# Signatures of the quantum nature of gravity in the differential motion of
two masses

Animesh Datta  Department of Physics, University of Warwick, Coventry CV4 7AL,
United Kingdom  Haixing Miao  School of Physics and Astronomy & Institute for
Gravitational Wave Astronomy, University of Birmingham, B15 2TT, United
Kingdom

###### Abstract

We show that a signature of the quantum nature of gravity is the quantum
mechanical squeezing of the differential motion of two identical masses with
respect to their common mode. This is because the gravitational interaction
depends solely on the relative position of the two masses. In principle, this
squeezing is equivalent to quantum entanglement between the masses. In
practice, detecting the squeezing is more feasible than detecting the
entanglement. To that end, we propose an optical interferometric scheme to
falsify hypothetical models of gravity.

_Introduction:_ Beginning with the work of Bronstein in 1936, thought
experiments have been a perennial tool in understanding the quantum nature of
gravity Bronstein (2011); Gorelik (2005). In his critical comments at the
Chapel Hill Conference on _The Role of Gravitation in Physics_ in 1957,
Feynman concluded that a real problem in a quantum mechanical theory of
gravitation is the lack of experimental guidance. “In this field since we are
not pushed by experiments we must be pulled by imagination.” he said R. P.
Feynman (1957). That era may be about to close.

The last few years have witnessed increasing numbers of experimental proposals
for detecting gravitational effects in quantum systems Kafri and Taylor
(2013); Anastopoulos and Hu (2015); Bose  _et al._ (2017); Marletto and Vedral
(2017); Marletto  _et al._ (2018); Al Balushi  _et al._ (2018); Haine (2021);
Howl  _et al._ (2019); Carlesso  _et al._ (2019); Carney  _et al._ (2019);
Krisnanda  _et al._ (2020); Miao  _et al._ (2020); Clarke  _et al._ (2020);
Matsumura and Yamamoto (2020); Liu  _et al._ (2021). There have followed
several clarifications and analyses Hall and Reginatto (2018); Anastopoulos
and Hu (2018); Belenchia  _et al._ (2018); Marletto and Vedral (2018, 2019);
Reginatto and Hall (2019); Christodoulou and Rovelli (2019); Marletto and
Vedral (2020); Bose and Morley (2018); Pedernales  _et al._ (2020); Chevalier
_et al._ (2020); Marshman  _et al._ (2020); van de Kamp  _et al._ (2020).
These presage the aftermath of the Page-Geilker experiment in the previous
generation Page and Geilker (1981); Hawkins (1982); Ballentine (1982); Page
(1982). While the present proposals are some years from being experimentally
implemented and the precise implications of their predictions still debated,
it behooves us to seek the simplest and most transparent route for detecting
gravitational effects between masses in the quantum regime.

In this Letter, we show the gravitational interaction between two identical
masses leads to quantum mechanical squeezing of their differential mode of
motion with respect to its common mode. Its origin lies in a shift in the
frequency of that former mode relative to the latter. In turn, this motional
squeezing is the genesis of the quantum entanglement between two masses
interacting gravitationally in the Newtonian limit Bose  _et al._ (2017);
Marletto and Vedral (2017). Experimentally, detecting the squeezing is less
challenging than detecting the entanglement. With that in mind, we propose an
optical interferometric scheme to test hypothetical models of gravity. We
close by suggesting an electromagnetic version of our scheme to hone the
experimental techniques.

Figure 1: Schematic of an optical interferometer for probing quantum
signatures of the gravity in the differential motion of two identical masses
$A,B$. The masses also form the end mirrors of optical cavities. Quantum
signatures of the gravitational interaction can thus be detected
optomechanically. For simplicity, we have omitted the local oscillator for the
homodyne measurement of the optical quadratures.

