# Scrambling and Entangling Spinning Particles

Ling-Yan Hung b Kaixin Ji d,e Tianheng Wang State Key Laboratory of Surface
Physics, Fudan University, 200433 Shanghai, ChinaDepartment of Physics and
Center for Field Theory and Particle Physics, Fudan University, Shanghai
200433, ChinaYau Mathematical Sciences Center (YMSC), Tsinghua University,
Beijing, 100084, ChinaInstitute of Theoretical Physics, Chinese Academy of
Science, 55 Zhongguancun Road East, 100190 Haidian District, Beijing,
ChinaInstitut für Physik und IRIS Adlershof, Humboldt-Universiät zu Berlin,
Zum Großen Windkanal 6, 12489 Berlin, Germany

###### Abstract

In this paper we revisit the gravitational eikonal amplitudes of two
scattering spinning particles and inspect their scrambling power in the spin
spaces that is quantified through the tripartite information. We found that in
the non-relativistic limit and a special high-energy limit the leading
contribution is a quantity that is universal and theory independent. The
minimal coupling is singled out with minimal scrambling in a different high
momenta limit. We also inspected the initial state dependence of entanglement
generation and found that the spin coherent state with vanishing spin may not
necessarily be the hardest to entangle. Interestingly, among a family of mixed
states, the only P-rep state there known to be the best approximation of
classical mixed states was singled out as one with minimal entanglement
generated.

##  1 Introduction

The study of gravitational-wave physics LIGOScientific:2016aoc ;
LIGOScientific:2017vwq ; LIGOScientific:2018mvr ; LIGOScientific:2020ibl ;
LIGOScientific:2021usb , in particular the dynamics of binary black holes or
neutron stars interacting via classical gravity, has seen a flurry of progress
both in analytical Buonanno:1998gg ; Goldberger:2004jt ; Kol:2007rx ;
Gilmore:2008gq ; Foffa:2011ub ; Foffa:2016rgu ; Porto:2017dgs ;
Blumlein:2019zku ; Foffa:2019rdf ; Foffa:2019yfl ; Blumlein:2020pog  and
numerical computations Pretorius:2005gq ; Campanelli:2005dd ; Baker:2005vv .
One fruitful direction is to extract classical dynamics perturbatively from
scattering amplitudes in quantum field theory Bjerrum-Bohr:2018xdl ;
Cheung:2018wkq ; Bern:2019nnu ; Bern:2019crd ; Neill:2013wsa ;
Cristofoli:2021vyo ; Bern:2020buy ; Bern:2021dqo . Modern unitarity-based
techniques allow computations of scattering amplitudes at high loop orders
Bern:1994zx ; Bern:1994cg ; Britto:2004nc ; Bjerrum-Bohr:2013bxa ;
Luna:2017dtq ; Kawai:1985xq ; Bern:2008qj ; Bern:2010ue ; Bern:2012uf ;
Bern:2019prr  and designated effective field theories Cheung:2018wkq ;
Bjerrum-Bohr:2002gqz  serve to connect quantum amplitudes and classical
observables. Such classical observables Kosower:2018adc ; Maybee:2019jus ;
Mogull:2020sak ; Jakobsen:2021smu ; Jakobsen:2021lvp ; Jakobsen:2021zvh ;
Jakobsen:2022fcj  can also be computed from generating functions. In
particular, one such generating function, the classical eikonal phase, is
closely related to scattering amplitudes Fourier transformed to impact
parameter space Parra-Martinez:2020dzs ; Bern:2021dqo ; DiVecchia:2021bdo ;
Heissenberg:2021tzo ; DiVecchia:2022nna .

In addition to being the generating function for physical observables, the
eikonal phase can also be viewed as the resummation of ladder diagrams in the
classical approximation Levy:1969cr ; Amati:1987uf ; DiVecchia:2019myk ;
DiVecchia:2020ymx . Currently the eikonal phase has been calculated up to the
third Post-Minkowskian(PM) order for non-spinning binary systems Parra-
Martinez:2020dzs ; DiVecchia:2021bdo ; Heissenberg:2021tzo , and the quadratic
order in spin vectors at the second PM order for spinning black holes
Jakobsen:2021zvh ; Jakobsen:2022fcj . The radiation reaction contribution to
the eikonal is also studied up to the third PM order for the spinning case
Alessio:2022kwv  using the leading soft theorem DiVecchia:2021ndb . Moreover,
the amplitude has been calculated up to the fifth order in spin at the second
PM order in for spinning binary systems Bern:2020buy ; Bern:2022kto ;
Chen:2021qkk , from which the 2PM eikonal phase can be extracted.111Recently
there have also been related works on the Compton scattering involving Kerr
black holes at the second PM order Aoude:2022trd ; Aoude:2022thd .

It is curious in this semi-classical limit whether one can dig out remnants of
quantum effects. One potentially fruitful line of considerations makes use of
tools in quantum information theory. For example it has been suggested that
should gravitational interaction increase the entanglement between interacting
particles, that would certainly suggest that gravity must not be a local
operation and classical communication  (LOCC)222There are controversies
questioning if generation of entanglement by gravity is a sufficient smoking
gun for the quantum nature of gravity. Currently it still appears possible
that if for sufficiently short wavelength of the mediating gravitational wave
the quantum nature of gravity might still be tested. A list of papers could be
found in vedral_ans .  Bose:2017nin ; Marletto:2017kzi . This line of thought
induced many studies of entanglement property of the S-matrix, which focused
particularly on the generation of entanglement in a scattering process, for
example in Cervera-Lierta:2017tdt ; Beane:2018oxh ; Afik:2020onf . (For a cute
use of the relative entropy see also Bose:2020shm ). Pertaining to
gravitational interaction, more recently, it was found numerically that the
generation of entanglement between two scattered spinning particles is minimal
at minimal coupling in a good range of kinematic space based on the eikonal
approximation Aoude:2020mlg ; Chen:2021huj .

In this paper we would like to study the eikonal phase in greater detail in
light of the quantum information perspective, and inspect a related but
orthogonal aspect, namely the scrambling behaviour of gravitational
scattering. A quantitative measure of the scrambling power of a unitary
evolution is given by the tripartite information Hosur:2015ylk . We will pay
special attention to expanding the result analytically in $G_{N}/|b|^{2}$
particularly for spin-$1/2$ and spin-$1$ scattering particles, where $G_{N}$
is the Newton’s constant, and $b$ is the impact parameter between the
scattering. Surprisingly, we find that bounds of the tripartite information
through computation of Renyi entropies show universal behaviours independent
of theories describing the gravitational background, in both the non-
relativistic and the high-energy limits. We compute a large-spin result that
confirms that scrambling tends to diminish in this limit, reducing to a more
classical behaviour. The tripartite information can be computed analytically
as an expansion at least for spin-$1/2$ and spin-$1$. With such analytic
control, we are able to identify a term logarithmic in the Newton’s constant.
This can be contrasted with logarithmic loop corrections to black hole
entropies notably discussed in Sen:2012dw . The scrambling power is a minimum
at minimal coupling only in the relativistic limit however. We obtain also an
analytical large-spin limit of the Renyi-2 tripartite entropy. It shows that
generically for large impact parameter $b$ and leading order in $G_{N}$,
scrambling power decreases exponentially with large spins. This is perhaps a
confirmation of intuition that large spins lead to more classical behaviour.
Although for spins much greater than the product of the impact parameter and
the particle mass, quantum behaviour would win and scrambling increases
exponentially instead.

We also revisit the question studied in Aoude:2020mlg ; Chen:2021huj , but
consider how entanglement generation is related to the initial states of the
scattering particles. It is customary that we take the spin coherent states as
the initial states of semi-classical spinning objects. (See for example
Aoude:2021oqj ; Gonzo:2022rfk .) Here, we explore a family of squeezed spin
coherent states, and see if the entanglement generation should single out a
squeezed state that is most resistant to getting entangled, and thus most
classical. This is inspired by considerations in the literature that study
squeezed coherent states in harmonic oscillators. (For a review, see Ma_2011
.) We again adopt this perturbative approach that allows analytic treatment.
We find that entanglement generation is at its minimum or maximum at vanishing
squeezing, depending on the relative orientation of the spins of the two
particles. While this analysis does not single out the zero-squeezed state, it
points to possibilities of amplifying quantum effects by appropriate adjusting
of initial states.

Our paper is organised as follows. In section 2, we review the tree-level
amplitude of the $2\rightarrow 2$ scattering process of two spinning
blackholes and the corresponding eikonal approach. In section 3, we study the
scrambling power of this scattering process. We display the bounds for the
tripartite information and its large-spin limit. We also compute the exact
tripartite information for the spin-$1/2$ case. In section 4, we consider
different initial states and study how classically they behave through the
lens of entanglement.

##  2 Gravity amplitudes and the Eikonal approximation

In this section, we first review the tree-level amplitude for the
$2\rightarrow 2$ scattering of a spinning binary system interacting via
gravity in the classical (soft) limit. The eikonal phase can be extracted from
the amplitude Fourier transformed to impact parameter space, sometimes called
the eikonal amplitude.

In the center-of-mass frame, the kinematics of the binary system is
parameterized by

$\displaystyle
p_{1}=(E_{1},0,0,p)\,,\;p_{2}=(E_{2},0,0,-p)\,,\;p^{\prime}_{1}=(E_{1},0,0,p^{\prime})\,,\;p^{\prime}_{2}=(E_{2},0,0,-p^{\prime})\,,$
(1)

and the total momentum transfer is then $q=(0,\vec{q})=p^{\prime}_{1}-p_{1}$.
In the classical limit $\hbar\rightarrow 0$, we consider only the long-range
interaction which is given by the soft graviton exchange, namely
$q^{2}\sim\mathcal{O}(\hbar^{2})$. Hence at tree level, only the $t$-channel
with the pole $q^{2}\rightarrow 0$ contributes to the $2\rightarrow 2$
scattering. Consequently we have $p_{1}\cdot q=p_{2}\cdot q=q^{2}/2\rightarrow
0$ in the classical limit.

For massive particles of generic spins $s_{1}$ and $s_{2}$, the tree-level
amplitude is computed in Arkani-Hamed:2017jhn ; Chung:2019duq ; Chung:2020rrz
and expressed in terms of the Wilson coefficients Levi:2015msa ; Levi:2016ofk
,

$\displaystyle M_{\text{tree}}(q^{2})=-8\pi G_{N}{m_{1}^{2}m_{2}^{2}\over
q^{2}}\sum_{\eta=\pm
1}e^{2\eta\Theta}\left[\varepsilon^{*}_{1^{\prime}}W(\eta\tau_{1})\varepsilon_{1}\right]\left[\varepsilon^{*}_{2^{\prime}}W(\eta\tau_{2})\varepsilon_{2}\right]+\mathcal{O}(q^{0})$
(2)

where $\varepsilon_{i}$ denotes the polarization tensor of the spinning
particle $i$ and the sum is taken over the helicities of the graviton
exchange. We have adopted the notations in Chung:2019duq ; Chung:2020rrz ;
Aoude:2020mlg ; Chen:2021huj  where $\tau_{i}={q\cdot S_{i}\over m_{i}}$ and
$\cosh\Theta={p_{1}\cdot p_{2}\over m_{1}m_{2}}$. Here $S_{i}$ denotes the
Pauli-Lubanskin spin vector and the function $W$ above is defined as

$\displaystyle W(\eta\tau_{i})=\sum_{n=0}^{2s_{i}}{C_{i,n}\over
n!}\left(\eta{q\cdot S_{i}\over m_{i}}\right)^{n}\,,$ (3)

where $C_{i,n}$ denotes the Wilson coefficients describing the multi-pole
moments of the spinning particles. Rotating black hole solutions correspond to
the minimal coupling given by $C_{1,n}=C_{2,n}=1$. We note that the spins here
are perceived in the classical sense, which we shall discuss in more detail
shortly. The spin vector can be transformed to an operator $\mathbb{S}^{\mu}$

