\displaystyle=\sum_{j=0}^{\infty}\frac{(-1)^{j}}{2j+1}\operatorname{Tr}\left(\left(\Omega(\cosh{(2s)}\mathbb{I}+\sinh{(2s)}M\right))^{-2j}\right)$
$\displaystyle=\sum_{j=0}^{\infty}\frac{(-1)^{j}}{2j+1}\operatorname{Tr}\left(\left(\cosh{(2s)}\Omega+\sinh{(2s)}\Omega
M\right)^{-2j}\right)$
$\displaystyle=\sum_{j=0}^{\infty}\frac{(-1)^{j}}{2j+1}\operatorname{Tr}\left(\left(\sinh^{2}{(2s)}M^{2}-\cosh^{2}{(2s)}\mathbb{I}\right)^{-j}\right).$
(A11)

We use the negative binomial theorem to expand Eq. A11. For non-negative
integers $j$ and real numbers $x$ and $a,$ we have

$\displaystyle(x+a)^{-n}=\sum_{k=0}^{\infty}(-1)^{k}\binom{n+k-1}{k}x^{k}a^{-n-k},$
(A12)

as discussed in Ref. [53]. This equation can be extended to matrices if $x$
and $a$ commute. Using $x=\sinh^{2}{(2s)}M^{2}$ and
$a=-\cosh^{2}{(2s)}\mathbb{I}$, we have

$\displaystyle S_{1}^{(2)}$
$\displaystyle=2k+\sum_{j=1}^{\infty}\frac{(-1)^{j}}{2j+1}\operatorname{Tr}\left(\sum_{i=0}^{\infty}(-1)^{i}\binom{j+i-1}{i}\sinh^{2i}(2s)M^{2i}(-\cosh^{2}(2s))^{-j-i}\right)$
(A13)
$\displaystyle=2k+\sum_{j=1}^{\infty}\left(\frac{1}{2j+1}\cdot\sum_{i=0}^{\infty}\binom{j+i-1}{i}\tanh^{2i}(2s)\text{sech}^{2j}(2s)\operatorname{Tr}(M^{2i})\right)$
$\displaystyle=2k\cosh{(2s)}\tanh^{-1}{(\text{sech}(2s))}+\sum_{i=1}^{\infty}\tanh^{2i}(2s)\operatorname{Tr}(M^{2i})\sum_{j=1}^{\infty}\frac{\text{sech}^{2j}(2s)}{2j+1}\binom{j+i-1}{i}$
$\displaystyle=2k\cosh{(2s)}\tanh^{-1}{(\text{sech}(2s))}+\frac{2}{3}\text{sech}^{2}(2s)\sum_{i=1}^{\infty}\tanh^{2i}(2s)\,_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},\text{sech}^{2}(2s)\right)\operatorname{Tr}(W^{i}).$
(A14)

Therefore,

$\displaystyle\begin{split}\underset{U\in
U(n)}{\mathbb{E}}S_{1}(U)&=\frac{k}{2}\log\\!\left[\frac{\sinh^{2}(2s)}{4}\right]-\sum_{j=1}^{\infty}\frac{1}{2j}\underset{U}{\mathbb{E}}\operatorname{Tr}W^{j}\\\
&\quad+k\cosh{(2s)}\tanh^{-1}{(\text{sech}(2s))}+\frac{\text{sech}^{2}(2s)}{3}\sum_{i=1}^{\infty}\tanh^{2i}(2s)\,_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},\text{sech}^{2}(2s)\right)\underset{U}{\mathbb{E}}\operatorname{Tr}(W^{i})\\\
\end{split}$ (A15)
$\displaystyle\begin{split}&=nr\left(\frac{1}{2}\log\\!\left[\frac{\sinh^{2}(2s)}{4}\right]+\cosh(2s)\tanh^{-1}[\text{sech}(2s)]\right)\\\
&\quad+\frac{\text{sech}^{2}(2s)}{3}\sum_{i=1}^{\infty}\left[\tanh^{2i}(2s)\,_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},\text{sech}^{2}(2s)\right)-\frac{3\cosh^{2}(2s)}{2i}\right]\underset{U}{\mathbb{E}}\operatorname{Tr}W^{i}\end{split}$
(A16)
$\displaystyle=\sum_{i=1}^{\infty}\left[\frac{1}{2i}-\frac{1}{3}\textup{sech}^{2}(2s)\tanh^{2i}(2s)\,_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},\textup{sech}^{2}(2s)\right)\right](nG_{i}(r)-H_{i}(r)+o(1)),$
(A17)

where the step from Eq. A16 to Eq. A17 arises from substituting
$\underset{U}{\mathbb{E}}\operatorname{Tr}W^{i}=nr-(nG_{i}(r)-H_{i}(r)+o(1))$
as derived in Ref. [39] and the subsequent cancellation of the constant term.
Finally, dividing by $n$ and noting the vanishing behavior of $H_{i}(r)$ and
$o(1)$ as $n\rightarrow\infty,$ the von Neumann Page curve is

$\displaystyle\begin{split}\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{n}S_{1}(U)&=\sum_{i=1}^{\infty}\left[\frac{1}{2i}-\frac{1}{3}\textup{sech}^{2}(2s)\tanh^{2i}(2s)\,_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},\textup{sech}^{2}(2s)\right)\right]G_{i}(r),\end{split}$
(A18)

which is given in Theorem 1 in the main text.