_Principle - Motional Squeezing:_ We consider two identical simple harmonic
oscillators $A$ and $B$ of mass $m$ and frequency $\omega_{m}$ each, separated
at equilibrium by a distance $d,$ and located at $\hat{q}_{A},\hat{q}_{B}$
with momenta $\hat{p}_{A},\hat{p}_{B}.$ The hamiltonian, _if_ the
gravitational interaction between the two masses is quantum, in the Newtonian
limit is,

$\hat{H}=\frac{1}{2m}\left(\hat{p}_{A}^{2}+\hat{p}_{B}^{2}\right)+\frac{m\omega_{m}^{2}}{2}\left(\hat{q}_{A}^{2}+\hat{q}_{B}^{2}\right)-\frac{Gm^{2}}{|d+\hat{q}_{A}-\hat{q}_{B}|},$
(1)

where $[\hat{q}_{i},\hat{p}_{j}]=\mathrm{i}\hbar\delta_{ij}$ for $i,j=A,B.$ If
$d\gg|\hat{q}_{A}-\hat{q}_{B}|,$ expanding the gravitational interaction up to
the second order in $\hat{q}_{A}-\hat{q}_{B},$ neglecting a constant and
absorbing a linear term gives

$\hat{H}\equiv\hat{H}_{+}+\hat{H}_{-}=\frac{\hat{p}_{+}^{2}}{2m}+\frac{m\omega_{m}^{2}}{2}\hat{q}_{+}^{2}+\frac{\hat{p}_{-}^{2}}{2m}+\frac{m\omega_{-}^{2}}{2}\hat{q}_{-}^{2},$
(2)

for the common ($+$) and differential ($-$) modes defined as

$\hat{p}_{\pm}=\frac{\hat{p}_{A}\pm\hat{p}_{B}}{\sqrt{2}},~{}~{}~{}\hat{q}_{\pm}=\frac{\hat{q}_{A}\pm\hat{q}_{B}}{\sqrt{2}},$
(3)

and

$\omega_{-}=\omega_{m}\sqrt{1-\delta^{2}},~{}~{}\delta=\frac{2\omega_{G}}{\omega_{m}},~{}~{}\omega_{G}=\sqrt{\frac{Gm}{d^{3}}}.$
(4)

The gravitational interaction is captured by the characteristic frequency
$\omega_{G}$ that depends on $m$ and $d.$ If $d$ is comparable to the size of
the masses, $\omega_{G}$ is independent of the mass. Rather, it depends on the
material density $\rho$ as Miao  _et al._ (2020).

$\omega_{G}=\sqrt{\Lambda G\rho},$ (5)

where $\Lambda$ is of the order of unity determined by the geometry of the
mass. For gold, $\omega_{G}\sim 10^{-3}$ Hz, implying $\delta\ll 1$ even for
Hz scale oscillators.

As $[\hat{H}_{+},\hat{H}_{-}]=0$ from Eq. (2), the $\pm$ modes are decoupled.
The gravitational interaction affects only the differential one and manifests
itself as a shift in the frequency of that mode relative to the common mode.
An optical interferometric setup for monitoring the differential mode is
presented in Fig. (1).

If the two masses are in thermal states of the simple harmonic hamiltonian at
temperature $T$ at the onset of the gravitational interaction, the
differential mode is in a Gaussian state with covariance matrix

${\bm{\sigma}}^{-}_{0}=\hbar
N_{\mathrm{th}}\left[\begin{array}[]{cccc}\dfrac{1}{m\omega_{m}}&0\\\
0&m\omega_{m}\end{array}\right],~{}~{}N_{\mathrm{th}}=\frac{1}{2}\coth\left(\frac{\hbar\omega_{m}}{2k_{B}T}\right),$
(6)

where $k_{B}$ is Boltzmann’s constant. As $\delta\ll 1$, after time $\tau$ the
differential mode evolves under
$\hat{U}_{-}=\exp(-\mathrm{i}\tau\hat{H}_{-}/\hbar)$ to

${\bm{\sigma}}^{-}_{\tau}=\hbar
N_{\mathrm{th}}\left[\begin{array}[]{cccc}\dfrac{1+\delta^{2}\sin^{2}(\omega_{m}\tau)}{m\omega_{m}}&\dfrac{\delta^{2}}{2}\sin(2\omega_{m}\tau)\\\
\dfrac{\delta^{2}}{2}\sin(2\omega_{m}\tau)&m\omega_{m}\left(1-\delta^{2}\sin^{2}(\omega_{m}\tau)\right)\end{array}\right].$
(7)

The fundamental signature of the quantum nature of the gravitational
interaction at the Newtonian limit is thus a squeezing of the differential
momentum and an equivalent anti-squeezing of the differential position
quadrature. Additionally, the gravitational interaction generates correlations
between the two quadratures. All these signatures are proportional to
$\delta^{2}$ and oscillate at double the frequency of the original harmonic
potential. Note that these signatures are a consequence of the frequency shift
of the differential mode relative to the common one which remains in its
original thermal state for the entire evolution.