$\displaystyle\mathbb{S}^{\mu}_{i}\equiv\varepsilon_{i}^{*}S^{\mu}\varepsilon_{i}=\left({\vec{p}_{i}\cdot\vec{\Sigma}\over
m_{i}},~{}\vec{\Sigma}+{\vec{p}_{i}\cdot\vec{\Sigma}\over
m_{i}(m_{i}+E_{i})}\vec{p}_{i}\right)\,,$ (4)

which acts on the little group indices with $\Sigma$ denoting the spin
operator in the rest frame. Re-expressing (3) in terms of $\mathbb{S}$
requires an extra Thomas-Wigner factor for each particle to compensate the
rotation in the little group space. (For detailed discussions, see
Chung:2019duq ; Chung:2020rrz .) The Thomas-Wigner factor reads

$\displaystyle U_{i}=\exp\left[-i{m_{1}m_{2}\mathbb{E}_{i}\over
r_{i}(E_{1}+E_{2})}\right]\,,$ (5)

where we have $r_{a}=1+E_{a}/m_{a}$, $\mathbb{E}_{1}\equiv{1\over
m_{1}^{2}m_{2}}\,\epsilon_{\mu\nu\rho\sigma}q^{\mu}p_{1}^{\nu}p_{2}^{\rho}\,\mathbb{S}^{\sigma}_{1}$
and $\mathbb{E}_{2}\equiv{1\over
m_{1}m_{2}^{2}}\,\epsilon_{\mu\nu\rho\sigma}q^{\mu}p_{1}^{\nu}p_{2}^{\rho}\,\mathbb{S}^{\sigma}_{2}$.
Consequently, the amplitude (2) is rewritten as follows,

$\displaystyle\bar{M}_{\text{tree}}(q^{2})=-8\pi G_{N}{m_{1}^{2}m_{2}^{2}\over
q^{2}}\sum_{\eta=\pm
1}e^{2\eta\Theta}W_{1}(\eta\mathbb{T}_{1})W_{2}(\eta\mathbb{T}_{2})U_{1}U_{2}+\mathcal{O}(q^{0})\,,$
(6)

where we have adopted the notation $\mathbb{T}_{i}={q\cdot\mathbb{S}_{i}\over
m_{i}}$.

In the classical limit $q^{2}\rightarrow 0$, the matrices $\mathbb{E}_{i}$ can
be analytically continued Chung:2020rrz ; Guevara:2017csg

$\displaystyle\mathbb{E}_{i}=-i\sinh\Theta\,\mathbb{T}_{i}\,.$ (7)

This relation allows us to trade $\mathbb{E}_{i}$ and $\mathbb{T}_{i}$ in (6)
and it turns out the most convenient to rewrite all the $\mathbb{E}_{i}$
factors in terms of $\mathbb{T}_{i}$ for our purposes. In later parts of the
paper, we have all implicitly adopted this treatment when referring to (6).

Generally speaking, in the classical limit, the so-called classical eikonal
phase, denoted as $\chi$ can be extracted from the amplitudes in impact
parameter space. The following relation is established in the non-spinning
case and it is conceivable that the same structure carries over to the
spinning case,

$\displaystyle 1+i\mathcal{M}(b)=\left(1+i\Delta(b)\right)e^{i\chi(b)}\,,$ (8)

where $\mathcal{M}$ denotes the Fourier transform of the amplitude and
$\Delta(b)$ denotes the quantum remainder DiVecchia:2021bdo ;
Heissenberg:2021tzo ; DiVecchia:2022nna . The quantum remainder enters at one
loop and therefore at tree level, the leading order of the classical eikonal
phase is simply the Fourier transform of the amplitude

$\displaystyle\chi^{(0)}(b)={1\over
4|p|E}\int{d^{2}\vec{q}\over(2\pi)^{2}}e^{i\vec{q}\cdot\vec{b}}\bar{M}_{\text{tree}}(q^{2})\,,$
(9)

where we note that both $b^{\mu}$ and $q^{\mu}$ are space-like vectors in the
$xy$-plane only. Hence the Fourier transform is performed in $2$ dimensions.
Beyond tree level, the relation between the classical eikonal phase and the
amplitude in impact parameter space is less explicit, but can be read off from
(8) order by order.

As we shall see in later sections, the hierarchy of scales encountered in the
scattering process is crucial to the validity of our analyses. Let us
elaborate on this by first restoring $\hbar$ explicitly in the quantities we
have encountered,

$\displaystyle q\rightarrow\hbar
q\,,~{}~{}~{}S^{\mu}\rightarrow{S^{\mu}\over\hbar}\,,~{}~{}~{}G_{N}\rightarrow{G_{N}\over\hbar}\,,~{}~{}~{}m_{1,2}\rightarrow
m_{1,2}\,,~{}~{}~{}p_{1,2}^{\mu}\rightarrow p_{1,2}^{\mu}\,.$ (10)

For notational brevity, keeping the above in mind, later in this paper we
still work in the unit $\hbar=1$ and the classical limit can be equivalently
viewed as a large-impact parameter limit. In other words, $1/|b|$ plays the
role of $\hbar$. Schematically, the amplitude in impact parameter space admits
a double expansion in both $G_{N}$ and $|b|$ as follows,

$\displaystyle
i\mathcal{M}(b)=G_{N}{f^{\text{cl}}_{0}(b,S)\over|b|^{-2\epsilon}}+G_{N}^{2}\left[{f^{\text{scl}}_{1}(b,S)\over|b|^{-4\epsilon}}+{f^{\text{cl}}_{1}(b,S)\over|b|^{1-4\epsilon}}+{f^{\text{q}}_{1}(b,S)\over|b|^{2-4\epsilon}}\right]+\mathcal{O}(G_{N}^{3})\,,$
(11)

where $f$’s are functions of the kinematics and spin degrees of freedom. The
superscripts indicate the so-called _superclassical_ , _classical_ and
_quantum_ contributions. It is straightforward to observe from the tree-level
amplitude (6) that in $f_{0}^{\text{cl.}}$ for each spin operator acquires a
factor of $(\hat{b}\cdot S)/(|b|m_{1,2})$ where $\hat{b}^{\mu}=b^{\mu}/|b|$.
This factor is taken to be of $\mathcal{O}(1)$. We have also checked that
$f_{1}^{\text{cl.}}(b,S)$ behaves the same way as the power of spin vectors
increases Bern:2020buy ; Bern:2022kto ; Chung:2019duq ; Chung:2020rrz ;
Chen:2021qkk . We believe that the behaviour of the spin-dependence in
$f^{\text{scl.}}_{1}(b,S)$ can be obtained from (8) in order to have the
consistent exponentiation DiVecchia:2021bdo ; Heissenberg:2021tzo ;
DiVecchia:2022nna . Consequently, although the phase $\chi$ is only an
expansion in $G_{N}$ and is universal in $\hbar$ or $1/|b|$, higher-order
corrections to the factor $e^{i\chi}$ contain superclassical terms which are
exponentiation of lower orders. They are usually discarded when discussing the
classical EFT of a quantum field theory, as they provide no new information.
However they are present in the eikonal amplitudes nonetheless and are in fact
stronger than the classical contributions. In the following sections of this
manuscript, we will see that these superclassical terms, together with those
arising from $\chi\chi^{\dagger}$ which are of the same order in both $G_{N}$
and $1/|b|$, give the leading contribution to entropies induced by the eikonal
amplitudes.

In order to discuss the non-relativistic and the high-energy limits of these
entropies, we take a closer look at the dependence of $m_{1,2}\sim m$ and
$p=|\vec{p}_{1}|=|\vec{p}_{2}|$ in the tree and one-loop amplitudes
Bern:2020buy ; Bern:2022kto ; Chung:2019duq ; Chung:2020rrz ; Chen:2021qkk .
In the non-relativistic regime $p\ll m$ ($\cosh\Theta\rightarrow 1$), we count
the powers of $m$ and $p$ in the amplitudes and have

$\displaystyle
i\mathcal{M}(b)={G_{N}m^{3}\over|b|^{-2\epsilon}p}\sum_{n}a_{0,n}^{\text{cl}}x^{n}+{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{G_{N}^{2}m^{6}\over|b|^{-4\epsilon}p^{2}}\sum_{n}a^{\text{scl.}}_{1,n}x^{n}}+{G_{N}^{2}m^{4}\over|b|^{1-4\epsilon}p}\sum_{n}a^{\text{cl.}}_{1,n}x^{n}+\cdots\,,$
(12)

where $x=(\hat{b}\cdot S)/(|b|m_{1,2})\sim\mathcal{O}(1)$. We note that here
the scaling behaviour of the one-loop superclassical terms in gray is inferred
from the consistency requirement that
$f^{\text{scl}}_{1}\propto(f_{0}^{\text{cl}})^{2}$. Demanding that the
superclassical contribution is indeed stronger than the classical one, we have
$m/p\gg 1/(|b|m)$ which is naturally satisfied in the non-relativistic regime.

Similarly in the high-energy (ultra-relativistic) regime $p\gg m$
($\cosh\Theta\sim p/m$), the amplitudes behave as follows,

$\displaystyle
i\mathcal{M}(b)={G_{N}p^{2}\over|b|^{-2\epsilon}}\sum_{n}a_{0,n}^{\text{cl}}x^{n}+{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{G_{N}^{2}p^{4}\over|b|^{-4\epsilon}}\sum_{n}a^{\text{scl.}}_{1,n}x^{n}}+{G_{N}^{2}mp^{2}\over|b|^{1-4\epsilon}p}\sum_{n}a^{\text{cl.}}_{1,n}x^{n}+\cdots\,.$
(13)

The same analysis gives that $(p/m)^{2}\gg 1/(|b|m)$ guarantees that the
classical contribution would not spoil our later computations.

##  3 Scrambling spinning particles via gravitational interactions

In this section, we would like to inspect the scrambling power of a
perturbative gravitational scattering event. As reviewed in the previous
section, in the classical limit, the eikonal amplitude $\exp(i\chi)$
approximates the evolution of a pair of particles scattered through soft
gravitons. When the particles carry non-trivial angular momenta, the eikonal
amplitude is a unitary matrix in spin space. Therefore it is natural to treat
$\exp(i\chi)$ as a unitary gate and inspect the scrambling power of such a
gate.

There are quite a number of quantitative measures of scrambling power. A
particularly handy one is the tri-partite information defined in Hosur:2015ylk
. The idea is that in the presence of scrambling, input information would be
highly delocalised by the evolution, and one needs the entirety of the output
qubits to recover the original state before the input qubits. Utilising this
defining property of scrambling, we consider treating the unitary gate as a
wavefunction, and the spin degrees of freedom of the scattering particles as
qubits. One can divide the input qubits made up of the in-coming scattering
particles into two sets $A$ and $B$, and the output qubits into two sets $C$
and $D$, and compute the tripartite information $I(A,C,D)$ defined as

$I(A,C,D)=S_{A}+S_{C}+S_{D}-S_{AC}-S_{AD}-S_{CD}+S_{ACD},$ (14)

where $S_{X}$ is the entanglement entropy of region $X$ with the rest of the
system. This quantity essentially measures the amount of information in $A$
that is stored non-locally in $C$ and $D$. We note that in the case at hand,
$A$ and $B$ would naturally be taken as the spin space of input particles 1
and 2 respectively, and $C$ and $D$ taken as the respective spin spaces of
particles 1 and 2 after the scattering event.