###  A.2 Limit as $s\rightarrow 0$

In this section, we show the validity of Theorem 2. Moreover, we elucidate the
odd behavior of the von Neumann Page curve scaling with $s^{2}\log(1/s^{2})$
instead of the traditional $s^{2}$ seen in the other Rényi-$\alpha$ entropies
with integer $\alpha\geq 2$. We also give our motivation for investigating the
$s^{2}\log(1/s^{2})$ dependence.

For small squeezing strengths, one might expect the Rényi entanglement entropy
to scale with the number of bosons in all modes. Furthermore, since the total
expected number of bosons in the state of $n$ modes is
$\sum_{i=1}^{n}\sinh^{2}{s}\approx ns^{2}$ for equal squeezing, one may guess
that all Rényi-$\alpha$ Page curves scale with $s^{2}$.

However,

$\displaystyle\lim_{s\rightarrow 0}\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{s^{2}n}S_{1}(U)=\infty.$ (A19)

One difficulty arises in evaluating the limit as $s\rightarrow\infty$ of the
first term in Eq. A16. Notably,

$\displaystyle\frac{1}{2s^{2}}\log\\!\left[\frac{\sinh^{2}(2s)}{4}\right]$
$\displaystyle=\frac{\log
s}{s^{2}}+\frac{2}{3}-\frac{4}{45}s^{2}+\frac{64}{2835}s^{4}-\frac{32}{4725}s^{6}+\frac{1024}{467775}s^{8}-\dots,$
(A20) $\displaystyle\frac{1}{s^{2}}\cosh(2s)\tanh^{-1}(\text{sech}(2s))$
$\displaystyle=-\frac{\log s}{s^{2}}+\frac{1-6\log s}{3}+\frac{53-60\log
s}{90}s^{2}+\frac{251-252\log s}{2835}s^{4}+\dots.$ (A21)

Summing Eqs. (A20) and (A21) yields a complicated series involving $\log{s},$
which diverges as $s\rightarrow 0.$ Therefore, we try evaluating
$\lim_{s\rightarrow 0}\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{(s^{2}\log{1/s^{2}})n}S_{1}(U)$ in hopes of
achieving a convergent result. Using the entanglement entropy expression
derived in Eq. A16,

$\displaystyle\begin{split}\lim_{s\rightarrow
0}\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{(s^{2}\log{1/s^{2}})n}S_{1}(U)&=\frac{r}{s^{2}\log{1/s^{2}}}\cdot\left(\frac{1}{2}\log\\!\left[\frac{\sinh^{2}(2s)}{4}\right]+\cosh(2s)\tanh^{-1}[\text{sech}(2s)]\right)\\\
&\quad+\frac{\text{sech}^{2}(2s)}{3s^{2}\log(1/s^{2})}\sum_{i=1}^{\infty}\left[\tanh^{2i}(2s)\,_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},\text{sech}^{2}(2s)\right)-\frac{3\cosh^{2}(2s)}{2i}\right]\\\
&\quad\cdot r^{i+1}C_{i}\,{}_{2}F_{1}(1-i;i;i+2;r).\end{split}$ (A22)

Note that $\frac{1}{s^{2}\log(1/s^{2})}\cdot
r\left(\frac{1}{2}\log\\!\left[\frac{\sinh^{2}(2s)}{4}\right]+\cosh(2s)\tanh^{-1}[\text{sech}(2s)]\right),$
the constant term in Eq. A22, approaches $r$ as $s\rightarrow 0.$ Now, we need
to find the limit as $s\rightarrow 0$ of the second term in Eq. A22:

$\displaystyle=\lim_{s\rightarrow
0}\frac{1}{s^{2}\log(1/s^{2})}\frac{\text{sech}^{2}(2s)}{3}\sum_{i=1}^{\infty}\left[\tanh^{2i}(2s)\,_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},\text{sech}^{2}(2s)\right)-\frac{3\cosh^{2}(2s)}{2i}\right]r^{i+1}C_{i}\,{}_{2}F_{1}(1-i;i;i+2;r)$
$\displaystyle=\sum_{i=1}^{\infty}\lim_{s\rightarrow
0}\frac{1}{s^{2}\log(1/s^{2})}\left(\frac{1}{3}\text{sech}^{2}(2s)\tanh^{2i}(2s)\,_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},\text{sech}^{2}(2s)\right)-\frac{1}{2i}\right)\cdot
r^{i+1}C_{i}\,{}_{2}F_{1}(1-i;i;i+2;r)$
$\displaystyle=\sum_{i=1}^{\infty}\lim_{s\rightarrow
0}\frac{1}{s^{2}\log(1/s^{2})}\left(\frac{1}{3}\cdot
1\cdot(2s)^{2i}\,_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},(1-2s^{2})^{2}\right)-\frac{1}{2i}\right)\cdot
r^{i+1}C_{i}\,{}_{2}F_{1}(1-i;i;i+2;r).$ (A23)