Any effort to detect the minuscule quantum mechanical squeezing of the
differential motional mode in the laboratory will be marred by decoherence.
Its specific nature will depend on the details of the experimental setup and
its regime of operation. For instance, under the Caldeira-Leggett model of
decoherence for an Ohmic spectral density with an infinite cutoff, the steady-
state covariance matrices of the differential and common motions are given by

${\bm{\sigma}}^{-}_{\infty}=\hbar
N_{\mathrm{th}}\left[\begin{array}[]{cccc}\dfrac{1}{m\omega_{-}}&0\\\
0&m\omega_{-}\end{array}\right],~{}{\bm{\sigma}}^{+}_{\infty}=\hbar
N_{\mathrm{th}}\left[\begin{array}[]{cccc}\dfrac{1}{m\omega_{m}}&0\\\
0&m\omega_{m}\end{array}\right],$ (8)

showing that the only remnant of the quantum aspect of gravity is the altered
variances or quantum mechanical squeezing of the differential motional mode -
a direct consequence of its shifted frequency due to the gravitational
interaction.

Interestingly, this squeezing of the differential motion manifests itself as
the entanglement between the two masses upon transformation as per Eq. (3).
Mathematically, the entanglement of the two masses $A,B$ is determined by the
product of the two smallest eigenvalues of the covariance matrix of the common
and differential modes Wolf  _et al._ (2003). Physically, this is identical to
the creation of two-mode entanglement by interfering a squeezed and an
unsqueezed thermal mode on a balanced beam splitter. The condition for
achieving steady-state quantum entanglement is identical (see Eq. (29)), to
within a factor of two, to that for observing squeezing.

Observing this squeezing requires, for $\delta\ll 1,$

$N_{\rm
th}\frac{\omega_{-}}{\omega_{m}}\leq\dfrac{1}{2},~{}\text{or}~{}\coth\left(\frac{\hbar\omega_{m}}{2k_{B}T}\right)\leq(1-\delta^{2})^{-1/2}\approx
1+\frac{\delta^{2}}{2}\,.$ (9)

This, in turn, requires

$\frac{k_{B}T}{\hbar\omega_{m}}\leq\frac{1}{\ln{(4/\delta^{2}})}\,,$ (10)

which may seem impossible to achieve. However, as this steady-state behaviour
does not depend on the mechanical damping or the mechanical quality factor
$Q_{m},$ feedback Mancini  _et al._ (1998); Cohadon  _et al._ (1999) or
optomechanical Marquardt  _et al._ (2007); Wilson-Rae  _et al._ (2007) cooling
can be invoked to reduce the effective temperature to $T/Q_{m}.$ Consequently,
final condition for observing the squeezing becomes

$\frac{k_{B}T}{\hbar\omega_{m}}\leq\frac{Q_{m}}{\ln{(4/\delta^{2}})}\,.$ (11)

This is within the reach of current technology using masses of gold or
tungsten with $\rho=19.3\times 10^{3}~{}\text{Kg/m}^{3},$ $\Lambda=2,$ $T=10$
mK, $\omega_{m}=2\pi\times 10~{}\mathrm{Hz}$ , and $Q_{m}=10^{9}$. However,
low frequency mechanical noise is expected to be a challenge. To that end, we
now estimate the impacts of gravity-gradient and seismic noise on our
proposal.

Gravity-gradient noise due to seismically-induced matter density fluctuation
is quite small at $10~{}\mathrm{Hz}$. Even for Advanced LIGO, this is
substantially smaller than thermal noise (Aasi and  _et al._ , 2015, Fig. 2).
A much larger contribution comes from anthropic activity, such as a person
walking close to the experiment. This was analysed in the context of the
gravitational wave detector Thorne and Winstein (1999). Converting the strain
noise to acceleration, and then force noise on a 1 gram mass gives $1.0\times
10^{-16}~{}\mathrm{N}/\sqrt{\mathrm{Hz}}$ at $10\,\rm Hz$ for a person is
moving 1 m away from the experiment. This is more than one order of magnitude
larger than the thermal Langevin force noise of about $6.0\times
10^{-18}~{}\mathrm{N}/\sqrt{\mathrm{Hz}}$ at 10 mK and $10\,\rm Hz.$ This may
be mitigated by restraining anthropic activity in the laboratory during data
collection.