To compute the entanglement entropies needed above, we need to construct the
reduced density matrix. Treating the eikonal as a “wavefunction”, we have

$|\chi\rangle\equiv\frac{1}{(2s_{1}+1)(2s_{2}+1)}\sum_{i_{1},i_{2},j_{1},j_{2}}\mathcal{E}_{i_{1},i_{2}}^{j_{1},j_{2}}|i_{1},i_{2},j_{1},j_{2}\rangle,\qquad\mathcal{E}\equiv\exp(i\chi),$
(15)

where we have restored the spin indices explicitly after the first equality,
and $i_{a},j_{a}\in\\{-s_{a},-s_{a}+1,\cdots,s_{a}\\}$ are the spin indices of
particle $a$ before and after scattering respectively. The density matrix is
thus given by

$\rho\equiv\sum_{i_{1},i_{2},j_{1},j_{2},i^{\prime}_{1},i^{\prime}_{2},j^{\prime}_{1},j^{\prime}_{2}}\mathcal{E}_{i_{1},i_{2},j_{1},j_{2}}|i_{1},i_{2},j_{1},j_{2}\rangle\langle
i^{\prime}_{1},i^{\prime}_{2},j^{\prime}_{1},j^{\prime}_{2}|\mathcal{E}^{*}_{i^{\prime}_{1},i^{\prime}_{2},j^{\prime}_{1},j^{\prime}_{2}}.$
(16)

Now suppose we would like to compute $S_{A}$, we obtain the reduced density
matrix by tracing out $B,C,D$. This gives

$\rho_{A}=\frac{1}{(2s_{1}+1)(2s_{2}+1)}\sum_{i_{1},i^{\prime}_{1}}\mathcal{E}_{i_{1},i_{2},j_{1},j_{2}}\mathcal{E}^{*}_{i_{1}^{\prime},i_{2},j_{1},j_{2}}|i_{1}\rangle\langle
i^{\prime}_{1}|=\frac{1}{(2s_{1}+1)}\sum_{i_{1},i_{1}^{\prime}}|i_{1}\rangle\langle
i_{1}|,$ (17)

where we have made use of the unitarity of $\mathcal{E}$ as a matrix, so that
$\mathcal{E}^{*}_{i_{1}^{\prime},i_{2},j_{1},j_{2}}=\mathcal{E}^{-1}_{j_{1},j_{2},i_{1}^{\prime},i_{2}}$.
This means that the spin $A$ is maximally entangled with the rest of the
spins. In this case the entanglement entropy has no dependence on the
kinematics and the couplings, and it is given simply by

$S_{A}=\ln(2s_{1}+1).$ (18)

By the same token, $S_{B},S_{D},S_{ABD}\equiv S_{C},S_{AB}\equiv S_{CD}$ all
take the form of $S_{X}=\log(d_{X})$, where $d_{X}$ is the dimension of the
reduced spin space $X$. $S_{AD}$ receives no $\mathcal{O}(G_{N}^{2})$
contributions either due to unitarity.

The only non-trivial contributions therefore come from $S_{AC}$. As discussed
in Section 2, $\mathcal{E}$ admits a double expansion in both $G_{N}$ and
$|b|$, and consequently we work in similar double expansions throughout this
paper. It is straightforward to check that the terms linear in $G_{N}$ cancel
out in all the reduced density matrices above, when the density matrices are
properly normalized. At $\mathcal{O}(G_{N}^{2})$, $\mathcal{E}$ receives the
superclassical ($\mathcal{O}(|b|^{4\epsilon})$) and classical
($\mathcal{O}(|b|^{-1+4\epsilon})$) contributions. Despite that the
superclassical contribution comes purely from the exponentiation of the tree-
level eikonal phase $\chi^{(0)}$, it is still the leading term at
$\mathcal{O}(G_{N}^{2})$ due to the classical limit. Therefore, in keeping
with the regime of validity of the expression, we ignore the contribution from
the one-loop amplitude and consider only the leading term which arises from
the tree-level amplitude. As we are going to show, it is possible to obtain
exact analytical results in a $G_{N}$ expansion.

As a quick summary, the tripartite information takes the following form to
order $G_{N}^{2}$

$\displaystyle I(A,C,D)=\ln[(2s_{1}+1)(2s_{2}+1)]-S_{AC}.$ (19)

The larger is $S_{AC}$, the smaller is $I(A,C,D)$, which signifies that
information is more scrambled.

###  3.1 An upper bound

It is quite cumbersome to compute the entanglement entropy, while one can more
readily recover analytic results for Renyi-$r$ entropy $S^{(r)}_{X}$ for a
subsystem $X$, defined as

$\displaystyle
S^{(r)}_{X}=\frac{\ln[\textrm{tr}(\rho_{X}^{r})/(\textrm{tr}(\rho_{X}))^{r}]}{1-r}.$
(20)

Note that in the limit $r\to 1$ the Renyi-$r$ entropy recovers the
entanglement entropy. They also satisfy a well known inequality

$\displaystyle S^{(1)}\geqslant S^{(2)}\geq S^{(3)}\geqslant\cdots
S^{(\infty)}.$ (21)

Therefore, we can give an upper bound to the tripartite information (19)

$\displaystyle I(A,C,D)\leqslant\ln[(2s_{1}+1)(2s_{2}+1)]-S^{(2)}_{AC}.$ (22)

In principle the Renyi-$r$ entropy $S_{AC}^{(r)}$ can be computed
straightforwardly, given the wavefunction
$\mathcal{E}_{i_{1}i_{2}}^{j_{1}j_{2}}$. Here we sketch out the derivation.
Schematically, the wavefunction $\mathcal{E}_{i_{1}i_{2}}^{j_{1}j_{2}}$ takes
the expansion below

$\displaystyle\mathcal{E}_{i_{1}i_{2}}^{j_{1}j_{2}}=\;$
$\displaystyle\delta^{j_{1}}_{i_{1}}\delta^{j_{2}}_{i_{2}}+G_{N}\sum_{n_{1}n_{2}}\sum_{I_{1}I_{2}}\mathbb{A}_{n_{1}n_{2}}^{I_{1}I_{2}}\left(O_{I_{1}}^{(n_{1})}\right)_{i_{1}}^{j_{1}}\otimes\left(O_{I_{2}}^{(n_{2})}\right)_{i_{2}}^{j_{2}}$
$\displaystyle+G_{N}^{2}\sum_{n_{1}m_{1}n_{2}m_{2}}\sum_{I_{1}J_{1}I_{2}J_{2}}\mathbb{B}^{I_{1}J_{1}I_{2}J_{2}}_{n_{1}m_{1}n_{2}m_{2}}\left(O^{(n_{1}m_{1})}_{I_{1}J_{1}}\right)_{i_{1}}^{j_{1}}\otimes\left(O^{(n_{2}m_{2})}_{I_{2}J_{2}}\right)_{i_{2}}^{j_{2}}+\mathcal{O}(G_{N}^{3})\,,$
(23)

where we have packaged the operators acting on the spin states as
$O_{0}=\mathbb{I}$, $O_{+,-}=\Sigma_{+,-}$ and the extra indices $i_{a},j_{a}$
denote the incoming and outgoing states. We have introduced the notations:
$O_{I}^{(n)}=\left(O_{I}\right)^{n},~{}O_{IJ}^{(nm)}=\left(O_{I}\right)^{n}\left(O_{J}\right)^{m}$.
The summations are taken over $n_{a},m_{a}=1,\cdots,2s_{a}$, for spin-$s_{a}$
and the operator summations labelled by $I$’s exclude $\Sigma_{z}$ , since
$\Sigma_{z}$ does not enter $\chi^{(0)}(b)$, which gives the terms linear in
$G_{N}$ above. This can be observed immediately from the definition of the
tree level eikonal phase (9). Moreover, mixed terms
$\Sigma_{+}^{n}\Sigma_{-}^{m}$ cancel out at $\mathcal{O}(G_{N})$. At
$\mathcal{O}(G_{N}^{2})$, $\Sigma_{z}$ contributes but only through
$(\Sigma_{+})^{n}(\Sigma_{-})^{m}$ for positive $n,m$. As we will see shortly,
the contribution linear in $G_{N}$ drops out in the Renyi-$r$ entropy and
therefore we keep the expansion up to the quadratic order. The coefficients
$\mathbb{A}_{n_{1}n_{2}}^{I_{1}I_{2}}$ and
$\mathbb{B}_{n_{1}m_{1}n_{2}m_{2}}^{I_{1}J_{1}I_{2}J_{2}}$ are functions of
kinematic variables and independent of the spin-related degrees of freedom.
They can be read off from the eikonal $\mathcal{E}$. Before diving into
details, we note the immediate simplifications following from the above
observations: $\mathbb{A}^{00}_{nm}=0$ for any $n,m>1$,
$\mathbb{A}^{0I}_{nm}=\mathbb{A}^{I0}_{mn}=0$ for $I=\pm$ and $n>1$, and
$\mathbb{A}^{IJ}_{nm}=0$ if $\\{I,J\\}=\pm$ and $I\neq J$. We also note that
$\mathbb{A}^{00}_{11}$ is multiplied by identity operators acting on both
spins, the same as the leading order. For our purpose, this term can be
absorbed into the leading term and in principle can affect the normalization.
Since we keep up to $\mathcal{O}(G_{N}^{2})$ only, this term does not
contribute at all and therefore can be discarded from now on. The quadratic
terms correspond to the superclassical contributions and arise from the
exponentiation of the tree-level eikonal phase. Hence the coefficients are
given by

$\displaystyle\mathbb{B}_{n_{1}m_{1}n_{2}m_{2}}^{I_{1}J_{1}I_{2}J_{2}}={1\over
2}\mathbb{A}_{n_{1}n_{2}}^{I_{1}I_{2}}\mathbb{A}_{m_{1}m_{2}}^{J_{1}J_{2}}\,.$
(24)

Now the reduced density matrix $\rho_{AC}$ reads

$\displaystyle\rho_{AC}=$
$\displaystyle\sum_{i_{1},j_{1},i^{\prime}_{1},j^{\prime}_{1}}\mathcal{E}_{i_{1}i_{2}j_{1}j_{2}}\mathcal{E}^{*}_{i^{\prime}_{1},i_{2},j^{\prime}_{1},j_{2}}|i_{1}j_{1}\rangle\langle
i^{\prime}_{1}j^{\prime}_{1}|$ $\displaystyle=$
$\displaystyle\left(\mathbb{P}+G_{N}\Omega_{1}+G_{N}^{2}\big{(}\Omega_{2}^{(1)}+\Omega_{2}^{(2)}\big{)}\right)_{i_{1}j_{1}i^{\prime}_{1}j^{\prime}_{1}}|i_{1}j_{1}\rangle\langle
i^{\prime}_{1}j^{\prime}_{1}|\,,$ (25)

where the leading term is a projector
$\mathbb{P}_{i_{1}j_{1}i^{\prime}_{1}j^{\prime}_{1}}=(2s_{2}+1)\delta_{i_{1}j_{1}}\delta_{i^{\prime}_{1}j^{\prime}_{1}}$
and the other operators read

$\displaystyle(\Omega_{1})_{i_{1}j_{1}i^{\prime}_{1}j^{\prime}_{1}}=$
$\displaystyle\sum_{\\{n_{a},I_{a}\\}}\left[\mathbb{A}_{n_{1}n_{2}}^{I_{1}I_{2}}\big{(}O^{(n_{1})}_{I_{1}}\big{)}_{i_{1}j_{1}}\delta_{i^{\prime}_{1}j^{\prime}_{1}}+\big{(}\mathbb{A}_{n_{1}n_{2}}^{I_{1}I_{2}}\big{)}^{*}\big{(}O^{(n_{1})}_{I_{1}}\big{)}_{i^{\prime}_{1}j^{\prime}_{1}}\delta_{i_{1}j_{1}}\right]\text{tr}\big{(}O_{I_{2}}^{(n_{2})}\big{)}\,,$
(26)
$\displaystyle(\Omega_{2}^{(1)})_{i_{1}j_{1}i^{\prime}_{1}j^{\prime}_{1}}=$
$\displaystyle\sum_{\\{n_{a},I_{a}\\}}\left[\mathbb{B}_{n_{1}m_{1}n_{2}m_{2}}^{I_{1}J_{1}I_{2}J_{2}}\big{(}O_{I_{1}J_{1}}^{(n_{1}m_{1})}\big{)}_{i_{1}j_{1}}\delta_{i^{\prime}_{1}j^{\prime}_{1}}+\big{(}\mathbb{B}_{n_{1}m_{2}n_{2}m_{2}}^{I_{1}J_{1}I_{2}J_{2}}\big{)}^{*}\big{(}O_{I_{1}J_{1}}^{(n_{1}m_{1})}\big{)}_{i^{\prime}_{1}j^{\prime}_{1}}\delta_{i_{1}j_{1}}\right]$
$\displaystyle\quad\quad\times\text{tr}\big{(}O_{I_{2}J_{2}}^{(n_{2}m_{2})}\big{)}\,,$
(27)
$\displaystyle(\Omega_{2}^{(2)})_{i_{1}j_{1}i^{\prime}_{1}j^{\prime}_{1}}=$
$\displaystyle\sum_{\\{n_{1},I_{a}\\}}\mathbb{A}_{n_{1}n_{2}}^{I_{1}I_{2}}\big{(}\mathbb{A}_{m_{1}m_{2}}^{J_{1}J_{2}}\big{)}^{*}\text{tr}\big{(}O_{I_{2}J_{2}}^{(n_{2}m_{2})}\big{)}\big{(}O_{I_{1}}^{(n_{1})}\big{)}_{i_{1}j_{1}}\big{(}O_{J_{1}}^{(m_{1})}\big{)}_{i^{\prime}_{1}j^{\prime}_{1}}\,,$
(28)

where $I_{a},J_{a}\in\\{\mathbb{I},\Sigma^{\pm}_{a}\\}$ and small letters
denote the power of these operators $\\{n_{a},m_{a}\\}=1,\cdots,2s_{a}$.
Recalling that $\text{tr}\big{(}\mathbb{I}\big{)}=2s_{2}+1$ when we trace out
$B$ and $D$ corresponding to particle $2$ and that
$\text{tr}\big{(}\Sigma^{n}_{\pm}\big{)}=0$, the remaining traces above are
readily computed. In addition to the identity matrix, the only non-vanishing
trace is given by