Note that, in the $i=1$ case, letting
$F(s)=\frac{1}{s^{2}\log(1/s^{2})}\frac{1}{3}\cdot 1\cdot(2s)^{2i},$ the left-
hand derivative $\frac{dF}{ds_{s=0^{-}}}=-\infty$, while the right-hand
derivative $\frac{dF}{ds_{s=0^{+}}}=\infty.$ This behavior deviates from all
other values of $i\geq 1,$ where $\frac{dF}{ds}=0$ evaluated at $s=0.$
Combined with the dependence of
${}_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},(1-2s^{2})^{2}\right)$ on $i,$
one can explicitly calculate that Section A.2 evaluates to $-r^{2},$ as the
$i=1$ case yields $-r^{2}$ and subsequent $i$ yield $0$ in the sum. Thus,

$\displaystyle\lim_{s\rightarrow 0}\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{(s^{2}\log(1/s^{2}))n}S_{1}(U)$
$\displaystyle=r(1-r).$ (A24)

It can easily be seen from Eq. A16 that the result is the same if we swap the
limits, so that by the Moore-Osgood theorem [54],

$\displaystyle\lim_{\begin{subarray}{c}n\to\infty\\\ s\rightarrow
0\end{subarray}}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{(s^{2}\log(1/s^{2}))n}S_{1}(U)$
$\displaystyle=r(1-r).$ (A25)

as displayed in Theorem 2 in the main text.

###  A.3 Limit as $s\rightarrow\infty$

In a similar vein as Section A.2, we also compute the von Neumann Page curve
for equal squeezing as $s\rightarrow\infty.$ The Page curve scales with $s$ in
the limit $s\to\infty$, so we compute
$\lim_{s\rightarrow\infty}\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{sn}S_{1}(U).$ We once again use the expression found
in Eq. A16 in our evaluations.

$\displaystyle\begin{split}\lim_{s\rightarrow\infty}\lim_{n\rightarrow\infty}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{sn}S_{1}(U)&=\lim_{s\rightarrow\infty}A_{1}+\lim_{s\rightarrow\infty}A_{2},\\\
\text{where
}A_{1}&=\frac{r}{s}\left(\frac{1}{2}\log\\!\left[\frac{\sinh^{2}(2s)}{4}\right]+\cosh(2s)\tanh^{-1}[\text{sech}(2s)]\right),\\\
A_{2}&=\frac{1}{s}\sum_{i=1}^{\infty}\left[\frac{1}{3}\text{sech}^{2}(2s)\tanh^{2i}(2s)\,_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},\text{sech}^{2}(2s)\right)-\frac{1}{2i}\right]\cdot
f_{i}(r),\\\ f_{i}(r)&=r^{i+1}C_{i}\,{}_{2}F_{1}(1-i;i;i+2;r).\end{split}$
(A26)

It turns out that

$\displaystyle\lim_{s\rightarrow\infty}A_{1}$
$\displaystyle=\lim_{s\rightarrow\infty}\left(\frac{r}{s}\frac{1}{2}\log\\!\left[\frac{\sinh^{2}(2s)}{4}\right]\right)+\lim_{s\rightarrow\infty}\left(\frac{r}{s}\cdot\cosh(2s)\tanh^{-1}[\text{sech}(2s)]\right)$
(A27)
$\displaystyle=\frac{r}{2}\lim_{s\rightarrow\infty}\left(\frac{1}{s}\log\\!\left[\frac{\sinh^{2}(2s)}{4}\right]\right)+0$
$\displaystyle=2\min(r,1-r).$ (A28)

Since the Page curve is symmetric about $r=\frac{1}{2},$ Eq. A28 contains a
$\min(r,1-r)$ term (as opposed to a simple $r$-dependence). Next, we calculate
$\lim_{s\rightarrow\infty}A_{2}$:

$\displaystyle\lim_{s\rightarrow\infty}A_{2}$
$\displaystyle=\lim_{s\rightarrow\infty}\frac{1}{s}\sum_{i=1}^{\infty}\left[\frac{1}{3}\text{sech}^{2}(2s)\tanh^{2i}(2s)\,_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},\text{sech}^{2}(2s)\right)-\frac{1}{2i}\right]\cdot
f_{i}(r)$ (A29)
$\displaystyle=\lim_{s\rightarrow\infty}\frac{1}{s}\sum_{i=1}^{\infty}\left[\frac{1}{3}\cdot
0\cdot
1\,_{2}F_{1}\left(\frac{3}{2},1+i,\frac{5}{2},0\right)-\frac{1}{2i}\right]\cdot
f_{i}(r)$ (A30) $\displaystyle=0.$ (A31)

Therefore,

$\displaystyle\lim_{\begin{subarray}{c}n\to\infty\\\
s\rightarrow\infty\end{subarray}}\underset{U\in
U(n)}{\mathbb{E}}\frac{1}{sn}S_{1}(U)$ $\displaystyle=2\min(r,1-r),$ (A32)

where we again used the same discussion as around Eq. A25

Eq. A32 is verified in Fig. B.2, alongside other Rényi-$\alpha$ Page curves in
the limit as $s\rightarrow\infty.$ Note that both the von Neumann and Rényi
Page curves approach the same limit of $2\min(r,1-r)$ (see Section B.3 for the
relevant derivation). This indicates that, at large squeezing values, the von
Neumann entropy behaves similarly to the larger class of Rényi entropies.