To counter seismic noise, the experiment can be seismically isolated to a
level close to $10^{-12}\,\rm m/\sqrt{\rm Hz}$ at $10\,\rm Hz$ using a
commercial active isolation platform (e.g., Accurion) in a reasonably quiet
lab ($10^{-10}\,\rm m/\sqrt{\rm Hz}$ at $10\,\rm Hz$). Furthermore, the
resonant frequency of two masses can be well matched such that a significant
part of the seismic noise acts on the common mode of the masses without
affecting the differential mode. To estimate the latter, a difference
$\Delta\omega$ in the frequency of two masses will induce on the differential
mode an effective force noise of $2m\omega_{-}\Delta\omega x_{\mathrm{GND}},$
where $x_{\mathrm{GND}}$ is the seismic motion. The corresponding single-sided
power spectral density is $4m^{2}\omega_{-}^{2}(\Delta\omega)^{2}S_{xx},$
where $S_{xx}$ is the single-sided power spectral density of the seismic
motion. Setting this to, say, $1\%$ of the thermal Langevin force noise
spectral density $S_{FF}=4\hbar m\gamma_{m}\omega_{-}N_{\rm th},$ where
$\gamma_{m}=\omega_{m}/Q_{m},$ gives

$4m^{2}\omega_{-}^{2}(\Delta\omega)^{2}S_{xx}=\frac{1}{100}S_{FF}.$ (12)

A seismic noise of $10^{-12}\,\rm m/\sqrt{\rm Hz}$ from active isolation gives
$S_{xx}=10^{-24}\,\rm m^{2}/Hz.$ A 1 gram, $10\,\rm Hz$ oscillator with
$Q_{m}=10^{9}$, then requires $\Delta\omega\sim 5\,\mu\rm Hz,$ which is about
80 times the mechanical linewidth $\gamma_{m}=63\,\rm nHz.$ It is also about
10 times the frequency stability of $0.4\,\mu\rm Hz$ required to observe
optical squeezing by integrating over one month (See next section). Achieving
this $\Delta\omega$ may require experimental innovation and development.

The outstanding challenge is resolving quantum squeezing on the order of
$\delta^{2}\sim 10^{-9}$ for a $10$ Hz mechanical oscillator. We address this
next.

_Practice - Optomechanical squeezing:_ A possible route to detecting this
minuscule motional squeezing is via optical interferometry that detects the
relative frequency shift, as illustrated in Fig. (1).

For low-frequency simple harmonic oscillators, the cavity bandwidth can be
much larger than the mechanical frequency - a regime amenable to feedback
cooling. This regime also allows the cavity mode can be eliminated
adiabatically, resulting in the input-output relation for the differential
_optical_ mode Chen (2013)

$\displaystyle\hat{X}_{-}^{\rm out}(t)$ $\displaystyle=\hat{X}_{-}^{\rm
in}(t)\,,$ (13) $\displaystyle\hat{Y}_{-}^{\rm out}(t)$
$\displaystyle=\hat{Y}_{-}^{\rm in}(t)+(\alpha/\hbar)\,\hat{q}_{-}(t)\,,$ (14)

where $\hat{X}_{-}~{}(\hat{Y}_{-})$ is the amplitude (phase) quadrature in the
two-photon formalism Caves and Schumaker (1985), $\alpha=8\sqrt{{\hbar
P\omega_{0}}/(c^{2}t_{m}^{2})}$ with $P$ the intra-cavity power, $\omega_{0}$
the laser frequency, and $t_{m}$ the amplitude transmissivity of the input
mirror of the optical cavity. The equation of motion of the masses’
differential mode is

$m\ddot{\hat{q}}_{-}(t)+m\gamma_{m}\dot{\hat{q}}_{-}(t)+m\omega_{-}^{2}{\hat{q}}_{-}(t)=\alpha\,\hat{X}_{-}^{\rm
in}(t)+\hat{F}_{\rm th}(t)\,,$ (15)

where $\hat{F}_{\rm th}$ is the Langevin force with a spectrum consistent with
the Caldeira-Leggett model.