$\displaystyle\text{tr}_{2}\left(O_{\pm\mp}^{(mm)}\right)=\text{tr}_{2}\left(\Sigma^{m}_{+}\Sigma^{m}_{-}\right)\equiv
t^{(m)}_{2}$ (29)

where the additional subscript indicates that the partial trace is taking over
indices belonging to particle 2.

Taking these traces can be carried out straightforwardly and the detailed
computation is demonstrated in Appendix A. Here we present the final
expression for the Renyi-$r$ entropy

$\displaystyle S^{(r)}_{AC}={G_{N}^{2}r\over
r-1}{1\over(2s_{1}+1)(2s_{2}+1)}\sum_{m,n}\sum_{I,J=\pm}\mathbb{A}_{mn}^{IJ}\left(\mathbb{A}_{mn}^{IJ}\right)^{*}t^{(m)}_{1}t^{(n)}_{2}\,,$
(30)

where $t^{(m)}_{a}=\text{tr}\,\Sigma_{+}^{m}\Sigma_{-}^{m}$ with
$m=1,\cdots,2s_{a}$. Notice that $\mathbb{B}$-terms, namely the superclassical
terms in $\mathcal{E}$, drop out in the Renyi-$r$ entropy when properly
normalised. The $G_{N}^{2}$ contribution above come from
$\chi^{(0)}\big{(}\chi^{(0)}\big{)}^{*}$ and although at the same order in the
soft expansion with these superclassical terms, are of a slightly different
nature.

For $s_{1}=s_{2}=1/2$, only the minimal coupling interactions are involved and
the Renyi-2 entropy $S_{AC}^{(2)}$ reads

$\displaystyle S_{AC,1/2}^{(2)}$ $\displaystyle={G_{N}^{2}\over
16}\left|\mathbb{A}_{1,1}^{++}\right|^{2}={16G^{2}_{N}\pi^{4}m_{1}^{2}m_{2}^{2}\over(-b^{2})E^{4}r_{1}^{2}r_{2}^{2}}\times$
(31)
$\displaystyle\left[8E\cosh\Theta\sinh^{2}\Theta(m_{1}r_{2}+m_{2}r_{2})-4\cosh
2\Theta\,\left(E^{2}r_{1}r_{2}+m_{1}m_{2}\sinh^{2}\Theta\right)\right]^{2}\,,$

Similarly, for $s_{1}=s_{2}=1$, plugging the values of the traces
$t_{a}^{(m)}$, (30) simply reads

$\displaystyle
S^{(2)}_{AC,1}=\frac{64}{9}G^{2}_{N}(|\mathbb{A}^{++}_{1,1}|^{2}+|\mathbb{A}^{++}_{1,2}|^{2}+|\mathbb{A}^{++}_{2,1}|^{2}+|\mathbb{A}^{++}_{2,2}|^{2})\,.$
(32)

We give the explicit expressions for the coefficients $\mathbb{A}_{mn}^{IJ}$
as follows,

$\displaystyle\mathbb{A}^{++}_{1,1}$
$\displaystyle=-\mathbb{A}^{--}_{1,1}=i\frac{4\pi^{2}}{b^{2}}(\frac{m_{1}m_{2}c_{2\Theta}s_{\Theta}}{E^{2}r_{1}r_{2}}+\frac{c_{2\Theta}}{s_{\Theta}}-\frac{2m_{2}c_{\Theta}s_{\Theta}}{Er_{1}}-\frac{2m_{1}c_{\Theta}s_{\Theta}}{Er_{2}})\,,$
$\displaystyle\mathbb{A}^{++}_{2,1}$
$\displaystyle=\mathbb{A}^{--}_{2,1}=\frac{4\pi^{2}}{b^{3}}\left(-\frac{C_{1,2}c_{2\Theta}}{Er_{2}}+\frac{2C_{1,2}c_{\Theta}}{m_{1}}-\frac{m_{2}^{2}c_{2\Theta}s_{\Theta}^{2}}{E^{3}r_{1}^{2}r_{2}}+\frac{4m_{2}c_{\Theta}s_{\Theta}^{2}}{E^{2}r_{1}r_{2}}-\frac{2m_{2}c_{2\Theta}}{Em_{1}r_{1}}+\frac{m_{2}^{2}s_{\Theta}s_{2\Theta}}{E^{2}m_{1}r_{1}^{2}}\right)\,,$
$\displaystyle\mathbb{A}^{++}_{1,2}$
$\displaystyle=\mathbb{A}^{++}_{1,2}=\frac{4\pi^{2}}{b^{3}}\left(-\frac{C_{2,2}c_{2\Theta}}{Er_{1}}+\frac{2C_{2,2}c_{\Theta}}{m_{2}}-\frac{m_{1}^{2}c_{2\Theta}s_{\Theta}^{2}}{E^{3}r_{1}r_{2}^{2}}+\frac{4m_{1}c_{\Theta}s_{\Theta}^{2}}{E^{2}r_{1}r_{2}}-\frac{2m_{1}c_{2\Theta}}{Em_{2}r_{2}}+\frac{m_{1}^{2}s_{\Theta}s_{2\Theta}}{E^{2}m_{2}r_{2}^{2}}\right)\,,$
$\displaystyle\mathbb{A}^{++}_{2,2}$
$\displaystyle=-\mathbb{A}^{--}_{2,2}=i\frac{6\pi^{2}}{b^{4}}\left(-\frac{m_{2}C_{2,2}c_{2\Theta}s_{\Theta}}{E^{2}m_{1}r_{1}^{2}}-\frac{m_{1}C_{1,2}c_{2\Theta}s_{\Theta}}{E^{2}m_{2}r_{2}^{2}}-\frac{C_{1,2}C_{2,2}c_{2\Theta}}{m_{1}m_{2}s_{\Theta}}+\frac{2C_{1,2}s_{2\Theta}}{Em_{2}r_{2}}\right.$
$\displaystyle\left.+\frac{2C_{2,2}s_{2\Theta}}{Em_{1}r_{1}}-\frac{m_{1}m_{2}c_{2\Theta}s_{\Theta}^{3}}{E^{4}r_{1}^{2}r_{2}^{2}}+\frac{4m_{2}c_{\Theta}s_{\Theta}^{3}}{E^{3}r_{1}^{2}r_{2}}+\frac{4m_{1}c_{\Theta}s_{\Theta}^{3}}{E^{3}r_{1}r_{2}^{2}}-\frac{4c_{2\Theta}s_{\Theta}}{E^{2}r_{1}r_{2}}\right)\,,$
(33)

where we have adopted the shorthand notations $c_{\Theta}=\cosh\Theta$ and
$s_{\Theta}=\sinh\Theta$.

There are several limits one can take. The most immediate ones are the non-
relativistic limit $\cosh\Theta\rightarrow 1$ and the opposite, namely the
high-energy limit $\cosh\Theta\gg 1$.

Recall that in the non-relativistic regime $\cosh\Theta\rightarrow 1$, we need
to have $m_{1,2}/p\gg 1/(|b|m_{1,2})$ as well. Hence, the Renyi entropies for
spin-$1/2$ and spin-$1$ read

$\displaystyle\left.S_{AC,1/2}^{(2)}\right|_{\text{non-
rel}}={16G_{N}^{2}\pi^{4}m_{1}^{2}m_{2}^{2}\over|b|^{4}(m_{1}+m_{2})^{2}p^{2}}\,,$
(34) $\displaystyle\left.S_{AC,1}^{(2)}\right|_{\text{non-
rel}}={256{G_{N}^{2}\pi^{4}m_{1}^{2}m_{2}^{2}}\over{9|b|^{4}(m_{1}+m_{2})^{2}p^{2}}}+{\color[rgb]{0.796875,0.25390625,0.328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.796875,0.25390625,0.328125}\frac{256G_{N}^{2}\pi^{4}C_{1,2}^{2}C_{2,2}^{2}}{|b|^{8}(m_{1}+m_{2})^{2}p^{2}}}\,.$
(35)

Here we note that in addition to the two inequalities above, this expansion
only makes sense when ${|b|m_{1,2}}\ll{m_{1,2}/p}$ is also satisfied. This
extra inequality guarantees that although the $G_{N}^{2}$ contribution comes
at $m_{1,2}^{2}/p^{2}$, it is still suppressed by the large impact parameter
and the weak coupling constant such that it would not overwhelm the
$\mathcal{O}(G_{N}^{0})$ term before taking the logarithm in (20). For
spin-$1/2$, we do not have unknown Wilson coefficients; while for spin-$1$, we
have two Wilson coefficients $C_{a,2},a=1,2$ that are theory-dependent. We
note that these two coefficients play no role in the non-relativistic limit as
they only feature in the first sub-leading term, colored red above, which is
suppressed by the large impact parameter. Therefore the non-relativistic
behaviour is theory-agnostic. Note that the Renyi entropy increases with
decreasing $p/m_{1,2}$, suggesting that the scrambling power increases with
larger masses in the non-relativistic limit. Figure 1 is a plot of the
entanglement entropy $S_{\textrm{AC}}$ in this regime, which shows the same
qualitative behaviour.

Figure 1: $S_{\text{AC},1}$(red points) and fitted surface vs.
$C_{1,2}$,$C_{2,2}$ with $m_{1}=m_{2}=m$, $G_{N}m^{2}=10^{-5}$, $|b|m=10$,
$p/m=0.01$. In this non-relativistic regime, the tripartite information is
insensitive to the theory. When the $C_{i,2}$ is comparable to $|b|m$ at the
corner, the entropy grows about $20\%$. The surface is fitted by $6.3\times
10^{-10}C_{1,2}^{2}C_{2,2}^{2}+2.4\times 10^{-6}$, which suggests (35) gives
the right form of the $|b|m$ suppressed theory-dependent sub-leading term. The
minimal occurs at $(0,0)$.

The high-energy regime $\cosh\Theta\gg 1$ (namely $p\gg m_{1,2}$), in which we
also need to demand $(p/m_{1,2})^{2}\gg 1/(|b|m_{1,2})$, is trickier and we
separate our discussions into different scenarios.