##  Appendix B General Rényi-$\alpha$ Entropies

In this section, we extend our derivation in Appendix A from the von Neumann
entropy to all general Rényi-$\alpha$ entropies, where $\alpha$ is a positive
integer greater than $1$. We find a closed formula for the Rényi-$\alpha$ Page
curve as a function of equal squeezing strength $s$ and partition ratio $r$ in
the asymptotic limit $n\rightarrow\infty.$ Similar to the von Neumann case, we
calculate squeezing limits of the Rényi-$\alpha$ entropies as $s\rightarrow 0$
and $s\rightarrow\infty$ and then discuss typicality of entanglement based on
the variance.

###  B.1 Page Curve for the Rényi-$\alpha$ entropy

Here, we present our full derivation for the Page curve of the Rényi-$\alpha$
entropy, where $\alpha$ is an integer great than 1, in terms of the partition
ratio $r$ and equal squeezing strength $s$.

Our starting point is the general expression relating the Rényi-$\alpha$
entropy with the symplectic eigenvalues $\nu_{j}$ of the covariance matrix
$\sigma$ (recall that $\sigma$ is precisely the second moment of $\rho$ [42]):

$\displaystyle\mathcal{S}_{\alpha}(\rho)=\frac{1}{\alpha-1}\sum_{j=1}^{k}\log\\!\left[(\nu_{j}+1)^{\alpha}-(\nu_{j}-1)^{\alpha}\right]-\frac{\alpha
k\log 2}{\alpha-1}.$ (B1)

If $\alpha$ is a positive integer, then this function has $\alpha-1$
singularities. We can solve for those singularities explicitly:

$\displaystyle(\nu_{0}+1)^{\alpha}=(\nu_{0}-1)^{\alpha}\Rightarrow\nu_{0}+1=(\nu_{0}-1)e^{i\frac{2\pi\ell}{\alpha}},$
(B2)
$\displaystyle\Rightarrow\nu_{0}=-i\cot\frac{\pi\ell}{\alpha},\quad\ell=1,\ldots,\alpha-1.$
(B3)

Note that we exclude the $\ell=\alpha$ case as Eq. (B2) does not have a root
in that case. Thus, the polynomial $(\nu_{j}+1)^{\alpha}-(\nu_{j}-1)^{\alpha}$
can be rewritten in terms of a product of factors as follows:

$\displaystyle(\nu_{j}+1)^{\alpha}-(\nu_{j}-1)^{\alpha}=2\alpha\nu_{j}^{\zeta}\prod\limits_{m=1}^{\lfloor\frac{\alpha-1}{2}\rfloor}(\nu_{j}^{2}+\cot^{2}\frac{\pi
m}{\alpha}),$ (B4)

where we define $\zeta:=1-(\alpha\bmod 2)$. The Rényi entropy is in turn given
by

$\displaystyle\mathcal{S}_{\alpha}(\rho)=\frac{1}{\alpha-1}\sum_{m=1}^{\lfloor\frac{\alpha-1}{2}\rfloor}\sum_{j=1}^{k}\log(\nu_{j}^{2}+\cot^{2}\frac{\pi
m}{\alpha})+\frac{\zeta\sum_{j=1}^{k}\log\nu_{j}^{2}}{2(\alpha-1)}+\frac{k\log\alpha}{\alpha-1}-k\log
2.$ (B5)

To work with the $\log\nu_{j}^{2}$ terms, note that the squares of the
symplectic eigenvalues of $\sigma$ are precisely the eigenvalues of
$(i\Omega\sigma)^{2}$ by matrix properties. Furthermore, since the eigenvalues
of $i\Omega\sigma$ come in plus-minus pairs but the symplectic eigenvalues
only consider positive numbers, the factor of $\frac{1}{2}$ in Eq. B5 arises.
Taking
$(i\Omega\sigma)^{2}=(i\Omega(\mathbb{I}\cosh(2s)+M\sinh(2s)))^{2}=\cosh^{2}(2s)-M^{2}\sinh^{2}(2s)$
into the above equation, we thus have

$\displaystyle\mathcal{S}_{\alpha}(\rho)=$
$\displaystyle\frac{1}{2(\alpha-1)}\sum_{m=1}^{\lfloor\frac{\alpha-1}{2}\rfloor}\text{Tr}\log(\cot^{2}\frac{\pi
m}{\alpha}+\cosh^{2}(2s)-M^{2}\sinh^{2}(2s))$
$\displaystyle+\frac{\zeta}{4(\alpha-1)}\text{Tr}\log(\cosh^{2}(2s)-M^{2}\sinh^{2}(2s))+\frac{k\log(\alpha)}{\alpha-1}-k\log
2.$ (B6)