Solving Eqs. (13)-(15) in the frequency domain gives

$\left[\begin{array}[]{c}\hat{X}_{-}^{\rm out}(\omega)\\\ \hat{Y}_{-}^{\rm
out}(\omega)\end{array}\right]=\left[\begin{array}[]{cc}1&0\\\
\dfrac{\alpha^{2}\chi_{-}(\omega)}{\hbar}&1\end{array}\right]\left[\begin{array}[]{c}\hat{X}_{-}^{\rm
in}(\omega)\\\ \hat{Y}_{-}^{\rm
in}(\omega)\end{array}\right]+\left[\begin{array}[]{c}0\\\
\dfrac{\alpha\chi_{-}(\omega)}{\hbar}\end{array}\right]\hat{F}_{\rm
th}(\omega)\,,$ (16)

where
$\chi_{-}(\omega)\equiv[-m(\omega^{2}+i\,\gamma_{m}\omega-\omega_{-}^{2})]^{-1}$
is the mechanical susceptibility of the differential motion. The corresponding
covariance matrix for the spectral densities is

$\left[\begin{array}[]{cc}S_{XX}(\omega)&S_{XY}(\omega)\\\
S_{YX}(\omega)&S_{YY}(\omega)\end{array}\right]=\left[\begin{array}[]{cc}1&\dfrac{\alpha^{2}\chi_{-}^{*}}{\hbar}\\\
\dfrac{\alpha^{2}\chi_{-}}{\hbar}&1+\dfrac{\alpha^{2}|\chi_{-}|^{2}}{\hbar^{2}}\left(\alpha^{2}+S_{FF}\right)\end{array}\right]\,,$
(17)

where $S_{FF}$ results in the steady-state covariance in Eq. (8) for the
differential motion. The correlation between the amplitude and phase
quadrature leads to optomechanical squeezing Kimble  _et al._ (2001); Purdy
_et al._ (2013) of the differential optical mode.

The common optical mode has a similar covariance matrix with the mechanical
susceptibility
$\chi_{+}(\omega)\equiv[-m(\omega^{2}+i\,\gamma_{m}\omega-\omega_{m}^{2})]^{-1},$
that is with the mechanical resonance at $\omega_{m}$ rather than
$\omega_{-}.$ The quantum nature of gravity thus manifests itself as the
difference in the ponderomotive squeezing for the common and differential
modes of the optical field.

This ponderomotive squeezing can be read out using a homodyne measurement. It
measures a general quadrature
$\hat{X}(\theta)=\hat{X}\cos\theta+\hat{Y}\sin\theta$ determined by $\theta,$
the phase of the local oscillator. The corresponding spectral density is
$S(\theta,\omega)=S_{XX}\cos^{2}\theta+(S_{XY}+S_{YX})\sin\theta\cos\theta+S_{YY}\sin^{2}\theta\,,$
where we have suppressed the $\omega$ argument on the RHS for brevity. Its
minimum value is

$S^{\rm
min}(\omega)=\frac{S_{XX}+S_{YY}}{2}-\frac{\sqrt{(S_{XY}+S_{YX})^{2}+(S_{XX}-S_{YY})^{2}}}{2}.$
(18)

For the common optical mode, $S_{XY}+S_{YX}=0$ at $\omega=\omega_{m},$ whereby

$S^{\rm min}_{+}(\omega_{m})>1.$ (19)

It thus shows no squeezing at the mechanical resonance. This is an instance of
the blindness of homodyne measurements to complex squeezing that is
ponderomotively generated Buchmann  _et al._ (2016).

The differential optical mode, on the contrary, does exhibit squeezing at the
$\omega=\omega_{m}$ due of the gravity-induced frequency shift on the
differential motional mode. Mathematically, as $\alpha\gg 1,$ $S_{YY}\gg
S_{XY},S_{YX}$ from Eq. (17), and $\alpha^{2}\gg S_{FF}$ in a regime where the
quantum radiation pressure dominates the thermal fluctuation,

$S^{\rm min}_{-}(\omega_{m})\approx
1-\left|\dfrac{\mathrm{Re}[\chi_{-}(\omega_{m})]}{\chi_{-}(\omega_{m})}\right|^{2}=\frac{1}{1+Q_{m}^{2}\delta^{4}}.$
(20)