#### Scenario I

The relevant scales satisfy the following inequalities: $(p/m_{1,2})^{2}\gg
1/(|b|m_{1,2})$, $\cosh\Theta\gg 1$, and $p/m_{1,2}\ll|b|m_{1,2}$. Within the
validity of these inequalities, we find that the Renyi-2 entropies behave as
follows for spin-$1/2$ and spin-$1$,

$\displaystyle\\!\\!\\!\\!\left.S_{AC,1/2}^{(2)}\right|_{\text{high-energy}}=$
$\displaystyle{256G_{N}^{2}\pi^{4}\over|b|^{4}}-{256G_{N}^{2}\pi^{4}(m_{1}+m_{2})\over|b|^{4}p}\,,$
(36) $\displaystyle\\!\\!\\!\\!\left.S_{AC,1}^{(2)}\right|_{\text{high-
energy}}=$ $\displaystyle{16384G_{N}^{2}\pi^{4}\over
9|b|^{4}}-{16384G_{N}^{2}\pi^{4}(m_{1}+m_{2})\over 9|b|^{4}p}\,.$ (37)

For both spin-$1/2$ and spin-$1$, the first sub-leading term is suppressed by
$m_{1,2}/p\ll 1$. For spin-$1$, the Wilson coefficients enter through the
second sub-leading term which we have not shown. We note that the leading
behaviour of the Renyi-2 entropy is theory independent and the scrambling
power increases with increasing energy. Moreover, the first sub-leading
behaviour is still theory-agnostic. Theory dependence only comes at a sub-sub-
leading term.

#### Scenario II

The relevant scales satisfy the following inequalities: $(p/m_{1,2})^{2}\gg
1/(|b|m_{1,2})$, $\cosh\Theta\gg 1$, and $p/m_{1,2}\gg|b|m_{1,2}$. The
expansion of the Renyi-2 entropies then reads,

$\displaystyle\\!\\!\\!\\!\left.S_{AC,1/2}^{(2)}\right|_{\text{high-energy}}=$
$\displaystyle{256G_{N}^{2}\pi^{4}\over|b|^{4}}-{256G_{N}^{2}\pi^{4}(m_{1}+m_{2})\over|b|^{4}p}\,,$
(38) $\displaystyle\\!\\!\\!\\!\left.S_{AC,1}^{(2)}\right|_{\text{high-
energy}}=$
$\displaystyle{\frac{4096G_{N}^{2}\pi^{4}p^{4}}{b^{8}m_{1}^{4}m_{2}^{4}}(C_{1,2}-1)^{2}(C_{2,2}-1)^{2}}$
$\displaystyle+{16384G_{N}^{2}\pi^{4}p^{2}\left(m_{2}^{4}(C_{1,2}-1)^{2}+m_{1}^{4}(C_{2,2}-1)^{2}\right)\over
9|b|^{6}m_{1}^{4}m_{2}^{4}}\,.$ (39)

The expansion takes the same form as in the previous scenario for spin-$1/2$
due to the absence of theory-dependent Wilson coefficients. For spin-$1$, the
leading and sub-leading contributions are both theory-dependent and the
scrambling has a minimum at the minimal coupling. Figure 2 is a plot of the
entanglement entropy in this regime.

Figure 2: $S_{\text{AC},1}$(red points) and fitted surface vs.
$C_{1,2}$,$C_{2,2}$ with $m_{1}=m_{2}=m$, $G_{N}m^{2}=10^{-5}$, $|b|m=10$,
$p/m=100$. In this Scenario II of the high-energy regime, the tripartite
information is sensitive to the theory. The surface is fitted by
$0.000115(C_{1,2}-1)^{2}(C_{2,2}-1)^{2}$, which suggests that (3.1) gives the
right form of the leading term. The minimal occurs at $(1,1)$.

In all cases we note that the scrambling ability always decrease with
increasing impact parameter, as expected. When there is hardly any interaction
between particles, they cannot be possibly scrambled.

###  3.2 Large-spin expansion

Here we discuss the Renyi entropies and the resulting tripartite information
for the minimal coupling in the large-spin limit.

The kinematics-dependent coefficients $\mathbb{A}_{mn}^{IJ}$ can be read off
from the definition given in (3)-(9). The remaining ingredient is the trace
$t^{(m)}$. Recall that for a given spin-$s$ and generic $m$, we have
$\Sigma_{\pm}|s,m\rangle=\sqrt{s(s+1)-m(m\pm 1)}|s,m\pm 1\rangle$ and hence

$\displaystyle t^{(m)}\equiv\sum_{n=-s}^{s}\langle
s,n|\Sigma_{+}^{m}\Sigma_{-}^{m}|s,n\rangle=\sum_{n=-(s-m)}^{s}{\Gamma(s+n+1)\Gamma(s+m-n+1)\over\Gamma(s-m+n+1)\Gamma(s-n+1)}\,,$
(40)

where in the second equal sign we have rewritten products of consecutive
integers using the gamma functions. Carrying out the sum, we obtain333Summing
up the summand (40) in terms of Gamma functions in Mathematica leads to a
missing overall ${1\over 2}$ which we have fixed numerically.

$\displaystyle t^{(m)}={-\sqrt{\pi}\,\csc(2s\pi)\Gamma(-m-{1\over
2})\Gamma(m+1)\over
2^{2m+1}\,\Gamma(-2s-m-1)\Gamma(2s-m+1)}={\sqrt{\pi}\,\Gamma(m+1)\Gamma(2s+m+2)\over
2^{2m+1}\,\Gamma(2s-m+1)\Gamma(m+{3\over 2})}\,.$ (41)

where in the second equal sign we have used
$\Gamma(x)\Gamma(1-x)=\pi/\sin(\pi(x+1))$ to remove the singularities of the
gamma functions.

In the large-spin limit $s\rightarrow\infty$, we note that
$\Gamma(m+1)/\Gamma(m+3/2)]\sim m^{-1/2}$ in the large $m$ limit. At the same
time $\Gamma(2s+m+2)/\Gamma(2s-m+1)$ is increasing with $m$ exponentially
quickly. Therefore we conclude that in the large-$s$ limit, the $m=2s$ term
should dominate exponentially. That is

$\displaystyle t^{(2s)}={\pi}(2s/e)^{4s}[1+\mathcal{O}(1/s)].$ (42)

Hence, we only need to consider the dominant contribution to the Renyi-$r$
entropies $S^{(r)}_{AC}$ and $S^{(r)}_{AD}$, namely the $m=n=2s$ term in
summation in the large-spin limit

$\displaystyle\left.S^{(r)}_{AC}\right|_{s\rightarrow\infty}={r\over
r-1}{G_{N}^{2}\over(2s+1)^{2}}\left(t^{(2s)}\right)^{2}\sum_{I,J=0}^{2}\mathbb{A}_{2s,2s}^{IJ}\left(\mathbb{A}_{2s,2s}^{IJ}\right)^{*}$
(43)

In the case of minimal coupling, the coefficients $\mathbb{A}_{2s,2s}^{IJ}$
can be readily extracted. It is given by

$\displaystyle\mathbb{A}_{2s,2s}^{II}$
$\displaystyle=-\frac{32\pi^{2}(4s-1)!}{m_{a}^{2s-2}m_{b}^{2s-2}((2s)!)^{2}b_{I}^{4s}}\left[e^{\Theta}(1-X_{a})^{2s}(1-X_{b})^{2s}+e^{-\Theta}(1+X_{a})^{2s}(1+X_{b})^{2s}\right]$
$\displaystyle\sim-\frac{8\pi^{2}2^{4s}}{\sqrt{2\pi
s^{3}}m_{a}^{2s-2}m_{b}^{2s-2}b_{I}^{4s}}\left[e^{\Theta}(1-X_{a})^{2s}(1-X_{b})^{2s}+e^{-\Theta}(1+X_{a})^{2s}(1+X_{b})^{2s}\right]\,,$

where $X_{a}=-m_{a}m_{b}\sinh\Theta/(E(m_{a}+E_{a}))$.

Altogether, this gives a large-$s$ behaviour of

$\displaystyle
S^{(2)}_{AC}\sim\frac{G_{N}^{2}\pi^{5}(4s/e)^{8s}\left[e^{\Theta}(1-X_{a})^{2s}(1-X_{b})^{2s}+e^{-\Theta}(1+X_{a})^{2s}(1+X_{b})^{2s}\right]^{2}}{2s^{5}m_{a}^{4s-4}m_{b}^{4s-4}}\left(\frac{1}{b_{+}^{8s}}+\frac{1}{b^{8s}_{-}}\right).$
(45)

We note that the result almost completely depends on the magnitude of
$s/(b\sqrt{m_{a}m_{b}})$ which is raised to a power of $8s$. If it is larger
than 1, $S^{(2)}_{AC}$ would grow rapidly with spins, and if it is less than
1, it would be suppressed exponentially by the spin. This suggests an
interesting competition between the number of degrees of freedom available for
scrambling and the strength of the gravitational interaction weakening with
increasing separation. When gravity is sufficiently suppressed, larger spins
make the system more classical and scrambling is further suppressed by spin.
This confirms with folklore that systems with large quantum numbers should
resemble classical ones. On the other hand, when there are sufficiently large
number of degrees of freedom supplied by the spin degrees of freedom, it can
actually exhibit more quantum behaviour.

###  3.3 An exact $G_{N}^{2}$ result at spin-$1/2$ and spin-$1$

When the spins of the particles are not too large, it is not difficult to
recover the leading $G_{N}$ contribution to the exact entanglement entropy
$S_{AC}$, which leads to exact results for the tripartite information. For
illustrative purpose, we present the result where the spins of the scatterinng
particles are both spin 1/2 and both spin 1.

To compute the entanglement entropy, we first solve for the eigenvalues of the
reduced density matrix (3.1) as a $G_{N}$ expansion. When both particles are
spin $1/2$, the eigenvalues are listed as follows:

$\displaystyle\lambda_{1}=1-\frac{1}{2}G_{N}^{2}|\mathbb{A}^{++}_{1,1}|^{2},\,\,\lambda_{2}=\frac{1}{4}G_{N}^{2}|\mathbb{A}^{++}_{1,1}|^{2},\,\,\lambda_{3}=\frac{1}{4}G_{N}^{2}|\mathbb{A}^{++}_{1,1}|^{2},\,\,\lambda_{4}=\mathcal{O}(G_{N}^{3}).$
(46)

The Renyi - information is then given by

$S_{AC,1/2}=\frac{1}{2}G_{N}^{2}|\mathbb{A}^{++}_{1,1}|^{2}[1+\ln{4}-2\ln{(G_{N}|\mathbb{A}^{++}_{1,1}|)}]$
(47)

We note that the $G_{N}^{2}$ correction to the largest eigenvalue
$\lambda_{1}$ is equal to half of the Renyi-$2$ entropy. This is expected
since other smaller eigenvalues of order $G_{N}^{2}$ cannot contribute at
$G_{N}^{2}$ to the Renyi entropy. In addition to the $G_{N}^{2}$ contribution
analogous to the Renyi entropy computation, we have a logarithmic correction.
The logarithmic correction follows from the fact that $\rho_{AC}$ is not a
full rank matrix to zeroth order in $G_{N}$ (in fact a rank 1 matrix for any
spins), which has already been pointed out earlier.

Using (31), the logarithmic term takes the form of
$+1/8G_{N}^{2}/(|b|^{2}\mathcal{F})^{2}\ln[|b|^{2}/G_{N}\times\mathcal{F}]$,
where $\mathcal{F}$ denotes dimensionless factors depending on the center of
mass energy and the masses of the particles. This can be contrasted with the
one-loop correction to the black hole entropy computed for example in
Sen:2012dw , which contains logarithmic correction of the form $\ln a$, where
$a$ is the horizon area measured in Planck units.

This computation can be done for the case of spin-1 particles. One would
expect that the logarithmic correction should persist. For spin-$1$, the
eigenvalues of $\rho_{AC}$ are given by

$\displaystyle\lambda_{1}=1-\Delta_{1},\,\,\lambda_{2}=\Delta_{1}(1+\Delta_{2})/4,\,\,\lambda_{3}=\Delta_{1}(1+\Delta_{2})/4,$
$\displaystyle\lambda_{4}=\Delta_{1}(1-\Delta_{2})/4,\,\,\lambda_{5}=\Delta_{1}(1-\Delta_{2})/4,\,\,\lambda_{i>5}=\mathcal{O}(G_{N}^{3})$
(48)

where

$\displaystyle\Delta_{1}$
$\displaystyle=\frac{32}{9}(|\mathbb{A}^{++}_{1,1}|^{2}+|\mathbb{A}^{++}_{1,2}|^{2}+|\mathbb{A}^{++}_{2,1}|^{2}+|\mathbb{A}^{++}_{2,2}|^{2})G_{N}^{2}$
(49) $\displaystyle\Delta_{2}$
$\displaystyle=\sqrt{1-4(\frac{|A^{++}_{1,1}A^{++}_{2,2}|+|A^{++}_{2,1}A^{++}_{1,2}|}{|\mathbb{A}^{++}_{1,1}|^{2}+|\mathbb{A}^{++}_{1,2}|^{2}+|\mathbb{A}^{++}_{2,1}|^{2}+|\mathbb{A}^{++}_{2,2}|})^{2}}$
(50)

Note that $\Delta_{1}$ is of order $G_{N}^{2}$, and $\Delta_{2}$ is of order
$G^{0}_{N}$.