Let us now expand all the logarithmic functions in the sum in Eq. (B6) in
orders of $M^{2}$. Noting that $M^{2}\leq 1$, one could show that each of the
following series always converge (for any integer $\ell$):

$\displaystyle\text{Tr}\log(\cosh^{2}(2s)-M^{2}\sinh^{2}(2s))$
$\displaystyle=4k\log\cosh(2s)-\sum_{\ell=1}^{\infty}\frac{\sinh^{2\ell}(2s)}{\ell\cosh^{2\ell}(2s)}\operatorname{Tr}(M^{2\ell}),$
(B7) $\displaystyle\text{Tr}\log(\cot^{2}\frac{\pi
m}{\alpha}+\cosh^{2}(2s)-M^{2}\sinh^{2}(2s))$
$\displaystyle=2k\log(\cot^{2}\frac{\pi
m}{\alpha}+\cosh^{2}(2s))-\sum_{\ell=1}^{\infty}\frac{\sinh^{2\ell}(2s)}{\ell(\cosh^{2}(2s)+\cot^{2}\frac{\pi
m}{\alpha})^{\ell}}\operatorname{Tr}(M^{2\ell}).$ (B8)

Note that the product in Eq. B4 vanishes when substituting $\alpha=2$.
Therefore, one gets from Eq. B5 that $S_{2}=\sum_{j=1}^{k}\log\nu_{j}^{2}$.
Combining the expansion in Eq. B7, we get that
$S_{2}=k\log\cosh(2s)-\sum_{\ell=1}^{\infty}\frac{\tanh^{2\ell}(2s)}{2\ell}\operatorname{Tr}(M^{2\ell}).$
This expression for $S_{2}$ is utilized in the following steps to formally
express the Rényi entropy with a positive integer $\alpha$ as

$\displaystyle\mathcal{S}_{\alpha}(\rho)=$ $\displaystyle
k\sum_{m=1}^{\lfloor\frac{\alpha-1}{2}\rfloor}\frac{\log(\cot^{2}\frac{\pi
m}{\alpha}+\cosh^{2}(2s))}{\alpha-1}+k\frac{\zeta\log\cosh(2s)+\log\alpha}{\alpha-1}-k\log
2$ (B9)
$\displaystyle-\frac{1}{\alpha-1}\sum_{\ell=1}^{\infty}\left[\sum_{m=1}^{\lfloor\frac{\alpha-1}{2}\rfloor}\frac{\sinh^{2\ell}(2s)}{\ell(\cosh^{2}(2s)+\cot^{2}\frac{\pi
m}{2\alpha})^{\ell}}+\zeta\frac{\sinh^{2\ell}(2s)}{2\ell\cosh^{2\ell}(2s)}\right]\frac{\operatorname{Tr}(M^{2\ell})}{2}$
$\displaystyle=$ $\displaystyle
k\sum_{m=1}^{\lfloor\frac{\alpha-1}{2}\rfloor}\frac{\log(\cot^{2}\frac{\pi
m}{\alpha}+\cosh^{2}(2s))}{\alpha-1}+\frac{k\log\alpha}{\alpha-1}-k\log
2+\frac{\zeta}{\alpha-1}S_{2}$
$\displaystyle-\frac{1}{\alpha-1}\sum_{\ell=1}^{\infty}\left[\sum_{m=1}^{\lfloor\frac{\alpha-1}{2}\rfloor}\frac{\sinh^{2\ell}(2s)}{\ell(\cosh^{2}(2s)+\cot^{2}\frac{\pi
m}{\alpha})^{\ell}}\right]\frac{\operatorname{Tr}(M^{2\ell})}{2}$ (B10)
$\displaystyle=$
$\displaystyle\frac{\zeta}{\alpha-1}S_{2}+\frac{1}{\alpha-1}\sum_{\ell=1}^{\infty}\left[\sum_{m=1}^{a}\frac{\sinh^{2\ell}(2s)}{\ell(\cosh^{2}(2s)+\cot^{2}\frac{\pi
m}{\alpha})^{\ell}}\right]\left[k-\operatorname{Tr}(W^{\ell})\right],$ (B11)

where $a\equiv\lfloor\frac{\alpha-1}{2}\rfloor=\frac{\alpha-1-\zeta}{2}$. We
have used $\operatorname{Tr}M^{2\ell}=2\operatorname{Tr}W^{\ell}$ in Eq. B11.
The step from Section B.1 to Eq. B11 utilizes the relation presented in Eq.
B8, where $M$ is substituted with the identity matrix. We can then use the
expression for $\mathbb{E}_{U\in\mathrm{U}(n)}\operatorname{Tr}W^{\ell}$ from
Ref. [39] to derive the Page curve.

###  B.2 Limit as $s\rightarrow 0$

We now analyze the Rényi-$\alpha$ entropy as the squeezing strength approaches
zero. This time, due to the analyticity of the Rényi entropies for integer
$\alpha\geq 2$, the Page curve scales with the number of bosons $\sim s^{2}$
instead of the $s^{2}\log(1/s^{2})$ dependence explained in Section A.2.