Estimating this squeezing in practice requires repeating the experiment - that
is reaching steady state and measuring $\nu$ times. This gives an estimate
with variance $S^{\rm min}_{-}(\omega_{m})/\nu$ which must equal $1-S^{\rm
min}_{-}(\omega_{m})$ to provide an SNR of unity. Thereby, total time for the
experiment is

$t=\frac{\nu}{\gamma_{m}}=\frac{1}{\omega_{m}Q_{m}\delta^{4}},$ (21)

after which time, the differential optical squeezing (in dB) is

${\cal S}_{-}\equiv-10\log_{10}\left[S^{\rm
min}_{-}(\omega_{m})\right]=10\log_{10}\left[1+Q_{m}^{2}\delta^{4}\right].$
(22)

Both ${\cal S}_{-}$ and $t$ are solely determined by the mechanical, and not
the optical properties of the setup. For the representative experimental
parameters chosen after Eq. (11), ${\cal S}_{-}\approx 9~{}\text{dB}$ and
$t\approx 1~{}\text{month}.$ This is challenging, but not prohibitive. Indeed,
its experimental accessibility is aided by the lack of any thermal fluctuation
limitations in a regime dominated by the quantum radiation pressure. An
observation of this squeezing shall provide an experimental witness of the
quantum nature of gravity - as we illustrate next for a semiclassical model of
gravity.

_Semiclassical gravity:_ A hypothetical model of gravity is a semiclassical
one - where a classical spacetime structure is sourced by matter of quantum
nature. The hamiltonian for two identical harmonically trapped masses
interacting as per the non-relativistic version of such a model - known as the
Schrödinger-Newton equation Yang  _et al._ (2013) is given by

$\displaystyle\hat{H}^{\mathrm{SN}}$ $\displaystyle=$
$\displaystyle\frac{1}{2m}\left(\hat{p}_{A}^{2}+\hat{p}_{B}^{2}\right)+\frac{m\omega_{m}^{2}}{2}\left(\hat{q}_{A}^{2}+\hat{q}_{B}^{2}\right)$
$\displaystyle-$ $\displaystyle
C_{1}(\hat{q}_{A}-\hat{q}_{B})-C_{2}\left((\hat{q}_{A}-\left\langle{\hat{q}_{B}}\right\rangle)^{2}+(\hat{q}_{B}-\left\langle{\hat{q}_{A}}\right\rangle)^{2}\right).$

where $C_{1,2}$ are constants depending on $m,d.$ Using Eq. (3),

$\hat{H}^{\mathrm{SN}}=\sum_{i={\pm}}\left(\frac{\hat{p}_{i}^{2}}{2m}+\frac{m\omega_{SN}^{2}}{2}\hat{q}_{i}^{2}\right),~{}~{}~{}\omega_{SN}=\omega_{m}\sqrt{1-\frac{2C_{2}}{m\omega_{m}^{2}}}.$
(24)

In this case, both the common and differential motional modes undergo
identical frequency shifts with respect to $\omega_{m}$ and quantum squeezing.
There will be no relative frequency shift between the two modes, both of which
after time $\tau$ will have identical covariance matrices given by Eq. (7)
with $\delta=\sqrt{2C_{2}/m\omega_{m}^{2}}.$ Consequently, there will be no
quantum mechanical squeezing in the differential mode relative to the common
mode. The quantum squeezing and correlations are transferred unchanged to the
motional quadratures of the two masses, as has been discussed in terms of the
transfer of quantum information and uncertainty Yang  _et al._ (2013).
Crucially, there will be no quantum entanglement between the motional states
of the masses - as is the case if two identically squeezed thermal states are
interfered on a balanced beam splitter.

_Conclusions:_ This Letter has three - the first two holding _if_ the
gravitational interaction between two masses, as given by the last term in Eq.
(1), is sufficient to test the quantum nature of gravity.

Firstly, the quantum nature of the gravitational interaction between two
masses can be probed via the squeezing of their differential motional mode.
This is an alternative to the experimental detection of quantum entanglement
generated by gravity Bose  _et al._ (2017); Marletto and Vedral (2017);
Marletto  _et al._ (2018); Al Balushi  _et al._ (2018); Carlesso  _et al._
(2019); Krisnanda  _et al._ (2020); Matsumura and Yamamoto (2020) which, as is
to be expected, is also proportional to $\delta^{2}.$ Which one is more
fundamental depends on which side of Eq. (3) one considers more intrinsic.