This gives $S_{AC}$ for spin-$1$

$S_{AC,1}=\Delta_{1}\left[1+\ln(\frac{4}{\sqrt{1-\Delta_{2}^{2}}\Delta_{1}})-\frac{\Delta_{2}}{2}\ln(\frac{1+\Delta_{2}}{1-\Delta_{2}})\right].$
(51)

##  4 Semi-classicality and coherent states

In the previous sections, we explored how quantum chaotic a gravitational
scattering process approximated by the eikonal in the semi-classical limit
could potentially be. The result depends on the properties of the scattering
matrix, and made no mention of the initial scattering states. In the current
section, we would like to ask a related but orthogonal question regarding the
classicality of the initial states. In the case of microscopic system, it is
possible to prepare the pure initial states. However for macroscopic systems
it is more natural to consider mixed states. Therefore, we will analyse these
two cases separately. In the first part, we will consider a family of squeezed
coherent state, and ask if vanisihing squeezing should minimize the final
entanglement after scattering. We will also consider a family of mixed states
and identify the most classical one.

###  4.1 Squeezed spin coherent states

It is well known that some states behave more classically than others, and
they are used as an important ingredient in recovering classical physics. One
very important class of examples are coherent states studied extensively in
the context of harmonic oscillators. The most important characterization of a
coherent state for the harmonic oscillator is that it is designed to minimize
the uncertainty in the non-commuting variables $\Delta X$ and $\Delta P$ such
that the Heisenberg uncertainty inequality is saturated. i.e.

$\displaystyle\Delta X\Delta P=\frac{\hbar}{2}.$ (52)

In addition to that, the coherent state distributes uncertainty equally
between $X$ and $P$ so that

$\displaystyle\Delta X=\Delta P=\sqrt{\frac{\hbar}{2}}.$ (53)

However, while saturating the Heisenberg uncertainty inequality is certainly a
sign of classicality, one could in principle distribute the remaining
uncertainty differently between $X$ and $P$. One can construct a family of
states satisfying

$\displaystyle\Delta X=\exp(-\eta)\sqrt{\frac{\hbar}{2}},\qquad\Delta
P=\exp(\eta)\sqrt{\frac{\hbar}{2}},$ (54)

and they are called the squeezed states. It is a curiosity if any one of these
states would be singled out and behave more “classically”. Indeed, the
original coherent state corresponding to $\eta=0$ distinguishes itself when
one considers an interacting system. It is shown in Zurek:1992mv  that
considering a generic Hamiltonian that couples the harmonic oscillator to a
heat bath of harmonic oscillators, the coherent state without squeezing is the
most resistant to getting entangled with the bath.

We would like to ask a similar question here. Is there a spin state that best
models macroscopic classical objects carrying angular momentum and interacting
via gravity? In the case of spin states, there is a corresponding notion of
spin coherent states. For spin $J$, spin coherent states are simply defined as
eigenstates of the angular momentum operator $\vec{J}$ which is aligned in the
direction the classical particle is spinning i.e. for a unit vector $\vec{n}$,

$\displaystyle|\vec{n}\rangle,\qquad\vec{n}\cdot\vec{J}|\vec{n}\rangle=J|\vec{n}\rangle\,.$
(55)

In the case of spins, it satisfies the following uncertainty inequality

$\displaystyle\Delta J_{x}\Delta J_{y}\leq\frac{\hbar\langle
J_{z}\rangle}{2}\,.$ (56)

One can readily check that the above choice of states with $\vec{n}=(0,0,1)$
saturates this inequality, with $\Delta J_{x}=\Delta
J_{y}=\sqrt{\frac{\hbar\langle J_{z}\rangle}{2}}$.

One can also construct squeezed states which distribute uncertainty unevenly
between $J_{x}$ and $J_{y}$. There are many ways to parametrize squeezed
states. One convenient family is obtained by

$\displaystyle|\zeta\rangle=\exp(-\zeta J_{x}^{2})|J,J\rangle\,.$ (57)

This gives $\langle J_{x}\rangle=\langle J_{y}\rangle$ =0, preserving the
axial symmetry along the $z$\- axis. We are working in the center of mass
frame of the scattering particles, where the in-coming particles have momenta
along the $z$-axis. The scattering matrix has axial symmetry along the
$z$-direction. We will thus pick an initial state of the form (57) for each of
the in-coming particles. We would like to explore the optimal amount of
squeezing that minimizes the entanglement after the scattering process. We
will present both analytic and numerical results.

To illustrate this analysis, we take spin-$1$ particles as an example. We
would like to consider an initial state that is a direct product state of the
two in-coming scattering particles. Moreover, the problem has axial symmetry
about the $z$-axis. Therefore we consider states where $J_{x}$ and $J_{y}$
have vanishing expectation value. A natural family of states would be given by
$|\zeta_{1}\rangle\otimes|\zeta_{2}\rangle$, i.e. in principle the two
particles could have different squeezing and they could lead to differing
amounts of final entanglement. We would like to look for universal statements
that have a cleaner physical interpretation. To that end, we consider small
deviations from the spin coherent state where $\zeta$ is vanishing. In the
small squeezing limit, we might as well introduce squeezing only to one of the
particles. Therefore, we would like to consider the two simple initial states
below,

$\displaystyle|\Psi_{1}(\zeta)\rangle$ $\displaystyle={e^{-2\zeta}+1\over
2}|+1\rangle\otimes|+1\rangle+{e^{-2\zeta}-1\over
2}|-1\rangle\otimes|+1\rangle\,,$ (58) $\displaystyle|\Psi_{2}(\zeta)\rangle$
$\displaystyle={e^{-2\zeta}+1\over
2}|+1\rangle\otimes|-1\rangle+{e^{-2\zeta}-1\over
2}|-1\rangle\otimes|-1\rangle\,.$ (59)

At $\zeta=0$, $|\Psi_{i}\rangle,\,\,i=1,2$ reduce to
$|+1\rangle\otimes|+1\rangle$ and $|+1\rangle\otimes|-1\rangle$ respectively.
For $i=1$, it describes an asymmetric initial condition where one spin is
aligned with their momentum and the other anti-aligned. In the second case, it
corresponds to a symmetric initial state in which both spins are aligned with
the momenta of the respective in-coming particle. At any other $\zeta$, the
initial states are squeezed. As they are given here, these squeezed coherent
states are not normalized. The proper normalization will be recovered in the
computations of entropies shortly.

For starter, we can inspect analytically the Renyi-2 entropy after the
scattering process. That is, we take the eikonal $\exp(i\chi)$ as the unitary
evolution the initial states undergo and first construct the partial density
matrix of the resulting final states by tracing out the spin degrees of
freedom of particle 2. The Renyi-2 entropy is then computed straightforwardly
from this partial density matrix. Similar to the scrambling case, we also see
that the terms linear in $G_{N}$ cancel out here, regardless of initial
states. The first non-trivial correction again comes in at
$\mathcal{O}(G_{N}^{2})$ and has the same power of $\hbar$ as the
superclassical contributions in $\exp(i\chi)$. The Renyi-2 entropies after
scattering with the two initial states can be calculated analytically.

$\displaystyle
S_{|\Psi_{1}\rangle}^{(2)}=4G^{2}_{N}(\text{sech}(2\zeta)+1)\left[|\mathbb{A}^{++}_{1,1}|^{2}+2|\mathbb{A}^{++}_{2,1}|^{2}+(\text{sech}(2\zeta)+1)(|\mathbb{A}^{++}_{1,2}|^{2}+2|\mathbb{A}^{++}_{2,2}|^{2})\right],\,\,$
(60) $\displaystyle
S_{|\Psi_{2}\rangle}^{(2)}=4G^{2}_{N}(-\text{sech}(2\zeta)+1)\left[|\mathbb{A}^{++}_{1,1}|^{2}+2|\mathbb{A}^{++}_{2,1}|^{2}+(-\text{sech}(2\zeta)+1)(|\mathbb{A}^{++}_{1,2}|^{2}+2|\mathbb{A}^{++}_{2,2}|^{2})\right].$
(61)

In the non-relativistic regime dictated by the conditions $m_{1,2}/p\gg
1/(|b|m_{1,2})$ and $\cosh\Theta\rightarrow 1$, we obtain the following
leading-order behavior of the Renyi-2 entropies

$\displaystyle\left.S_{|\Psi_{1}\rangle}^{(2)}\right|_{\text{non-rel}}=$
$\displaystyle{64\pi^{4}G_{N}^{2}m_{1}^{2}m_{2}^{2}\left(1+\text{sech}(2\zeta)\right)\over|b|^{4}p^{2}(m_{1}+m_{2})^{2}}\,,$
(62) $\displaystyle\left.S_{|\Psi_{2}\rangle}^{(2)}\right|_{\text{non-rel}}=$
$\displaystyle{64\pi^{4}G_{N}^{2}m_{1}^{2}m_{2}^{2}\left(1-\text{sech}(2\zeta)\right)\over|b|^{4}p^{2}(m_{1}+m_{2})^{2}}\,,$
(63)

where the subscript indicates which initial state the entropy is computed
with. Here the leading behaviours are theory-agnostic for both initial states.
The sub-leading terms however depend on the Wilson coefficients. We observe
that the minimal coupling does not appear to be correlated with extremums in
this case.

In the high energy regime defined by conditions $(p/m_{1,2})^{2}\gg
1/(|b|m_{1,2})$ and $\cosh\Theta\gg 1$, we again divide our analysis into two
scenarios according to the relative ratio of the competing scales $p/m_{1,2}$
and $|b|m_{1,2}$, as discussed in the previous section.

#### Scenario I:

in addition to the two inequalities above, we also demand $|b|m_{1,2}\gg
p/m_{1,2}$. To leading order, the Renyi-2 entropies read

$\displaystyle\left.S_{|\Psi_{1}\rangle}^{(2)}\right|_{\text{high-energy}}=$
$\displaystyle{1024\pi^{4}G_{N}^{2}\left(1+\text{sech}(2\zeta)\right)\over|b|^{4}}\,,$
(64) $\displaystyle\left.S_{|\Psi_{2}\rangle}^{(2)}\right|_{\text{high-
energy}}=$
$\displaystyle{1024\pi^{4}G_{N}^{2}\left(1-\text{sech}(2\zeta)\right)\over|b|^{4}}\,.$
(65)

We note that in this case the leading behaviour is theory independent and the
first sub-leading terms, which we have not displayed, vanish at the minimal
coupling.

#### Scenario II:

in the case $|b|m_{1,2}\ll p/m_{1,2}$ we have to leading order the following,

$\displaystyle\\!\\!\\!\\!\\!\left.S_{|\Psi_{1}\rangle}^{(2)}\right|_{\text{high-
energy}}=$
$\displaystyle{\frac{4608G_{N}^{2}\pi^{4}p^{4}\left(1+\text{sech}(2\zeta)\right)^{2}}{b^{8}m_{1}^{4}m_{2}^{4}}(C_{1,2}-1)^{2}(C_{2,2}-1)^{2}}\,,$
(66)
$\displaystyle\\!\\!\\!\\!\\!\left.S_{|\Psi_{2}\rangle}^{(2)}\right|_{\text{high-
energy}}=$
$\displaystyle{\frac{4608G_{N}^{2}\pi^{4}p^{4}\left(1-\text{sech}(2\zeta)\right)^{2}}{b^{8}m_{1}^{4}m_{2}^{4}}(C_{1,2}-1)^{2}(C_{2,2}-1)^{2}}\,.$
(67)

We observe that only in this case the leading behaviours have theory-
dependences. The sub-leading also depends on the Wilson coefficients. Both the
leading and the first sub-leading terms vanish at the minimal coupling.