Recall that $G_{\ell}(r)$ is a function symmetric about $r=\frac{1}{2}$
defined as $r-r^{\ell+1}C_{\ell}\,{}_{2}F_{1}(1-\ell;\ell;\ell+2;r)$. Using
Eq. B11 and substituting $k-\operatorname{Tr}(W^{\ell})=nG_{\ell}(r)$ to
investigate the $s\rightarrow 0$ limit,

$\displaystyle\lim_{s\to
0}\lim_{n\to\infty}\frac{1}{s^{2}n}\operatorname*{\mathbb{E}}_{U}S_{\alpha}(U)$
$\displaystyle=\lim_{s\to
0}\frac{1}{s^{2}}\frac{1}{\alpha-1}\sum_{\ell=1}^{\infty}\frac{1}{\ell}(2s)^{2\ell}G_{\ell}(r)\sum_{m=1}^{a}\left(1+\cot^{2}\left(\frac{\pi
m}{\alpha}\right)\right)^{-\ell}+\frac{\zeta}{\alpha-1}2r(1-r)$ (B12)
$\displaystyle=\lim_{s\to
0}\frac{1}{s^{2}}\frac{1}{\alpha-1}(2s)^{2}G_{1}(r)\sum_{m=1}^{a}\left(\cot^{2}\left(\frac{\pi
m}{\alpha}\right)+1\right)^{-1}+\frac{\zeta}{\alpha-1}2r(1-r)$
$\displaystyle=\frac{4}{\alpha-1}r(1-r)\sum_{m=1}^{a}\left(\cot^{2}\left(\frac{\pi
m}{\alpha}\right)+1\right)^{-1}+\frac{\zeta}{\alpha-1}2r(1-r)$
$\displaystyle=\frac{r(1-r)}{\alpha-1}\left(4\sum_{m=1}^{a}\left(\cot^{2}\left(\frac{\pi
m}{\alpha}\right)+1\right)^{-1}+2\zeta\right)$
$\displaystyle=\frac{r(1-r)}{\alpha-1}\left(\left(-\csc\left(\frac{\pi}{\alpha}\right)\sin\left(\frac{2\pi
a+\pi}{\alpha}\right)+2a+1\right)+2\zeta\right).$ (B13)

Next, we use that
$a=\left\lfloor(\alpha-1)/2\right\rfloor=(\alpha-1-\zeta)/2$. Then

$\displaystyle\lim_{s\to
0}\lim_{n\to\infty}\frac{1}{s^{2}n}\operatorname*{\mathbb{E}}_{U}S_{\alpha}(U)$
$\displaystyle=\frac{r(1-r)}{\alpha-1}\left(\alpha+\zeta-\frac{\sin(\pi\zeta/\alpha)}{\sin(\pi/\alpha)}\right)$
(B14) $\displaystyle=\frac{\alpha}{\alpha-1}\cdot r(1-r).$ (B15)

As in the other cases, this results holds as a double limit
$\lim_{\begin{subarray}{c}n\to\infty\\\ s\to 0\end{subarray}}$. Evidently, the
Rényi-$\alpha$ Page Curve with small squeezing approaches $r(1-r)\cdot s^{2}$
as $\alpha\rightarrow\infty.$ We demonstrate Eq. B15 numerically in Fig. B.1.

$0.1$$0.2$$0.3$$0.4$$0.5$$0.3$$0.4$$0.5$$\alpha=2$$\alpha=3$$\alpha=4$$\alpha=5$$\alpha=15$$s$$\frac{1}{s^{2}}\operatorname*{\mathbb{E}}_{U}\frac{1}{n}S_{\alpha}(U)$
Figure B.1: The Rényi-$\alpha$ Page curves, for $\alpha\in\\{2,3,4,5,15\\},$
divided by $s^{2}$, as a function of the squeezing strength $s.$ These
functions are displayed as the solid lines. The Rényi-$\alpha$ Page curve
limit as $s\rightarrow 0,$ presented in Eq. B15, is shown by the dashed lines
in corresponding color. As expected, the Rényi-$\alpha$ Page curve approaches
$\frac{\alpha}{\alpha-1}r(1-r)$ in order $s^{2},$ where we take $r=0.5$ and
$n=400$.

###  B.3 Limit as $s\rightarrow\infty$

Now, we investigate the squeezing limit $s\rightarrow\infty.$

We have

$\displaystyle\begin{split}\lim_{s\to\infty}\lim_{n\to\infty}\frac{1}{sn}\operatorname*{\mathbb{E}}_{U}S_{\alpha}(U)&=\lim_{s\to\infty}\frac{1}{s}\frac{1}{\alpha-1}\sum_{\ell=1}^{\infty}\frac{1}{\ell}\sinh^{2\ell}(2s)G_{\ell}(r)\sum_{m=1}^{a}\left(\cot^{2}\left(\frac{\pi
m}{\alpha}\right)+\cosh^{2}(2s)\right)^{-\ell}\\\
&\qquad+\frac{\zeta}{\alpha-1}2\text{ min}(r,1-r)\end{split}$ (B16)
$\displaystyle=\lim_{s\to\infty}\frac{1}{s}\frac{2a}{\alpha-1}\left(\sum_{\ell=1}^{\infty}\frac{\tanh^{2\ell}(2s)}{2\ell}G_{\ell}(r)\right)+\frac{\zeta}{\alpha-1}2\text{
min}(r,1-r)$ $\displaystyle=\frac{2a}{\alpha-1}2\text{
min}(r,1-r)+\frac{\zeta}{\alpha-1}2\text{ min}(r,1-r)$
$\displaystyle=\frac{4a+2\zeta}{\alpha-1}\text{ min}(r,1-r)$
$\displaystyle=2\text{ min}(r,1-r).$ (B17)

As in the other cases, this results holds as a double limit
$\lim_{\begin{subarray}{c}n\to\infty\\\ s\to 0\end{subarray}}$.