Secondly, the signature of the quantum nature of gravitational interaction is
imprinted in the relative frequency shift of the differential motional mode.
In principle, such a frequency shift may be inferred from a standard transfer
function measurement by introducing a strong classical force to drive the
masses. In practice, the optical squeezing provides direct evidence of the
quantum correlation between the optical fields in the two cavities meditated
by gravity Miao  _et al._ (2020). This being absent for some hypothetical
models of gravity such as the Schrödinger-Newton model allows them to be
falsified experimentally.

Thirdly, the proposed experiment in Fig. (1) for detecting the squeezing of
the differential optical mode is less demanding than one for detecting
entanglement - either in the motional or the optical modes. The physical
reason lies in the negligibility of the thermal fluctuations in the quantum
radiation pressure dominated regime. An exhaustive mathematical argument is
hindered by the variety of experimental setups for detecting gravitationally-
induced entanglement. Consequently, we illustrate this conclusion by
considering the entanglement between the optical fields in the two cavities
identified with the two masses in Fig. (1).

We begin by defining effective single-mode optical quadratures at $\omega_{m}$
as Miao  _et al._ (2020) $\hat{\cal
X}_{k}\equiv\hat{X}_{k}(\omega_{m})\sqrt{\Delta\omega/\pi},\hat{\cal
Y}_{k}\equiv\hat{Y}_{k}(\omega_{m})\sqrt{\Delta\omega/\pi}\,,$ such that
$[\hat{\cal X}_{k},\hat{\cal
Y}^{{\dagger}}_{k^{\prime}}]=2i\delta_{k,k^{\prime}}$ and
$\hat{X}_{k}~{}(\hat{Y}_{k})$ are the amplitude (phase) quadratures of cavity
$k=A,B.$ The bandwidth $\Delta\omega$ is determined by the integration time
and smaller than $\gamma_{m}$. The entanglement between the optical fields in
the $A,B$ cavities is determined solely by the off-diagonal block of the
optical covariance matrix $\bm{\sigma}^{\rm opt}$ for $\\{\hat{\cal
X}_{A},\hat{\cal Y}_{A},\hat{\cal X}_{B},\hat{\cal Y}_{B}\\}$ quadratures,
that is

${\bm{\sigma}}_{AB}^{\rm
opt}=\frac{\alpha^{2}}{2\hbar}\left[\begin{array}[]{cc}0&\chi^{*}_{+}(\omega_{m})-\chi^{*}_{-}(\omega_{m})\\\
\chi_{+}(\omega_{m})-\chi_{-}(\omega_{m})&\dfrac{\alpha^{2}}{\hbar}(|\chi_{+}(\omega_{m})|^{2}-|\chi_{-}(\omega_{m})|^{2})\end{array}\right]\,.$
(25)

Evidently, it is the difference in the mechanical susceptibilities of the
common and differential motion that generates the entanglement111The
Schrödinger-Newton model predicts $\chi_{+}(\omega)=\chi_{-}(\omega),$ whereby
${\bm{\sigma}}_{AB}^{\rm opt}={\bm{0}}_{2\times 2},$ and therefore no
entanglement. . The condition for entanglement between the optical modes, as
per the logarithmic negativity, gives

$\coth\left(\frac{\hbar\omega_{m}}{2k_{B}T}\right)\leq
Q_{m}\delta^{2},~{}\text{or}~{}\frac{k_{B}T}{\hbar\omega_{m}}\leq\left[\ln\left(\frac{Q_{m}\delta^{2}+1}{Q_{m}\delta^{2}-1}\right)\right]^{-1}.$
(26)

Once again, this is solely determined by the mechanical, and not the optical
properties of the setup. More importantly, this condition on the temperature
is ‘exponentially’ more demanding than Eq. (11) as well as Eq. (21),
substantiating our second conclusion. The experimental requirements are
summarised in Table (1).

Probe Signature |  Squeezing |  Entanglement
---|---|---
Motional |  Eq. (11) |  Eq. (30)
Optical |  Eq. (21) |  Eq. (26)
Table 1: Experimental requirements necessary to detect signatures of the
quantum nature of gravity. Squeezing corresponds to that of the differential
mode while entanglement is between the two masses. Only the optical squeezing
imposes no restrictions on the temperature. Thus, in principle, the experiment
can be performed at room temperature if the radiation pressure limited regime
can be achieved with low frequency, high quality factor mechanical
oscillators.