One can see that the symmetric vs asymmetric configurations give opposite
results. In the symmetric initial configuration, squeezing reduces final
entanglement, making vanishing squeezing the minimum and most classical point,
whereas in the anti-symmetric configuration, vanishing squeezing leads to
maximum final entanglement after scattering, making the spin-coherent state
the least classical configuration! A global plot of the final entanglement
entropy plotting against a pair of parameters that could smoothly interpolate
between different squeezing and also relative orientations of the spins is
shown in figure 3. The colored lines correspond to symmetric and asymmetric
relative orientations considered above.

While these are lower bounds of the entanglement entropy, exact numerical
results can be obtained for example for spin-1, confirming the above
statement. This is an interesting observation. While it is not entirely
unexpected that final entanglement has dependence on details of the initial
state, it is quite amusing that the spin coherent state can in fact be the
most classical or the least classical initial state, depending on relative
orientation between the two scattering particles. This opens the door to
adjusting and fine-tuning initial states in future experiments to amplify the
quantum effects of scattering event.

Figure 3: The entanglement entropy of the outcome states vs $\theta_{1}$ and
$\theta_{2}$ with the initial states
$(\text{cos}\theta_{1}|+1\rangle+\text{sin}\theta_{1}|-1\rangle)\otimes(\text{cos}\theta_{2}|+1\rangle+\text{sin}\theta_{2}|-1\rangle)$,
$m_{1}=m_{2}=m$, $G_{N}m^{2}=10^{-5}$, $|b|m=10$, $p/m=100$,
$C_{1,2}=C_{2,2}=1.5$. This graph contains the configurations of squeezed
states. The blue line shows when we squeeze both particles starting from
parallel configurations ($|+1\rangle\otimes|+1\rangle$ or
$|-1\rangle\otimes|-1\rangle$) to the middle point, the entanglement entropy
decreases. The red line shows when we squeeze both particles starting from
anti-parallel configurations ($|+1\rangle\otimes|-1\rangle$ or
$|-1\rangle\otimes|+1\rangle$) to the middle point, the entanglement entropy
increases.

###  4.2 Mixed initial states

By the same token, we are curious if any initial mixed state with the same
expectation value of $J_{x},J_{y}$ and $J_{z}$ can be entangled by different
amount under gravitational scattering.

As it is well known, entanglement entropy is a good measure of quantum
entanglement if we work with pure quantum states. To deal with quantum
entanglement of mixed states, it is more appropriate to work with the notion
of negativity $\mathcal{N}$, which is defined as

$\displaystyle\mathcal{N}_{A}=||\rho^{\Gamma_{A}}||_{1},\qquad||\mathcal{O}||_{1}\equiv\textrm{tr}\sqrt{\mathcal{O}^{\dagger}\mathcal{O}}$
(68)

where $\rho$ is the density matrix of the full system, and $\Gamma_{A}$
denotes taking transpose of the density matrix only over degrees of freedom of
the subsystem $A$. One then computes the norm of the resultant matrix.

We pick an initial mixed state that again preserves axial symmetry along the
$z$ direction. For spin-$1$, consider a family of states that is parameterized
as

$\displaystyle|\psi\rangle_{\phi}=\sqrt{\frac{1}{2}}(|+1\rangle+e^{i\phi}|-1\rangle)\otimes|0\rangle,$
(69)

which has vanishing expectation value for spin angular momentum in all
directions. We choose
$\rho(\phi,\epsilon)=(1-\epsilon)|\psi\rangle_{0}\langle\psi|_{0}+\epsilon|\psi\rangle_{\phi}\langle\psi|_{\phi}$
to be the set of initial states. When $\phi=\pi$ and $\epsilon=1/2$, $\rho$
coincides with the equal mixture of the two coherent states $|+1\rangle$ and
$|-1\rangle$.

To quantify quantum entanglement in mixed states, we plot the negativity of
the final states in Fig. 4. The quantum entanglement of the final states is
minimal at $\phi=\pi$ and $\epsilon=1/2$.

Figure 4: The negativity of the outcome states vs $\phi$ and $\epsilon$ with
the initial states $\rho(\phi,\epsilon)$, $m_{1}=m_{2}=m$,
$G_{N}m^{2}=10^{-5}$, $|b|m=10$, $p/m=100$, $C_{1,2}=C_{2,2}=1$.

Mixed spin states that are closest to classical states are considered as those
that can be expressed in terms of a positive combination of spin-coherent
states, known as a P-rep Giraud_2008 . i.e.

$\displaystyle\rho_{P}=\int d\alpha
P(\alpha)|\alpha\rangle\langle\alpha|,\qquad P(\alpha)\geq
0\,\,\forall\alpha,\,\,\int d\alpha P(\alpha)=1.$ (70)

where $\alpha$ parametrizes the spin coherent states, and $d\alpha$ denotes
the corresponding measure Giraud_2008 . We note that the initial state that
end up with minimal negativity is also the only P-rep states in the family of
mixed states considered above.

##  5 Conclusions and Outlook

We would like to revisit the question of the emergence of a sense of
classicality in the context of particles interacting through gravity. Our
study is two-folded. We first look into the scrambling power of a scattering
event between two particles for given initial kinematics. We measure the power
of scrambling power in spin space using the tripartite information computed
from the eikonal amplitude, which is expected to diminish in a classical
interaction – since the tripartite information is measuring how the
information gets delocalized through entanglement between widely separated
particles generated under the scattering process. Our approach based on the
tripartite information is independent of the initial spin polarization of the
scattering particles, and depend only on the coupling of the particles to
gravity, and the initial kinematics. Since we are interested in the classical
limit we are expanding the tripartite information to leading order in $G_{N}$
and the inverse impact parameter $1/|b|$. We carefully isolate regions of
kinematics so that the loop corrections remain sub-leading, ensuring the
validity of our approximation. We found that the tripartite information, or
more precisely an upper bound of the tripartite information based on the Renyi
entropy, shows universal behaviour independent of theory both in the non-
relativistic and a relativistic limit in which the impact parameter remains
the most dominant scale. The scrambling power in these limits do diminish with
increasing $|b|$. We note also that since they are theory agnostic, the
minimal coupling does not play a visible role. Theory dependence comes in a
somewhat different relativistic limit, in which the minimal coupling is
singled out to lead to minimal scrambling, as is observed in Aoude:2020mlg ;
Chen:2021huj , supporting at least in this limit that the minimal coupling is
a special point that behaves most classically. The exact tripartite
information can be computed more readily for small spins, and we obtained
explicit results for spin 1/2 and spin 1. Interestingly, they receive
corrections logarithmic in $G_{N}$. There is a logorithmic boost to the
scrambling power that would be otherwise missed in the Renyi entropy bound,
and it bears some resemblance to the logarithmic corrections to the black hole
entropy. We also computed the large spin limit of the bound of the tripartite
information, which can be obtained analytically at minimal coupling. Perhaps
somewhat as expected, it confirms the folklore that scrambling should diminish
and thus the system appearing more classical as spins approach infinity while
keeping $|s|/(bm)$ of order 1.

On a different note, we explore the initial spin state dependence of
entanglement generation. We would like to explore the folklore that the
coherent state is the most classically behaving state. In the case of harmonic
oscillators, it was found that among squeezed coherent states that all
saturate uncertainty inequalities, the amount of entanglement generation is
minimal when there is no squeezing under generic local Hamiltonian evolution
Zurek:1992mv . In this paper we inspect squeezed spin coherent states as
initial states and found that the state with vanishing squeezing happens to be
an extremal point in the entanglement generation under gravitational
scattering. However it could be either the minimal or the maximal depending on
the initial relative orientation of the spins of the particles. While the
result is not definitive selecting the usual spin coherent states, it suggests
that careful choice of initial kinematics and orientations could amplify
quantum signatures of a scattering event. We explored also a family of mixed
initial states and used negativity as a measure of entanglement generation in
the scattered final states. Interestingly, we found that in this case the
minimally entangled final state do single out the so-called P-rep states which
are considered as the best approximation of classical mixed spin states.

To conclude, the study of scrambling and quantum entanglement generation
provide some more intuition to the emergence of classicality in various
corners of kinematic space and theory space, confirming a number of folklores.
In appropriate limits, we found universal behaviour independent of theories,
which is a bonus to our study. It would be interesting to explore how other
forms of interactions would show different universal behaviour. If so, it
gives us an extra handle to detect and distinguish - perhaps even
experimentally in the future - new fundamental forces.

Our study scratches only the surface of the subject – exploring the signature
of quantum-ness/classicality of field theory in particular scattering
amplitudes, through the lens of quantum information. For example, it would be
interesting to compare our results with interactions via other spins, such as
spin-$1$ and higher spin gauge fields and seek universal scaling behaviour. It
would also be interesting to study the contribution to the amplitudes through
the quantum remainder, which shows up starting from one loop – to see if
genuine quantum effects would produce qualitatively different quantum effects.
Higher-point functions beyond four-point scattering would allow us to explore
intricate entanglement pattern that gravity or other forces are capable of
producing – for example for 3 qubits there are exactly 2 different classes of
entanglement not connected by LOCC – the W class and the GHZ class Dur:2000zz
. There is a wealth of tools from the quantum information literature to be
tapped into. We will leave the various questions we raised above to future
work.

## Acknowledgements

We thank Aninda Sinha, Si-Wei Zhong and Parthiv Halder for initial
collaboration. We also thank Yu-tin Huang for many helpful discussions. LYH
acknowledges the support of NSFC (Grant No. 11922502, 11875111) and the
Shanghai Municipal Science and Technology Major Project (Shanghai Grant
No.2019SHZDZX01) TW is supported by in part by the Special Research
Assistantship of the Chinese Academy of Sciences and Humboldt Universität zu
Berlin. TW is also supported by the Fellowship of China Postdoctoral Science
Foundation (No. 2022M713228), the Key Research Program of CAS, Grant No.
XDPB15 and National Natural Science Foundation of China, Grants No. 11935013,
No. 11947301, No. 12047502, and No. 12047503.

##  Appendix A Renyi Entropy

We compute the Renyi entropy $S_{AC}$ explicitly. We note that the leading
order of the reduced density matrix $\rho_{AC}$ is a projector and this leads
to a discontinuity in $\text{tr}\rho^{r}_{AC}$ as $r\rightarrow 1$. Therefore
for $r\geq 2$, we need to compute $\text{tr}\rho^{r}_{AC}$ and the
normalisation $(\text{tr}\rho_{AC})^{r}$ separately.