We numerically validate Eq. B17 and Eq. A32 in Fig. B.2 below. As shown, both
the von Neumann and Rényi-$\alpha$ entropies indeed approach $2\min(r,1-r)$ in
order $s$ as $s\rightarrow\infty.$

$2$$4$$6$$8$$10$$12$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$2\min(r,1-r)$$\alpha=1$$\alpha=2$$\alpha=3$$s$$\frac{1}{s}\operatorname*{\mathbb{E}}_{U}\frac{1}{n}S_{\alpha}(U)$
Figure B.2: Plots of the von Neumann, Rényi-2, and Rényi-3 Page curves divided
by $s$, as a function of the squeezing strength $s.$ These functions, which
are presented more generally in Theorem 3 and Theorem 5, are shown by the
solid red, blue, and brown lines, respectively. The von Neumann and Rényi Page
curves in the limit as squeezing strength approaches infinity, derived in
Section A.3 and Section B.3, have coinciding values of $2\min(r,1-r)$
displayed as the black dotted line. We take $r=0.5$.

###  B.4 Variance and typicality of entanglement

In this section, we prove weak typicality of entanglement described in Section
4.4. We consider the general expression for the Rényi-$\alpha$ entropy as
found in Eq. B11:

$\displaystyle\mathcal{S}_{\alpha}(\rho)$
$\displaystyle=\frac{\zeta}{\alpha-1}S_{2}+\frac{1}{\alpha-1}\sum_{\ell=1}^{\infty}\left[\sum_{m=1}^{a}\frac{\sinh^{2\ell}(2s)}{\ell(\cosh^{2}(2s)+\cot^{2}\frac{\pi
m}{\alpha})^{\ell}}\right]\left[k-\operatorname{Tr}(W^{\ell})\right]$ (B18)
$\displaystyle=\frac{1}{\alpha-1}\sum_{\ell=1}^{\infty}\left[\sum_{m=1}^{\lfloor\frac{\alpha-1}{2}\rfloor}\frac{\sinh^{2\ell}(2s)}{\ell(\cosh^{2}(2s)+\tan^{2}\frac{\pi
m}{2\alpha})^{\ell}}+\zeta\frac{\sinh^{2\ell}(2s)}{2\ell\cosh^{2\ell}(2s)}\right]\left[k-\operatorname{Tr}(W^{\ell})\right],$
(B19)

with the same definition
$a:=\lfloor\frac{\alpha-1}{2}\rfloor=\frac{\alpha-1-\zeta}{2}$. More
specifically, the $\alpha=2$ Rényi entropy is given by

$\displaystyle\mathcal{S}_{\alpha=2}(\rho)=\sum_{\ell=1}^{\infty}\frac{\sinh^{2\ell}(2s)}{2\ell\cosh^{2\ell}(2s)}\left[k-\operatorname{Tr}(W^{\ell})\right].$
(B20)

From Ref. [39], we know that the following covariance is a non-zero constant
independent of $n$:

$\displaystyle\mathcal{V}_{d}\equiv\sum_{\ell=1}^{d-1}\frac{1}{\ell(d-\ell)}(\overline{\operatorname{Tr}(W^{\ell})\operatorname{Tr}(W^{d-\ell})}-\overline{\operatorname{Tr}(W^{\ell})}\cdot\overline{\operatorname{Tr}(W^{d-\ell})}).$
(B21)

Then the variance of $\mathcal{S}_{\alpha=2}(\rho)$ can also be proved to be a
constant. Letting the fluctuation in the Rényi entropy be defined as
$\delta\mathcal{S}_{\alpha=2}=\mathcal{S}_{\alpha=2}(\rho)-\overline{\mathcal{S}_{\alpha=2}(\rho)}$,
we get
$\text{Var}\,S_{\alpha=2}(\rho)=\overline{\delta\mathcal{S}_{\alpha=2}^{2}}$
as

$\displaystyle\overline{\delta\mathcal{S}_{\alpha=2}^{2}}$
$\displaystyle=\frac{1}{4}\sum_{d=2}^{\infty}\tanh^{2d}(2s)\sum_{\ell=1}^{d-1}\frac{1}{\ell(d-\ell)}(\overline{\operatorname{Tr}(W^{\ell})\operatorname{Tr}(W^{d-\ell})}-\overline{\operatorname{Tr}(W^{\ell})}\cdot\overline{\operatorname{Tr}(W^{d-\ell})})$
(B22)
$\displaystyle=\frac{1}{4}\sum_{d=2}^{\infty}\tanh^{2d}(2s)\mathcal{V}_{d}.$
(B23)