Let us consider the normalisation first. Up to $\mathcal{O}(G_{N}^{2})$, we
have

$\displaystyle(\text{tr}\rho_{AC})^{r}=$ $\displaystyle
d_{S_{2}}^{r}\big{(}\text{tr}\mathbb{P}\big{)}^{r}+G_{N}\,rd_{S_{2}}^{r-1}\left(\text{tr}\Omega_{1}\right)\big{(}\text{tr}\mathbb{P}\big{)}^{r-1}$
$\displaystyle+G_{N}^{2}\,rd_{S_{2}}^{r-1}\left(\text{tr}\Omega_{2}^{(1)}+\text{tr}\Omega_{2}^{(2)}\right)\big{(}\text{tr}\mathbb{P}\big{)}^{r-1}+G_{N}^{2}\,{r(r-1)\over
2}d_{S_{2}}^{r-2}\left(\text{tr}\Omega_{1}\right)^{2}\big{(}\text{tr}\mathbb{P}\big{)}^{r-2}$
$\displaystyle=$ $\displaystyle
d_{S_{1}}^{r}d_{S_{2}}^{r}+G_{N}^{2}\,rd_{S_{1}}^{r-1}d_{S_{2}}^{r-1}\left(\text{tr}\Omega_{2}^{(1)}+\text{tr}\Omega_{2}^{(2)}\right)\,,$
(71)

where we denote $d_{s_{a}}=2s_{a}+1$ and $\left(\text{tr}\Omega_{1}\right)=0$
since $\text{tr}\Sigma^{n}_{\pm}=0$ and $\mathbb{A}_{00}^{nm}=0$ for any
positive integers $n,m$. The two remaining trace terms
$\text{tr}\Omega_{2}^{(1)}$ and $\text{tr}\Omega_{2}^{(2)}$ can be computed
straightforwardly,

$\displaystyle\text{tr}\Omega_{2}^{(1)}$
$\displaystyle=\sum_{\\{n_{a}\\}}\sum_{\\{I_{a}\\}}{1\over
2}\left[\mathbb{A}^{I_{1}I_{2}}_{n_{1}n_{2}}\mathbb{A}^{J_{1}J_{2}}_{m_{1}m_{2}}+\text{c.c.}\right]\text{tr}\big{(}O_{I_{1}J_{1}}^{(n_{1}m_{1})}\big{)}\text{tr}\big{(}O_{I_{2}J_{2}}^{(n_{2}m_{2})}\big{)}\,,$
(72) $\displaystyle\text{tr}\Omega_{2}^{(1)}$
$\displaystyle=\sum_{\\{n_{a}\\}}\sum_{\\{I_{a}\\}}\mathbb{A}^{I_{1}I_{2}}_{n_{1}n_{2}}\left(\mathbb{A}^{J_{1}J_{2}}_{m_{1}m_{2}}\right)^{*}\text{tr}\big{(}O_{I_{1}J_{1}}^{(n_{1}m_{1})}\big{)}\text{tr}\big{(}O_{I_{2}J_{2}}^{(n_{2}m_{2})}\big{)}\,.$
(73)

Now we compute $\text{tr}\rho^{r}_{AC}$ up to $\mathcal{O}(G_{N}^{2})$. For
$r\geqslant 2$, we have

$\displaystyle\text{tr}\rho^{r}_{AC}=$
$\displaystyle\,\text{tr}\left(\mathbb{P}^{r}\right)+G_{N}\,r\,\text{tr}\left(\Omega_{1}\mathbb{P}^{r-1}\right)+G_{N}^{2}\,r\,\text{tr}\left[\big{(}\Omega_{2}^{(1)}+\Omega_{2}^{(2)}\big{)}\mathbb{P}^{r-1}\right]$
$\displaystyle+G_{N}^{2}{r\over 2}\sum_{1<p\leqslant
r}\text{tr}\left[\Omega_{1}\mathbb{P}^{p-2}\Omega_{1}\mathbb{P}^{r-p}\right]\,.$
(74)

Recall that $\mathbb{P}^{r}=d_{s_{1}}^{r-1}d_{s_{2}}^{r-1}\mathbb{P}$ and
therefore the first line above can be calculated immediately,

$\displaystyle\text{tr}\left(\mathbb{P}^{r}\right)=d_{S_{1}}^{r}d_{S_{2}}^{r}\,,~{}~{}~{}~{}\text{tr}\left(\Omega\mathbb{P}^{r-1}\right)=d_{S_{1}}^{r-2}d_{S_{2}}^{r-2}\text{tr}\left(\Omega\mathbb{P}\right)\,,$
(75)

where $\Omega=\\{\Omega_{1},\Omega_{2}^{(1)},\Omega_{2}^{(2)}\\}$ . These
traces read

$\displaystyle\text{tr}\big{(}\Omega_{1}\mathbb{P}\big{)}$
$\displaystyle=d_{s_{1}}d_{s_{2}}\sum_{\\{n_{a}\\}}\sum_{\\{I_{a}\\}}\left[\mathbb{A}_{n_{1}n_{2}}^{I_{1}I_{2}}+\text{c.c.}\right]\text{tr}\big{(}O_{I_{1}}^{(n_{1})}\big{)}\text{tr}\big{(}O_{I_{2}}^{(n_{2})}\big{)}\,,$
(76) $\displaystyle\text{tr}\big{(}\Omega_{2}^{(1)}\mathbb{P}\big{)}$
$\displaystyle={d_{s_{1}}d_{s_{2}}\over
2}\sum_{\\{n_{a}\\}}\sum_{\\{I_{a}\\}}\left[\mathbb{A}_{n_{1}n_{2}}^{I_{1}I_{2}}\mathbb{A}_{m_{1}m_{2}}^{J_{1}J_{2}}+\text{c.c.}\right]\text{tr}\big{(}O_{I_{1}J_{1}}^{(n_{1}m_{1})}\big{)}\text{tr}\big{(}O_{I_{2}J_{2}}^{(n_{2}m_{2})}\big{)}\,,$
(77) $\displaystyle\text{tr}\big{(}\Omega_{2}^{(1)}\mathbb{P}\big{)}$
$\displaystyle=d_{s_{2}}\sum_{\\{n_{a}\\}}\sum_{\\{I_{a}\\}}\mathbb{A}_{n_{1}n_{2}}^{I_{1}I_{2}}\left(\mathbb{A}_{m_{1}m_{2}}^{J_{1}J_{2}}\right)^{*}\text{tr}\big{(}O_{I_{1}}^{(n_{1})}\big{)}\text{tr}\big{(}O_{J_{1}}^{(m_{1})}\big{)}\text{tr}\big{(}O_{I_{2}J_{2}}^{(n_{2}m_{2})}\big{)}\,.$
(78)

Similar to the normalisation, we have
$\text{tr}\big{(}\Omega_{1}\mathbb{P}\big{)}=0$. The second line of (A)
involves several structures. For $r=2$, there is only one term in the sum,
namely $\sum_{1<p\leqslant
r}\text{tr}\left[\Omega_{1}\mathbb{P}^{p-2}\Omega_{1}\mathbb{P}^{r-p}\right]=\text{tr}\Omega_{1}^{2}$
and we have

$\displaystyle\text{tr}\Omega_{1}^{2}=2d_{s_{1}}\sum_{\\{n_{a},I_{a}\\}}\mathbb{A}_{n_{1}n_{2}}^{I_{1}I_{2}}\left(\mathbb{A}_{m_{1}m_{2}}^{J_{1}J_{2}}\right)^{*}\text{tr}\big{(}O_{I_{1}J_{1}}^{(n_{1}m_{1})}\big{)}\text{tr}\big{(}O_{I_{2}}^{(n_{2})}\big{)}\text{tr}\big{(}O_{J_{2}}^{(m_{2})}\big{)}\,.$
(79)

For $r=3$, there is only one structure $\sum_{1<p\leqslant
r}\text{tr}\left[\Omega_{1}\mathbb{P}^{p-2}\Omega_{1}\mathbb{P}^{r-p}\right]=2\text{tr}\big{(}\Omega_{1}^{2}\mathbb{P}\big{)}$
and

$\displaystyle\text{tr}\big{(}\Omega_{1}^{2}\mathbb{P}\big{)}=d_{s_{1}}^{2}d_{s_{2}}\sum_{\\{n_{a},I_{a}\\}}\mathbb{A}_{n_{1}n_{2}}^{I_{1}I_{2}}\left(\mathbb{A}_{m_{1}m_{2}}^{J_{1}J_{2}}\right)^{*}\text{tr}\big{(}O_{I_{1}J_{1}}^{(n_{1}m_{1})}\big{)}\text{tr}\big{(}O_{I_{2}}^{(n_{2})}\big{)}\text{tr}\big{(}O_{J_{2}}^{(m_{2})}\big{)}$
(80)

For $r\geqslant 4$, there are two structures as follows,

$\displaystyle\sum_{1<p\leqslant
r}\text{tr}\left[\Omega_{1}\mathbb{P}^{p-2}\Omega_{1}\mathbb{P}^{r-p}\right]=2d^{r-3}_{s_{1}}d_{s_{2}}^{r-3}\text{tr}\big{(}\Omega_{1}^{2}\mathbb{P}\big{)}+(r-3)d^{r-4}_{S_{1}}d_{S_{2}}^{r-4}\text{tr}\big{(}\Omega_{1}\mathbb{P}\Omega_{1}\mathbb{P}\big{)}\,.$
(81)

It is straightforward to see that
$\text{tr}\big{(}\Omega_{1}\mathbb{P}\Omega_{1}\mathbb{P}\big{)}$ vanishes in
the same way as the terms linear in $G_{N}$. Hence the second line of (A)
takes the following expression for $r\geqslant 2$,

$\displaystyle{r\over 2}\sum_{1<p\leqslant
r}\text{tr}\left[\Omega_{1}\mathbb{P}^{p-2}\Omega_{1}\mathbb{P}^{r-p}\right]$
$\displaystyle=\,$ $\displaystyle
rd_{s_{1}}^{r-1}d_{s_{2}}^{r-2}\sum_{\\{n_{a},I_{a}\\}}\mathbb{A}_{n_{1}n_{2}}^{I_{1}I_{2}}\left(\mathbb{A}_{m_{1}m_{2}}^{J_{1}J_{2}}\right)^{*}\text{tr}\big{(}O_{I_{1}J_{1}}^{(n_{1}m_{1})}\big{)}\text{tr}\big{(}O_{I_{2}}^{(n_{2})}\big{)}\text{tr}\big{(}O_{J_{2}}^{(m_{2})}\big{)}$
(82)

The Renyi entropy can now be readily computed as follows,

$\displaystyle{\text{tr}\left(\rho^{r}_{AC}\right)\over\left(\text{tr}\rho_{AC}\right)^{r}}$
$\displaystyle=\frac{d_{s_{1}}^{r}d_{s_{2}}^{r}+G_{N}^{2}rd_{S_{1}}^{r-2}d_{S_{2}}^{r-2}\left[\text{tr}\big{(}\Omega_{2}^{(1)}\mathbb{P}\big{)}+\text{tr}\big{(}\Omega_{2}^{(2)}\mathbb{P}\big{)}\right]+{G_{N}^{2}r\over
2}\sum\limits_{p}\text{tr}\left[\Omega_{1}\mathbb{P}^{p-2}\Omega_{1}\mathbb{P}^{r-p}\right]}{d_{S_{1}}^{r}d_{S_{2}}^{r}+G_{N}^{2}rd_{S_{1}}^{r-1}d_{S_{2}}^{r-1}\left[\text{tr}\Omega_{2}^{(1)}+\text{tr}\Omega_{2}^{(2)}\right]}$
$\displaystyle=1+{G_{N}^{2}r\over
d_{s_{1}}^{2}d_{s_{2}}^{2}}\left[\text{tr}\big{(}\Omega_{2}^{(2)}\mathbb{P}\big{)}-d_{S_{1}}d_{S_{2}}\text{tr}\Omega_{2}^{(2)}\right]+{G_{N}^{2}r\over
2d_{s_{1}}^{r}d_{s_{2}}^{r}}\sum\limits_{p}\text{tr}\left[\Omega_{1}\mathbb{P}^{p-2}\Omega_{1}\mathbb{P}^{r-p}\right]$
$\displaystyle=1-{G_{N}^{2}r\over
d_{s_{1}}d_{s_{2}}}\sum_{n_{a}=1}^{2S_{a}}\sum_{I_{a}=\pm}\mathbb{A}_{n_{1}n_{2}}^{I_{1}I_{2}}\big{(}\mathbb{A}_{n_{1}n_{2}}^{\bar{I}_{1}\bar{I}_{2}}\big{)}^{*}t_{1}^{(n_{1})}t_{2}^{(n_{2})}\,,$
(83)

where in the second line we have used
$\text{tr}\big{(}\Omega_{2}^{(1)}\mathbb{P}\big{)}=d_{S_{1}}d_{S_{2}}\text{tr}\Omega_{2}^{(1)}$
and in the third line we have plugged in the explicit expressions for the
remaining traces. Taking into account that
$\text{tr}\big{(}\Sigma_{\pm}^{n}\big{)}=0$ and $\mathbb{A}_{nm}^{00}=0$ for
any positive integers $n,m$, it is straightforward to find the final
expression in which $\bar{I}_{a}=\mp$ for $I_{a}=\pm$.

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