On the other hand, we can also expand the Rényi entropy as

$\displaystyle\mathcal{S}_{\alpha}(\rho)$
$\displaystyle=\frac{1}{\alpha-1}\sum_{m=1}^{\lfloor\frac{\alpha-1}{2}\rfloor}\mathcal{C}_{m}(\alpha)+\frac{\zeta}{\alpha-1}\mathcal{S}_{2}(\rho)$
(B24)
$\displaystyle=\frac{1}{\alpha-1}\left[\sum_{m=1}^{\lfloor\frac{\alpha-1}{2}\rfloor}\sum_{\ell=1}^{\infty}\left(\frac{\sinh^{2}(2s)}{\cosh^{2}(2s)+\cot^{2}(\frac{\pi
m}{2\alpha})}\right)^{\ell}\frac{\left[k-\operatorname{Tr}(W^{\ell})\right]}{\ell}+\frac{\zeta}{2}\sum_{\ell=1}^{\infty}\tanh^{2\ell}(2s)\frac{\left[k-\operatorname{Tr}(W^{\ell})\right]}{\ell}\right],$
(B25)

where

$\displaystyle\mathcal{C}_{m}(\alpha)=\sum_{\ell=1}^{\infty}\left(\frac{\sinh^{2}(2s)}{\cosh^{2}(2s)+\cot^{2}(\frac{\pi
m}{2\alpha})}\right)^{\ell}\frac{\left[k-\operatorname{Tr}(W^{\ell})\right]}{\ell},\quad
m=1,2,\ldots,\left\lfloor\frac{\alpha-1}{2}\right\rfloor.$ (B26)

Note that all quantities $\mathcal{C}_{m}(\alpha)$ are well-defined and
converge for all integers $m$, $\alpha$, and $s$, and, by definition,
$\mathcal{C}_{0}(\alpha)=\mathcal{S}_{\alpha=2}(\rho)$. Further, for each
$\mathcal{C}_{m}(\alpha)$, if we introduce its fluctuation
$\delta\mathcal{C}_{m}(\alpha)=\mathcal{C}_{m}(\alpha)-\overline{\mathcal{C}_{m}(\alpha)}$
, then

$\displaystyle\overline{\delta\mathcal{C}^{2}_{m}(\alpha)}$
$\displaystyle=\frac{1}{4}\sum_{d=2}^{\infty}\left(\frac{\sinh^{2}(2s)}{\cosh^{2}(2s)+\cot^{2}(\frac{\pi
m}{2\alpha})}\right)^{d}\sum_{\ell=1}^{d-1}\frac{1}{\ell(d-\ell)}(\overline{\operatorname{Tr}(W^{\ell})\operatorname{Tr}(W^{d-\ell})}-\overline{\operatorname{Tr}(W^{\ell})}\cdot\overline{\operatorname{Tr}(W^{d-\ell})})$
(B27)
$\displaystyle=\frac{1}{4}\sum_{d=2}^{\infty}\left(\frac{\sinh^{2}(2s)}{\cosh^{2}(2s)+\cot^{2}(\frac{\pi
m}{2\alpha})}\right)^{d}\mathcal{V}_{d}$ (B28)
$\displaystyle\leq\frac{1}{4}\sum_{d=2}^{\infty}\tanh^{2d}(2s)\mathcal{V}_{d},$

meaning that each $C_{m}$ has smaller variance than that of $C_{0}$.
Furthermore, by the Cauchy-Schwarz inequality, we have that

$\displaystyle\overline{\delta\mathcal{C}_{m}(\alpha)\delta\mathcal{C}_{m^{\prime}}(\alpha)}\leq\sqrt{\overline{\delta\mathcal{C}_{m}(\alpha)\delta\mathcal{C}_{m}(\alpha)}\cdot\overline{\delta\mathcal{C}_{m^{\prime}}(\alpha)\delta\mathcal{C}_{m^{\prime}}(\alpha)}}\quad\forall
m,m^{\prime}\in{0,1,\ldots,\left\lfloor\frac{\alpha-1}{2}\right\rfloor},$
(B29)

so that covariances of the form
$\overline{\delta\mathcal{C}_{m}(\alpha)\delta\mathcal{C}_{m^{\prime}}(\alpha)}$
$\forall m,m^{\prime}$ must also be bounded by a constant as $n$ increases.
Finally, since we have found the variance of $\mathcal{C}_{m}(\alpha)$ and
$S_{2}(\rho)$ to be constant in $n,$ we can prove the concentration of Rényi
entropy with integer $\alpha\geq 2$:

$\displaystyle\mathcal{S}_{\alpha}(\rho)=\frac{1}{\alpha-1}\sum_{m=1}^{\lfloor\frac{\alpha-1}{2}\rfloor}\mathcal{C}_{m}(\alpha)+\frac{\zeta}{\alpha-1}\mathcal{S}_{2}(\rho)\Rightarrow\overline{\delta\mathcal{S}_{\alpha}^{2}}=\text{const.}$
(B30)

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