Finally, the scheme in Fig. (1) accumulates the quantum information about
$\delta$ in the squeezing of the optical differential mode quantum
mechanically, while the conditional squeezing in Ref. Miao  _et al._ (2020)
combines that same quantum information from the readouts of two optomechanical
cavities classically. The coherence of our scheme leads to a well-known
improvement in the detection of $\delta$ by a factor of $\sqrt{2}.$ Given the
quartic dependences in Eqs. (21), (22), our scheme offers a very consequential
four-fold improvement over Ref. Miao  _et al._ (2020) 222Eqs. (2), (3) in Ref.
Miao  _et al._ (2020) are incorrect by a factor of 4..

Nevertheless, the experiment proposed here is an exacting one. We therefore
suggest, as a point of embarkation, an endeavour to detect the relative
frequency shift between the common and differential modes due to an
electromagentic interaction between the masses. Given its strength, larger
values of $\delta$ would be possible with significantly higher $\omega_{m}.$
This will enable the identification - and suppression, of various noise
sources as the electromagnetic strength and $\omega_{m}$ are progressively
reduced. A course to experimentally detecting signatures of the quantum nature
of gravity may thus be charted.

_Appendix - Quantum entanglement between the two masses:_ The steady-state
covariance matrix of the masses $A$, $B$, that is for
$\\{\hat{q}_{A},\hat{p}_{A},\hat{q}_{B},\hat{p}_{B}\\}$ quadratures, is

${\bm{\sigma}}=\left[\begin{array}[]{cc}{\bm{\sigma}}_{A}&{\bm{\sigma}}_{AB}\\\
{\bm{\sigma}}_{AB}&{\bm{\sigma}}_{B}\end{array}\right]={\bf
M}\left[\begin{array}[]{cc}{\bm{\sigma}}^{+}_{\infty}&{\bm{0}}_{2\times 2}\\\
{\bm{0}}_{2\times 2}&{\bm{\sigma}}^{-}_{\infty}\end{array}\right]{\bf M}\,,$
(27)

where the transformation matrix $\bf M$ is given by

${\bf
M}=\left[\begin{array}[]{cccc}\sqrt{\frac{m\omega_{m}}{\hbar}}&0&\sqrt{\frac{m\omega_{m}}{\hbar}}&0\\\
0&\sqrt{\frac{1}{\hbar m\omega_{m}}}&0&\sqrt{\frac{1}{\hbar m\omega_{m}}}\\\
\sqrt{\frac{m\omega_{m}}{\hbar}}&0&-\sqrt{\frac{m\omega_{m}}{\hbar}}&0\\\
0&\sqrt{\frac{1}{\hbar m\omega_{m}}}&0&-\sqrt{\frac{1}{\hbar
m\omega_{m}}}\end{array}\right]\,.$ (28)

The quantum entanglement of this bipartite system, as per the logarithmic
negativity Adesso and Illuminati (2007), is ${\cal E}_{N}={\rm
max}\left\\{-(1/2)\log_{2}\left[\left(\Sigma-\sqrt{\Sigma^{2}-4{\det\bm{\sigma}}}\right)/2\right],\,0\right\\}\,,$
where
$\Sigma\equiv\det{\bm{\sigma}}_{A}+\det{\bm{\sigma}}_{B}-2\,\det{\bm{\sigma}}_{AB}$.
Using Eq. (8), ${\cal E}_{N}={\rm max}\left\\{-\log_{2}\left[2N_{\rm
th}(1-\delta^{2})^{1/4}\right],\,0\right\\}.$ Achieving steady-state quantum
entanglement thus requires

$\coth\left(\frac{\hbar\omega_{m}}{2k_{B}T}\right)\leq(1-\delta^{2})^{-1/4}\approx
1+\frac{\delta^{2}}{4}\,,$ (29)

which matches Eq. (9) to within a factor of 2. Invoking feedback or
optomechanical cooling leads to

$\frac{k_{B}T}{\hbar\omega_{m}}\leq\frac{Q_{m}}{\ln{(8/\delta^{2}})}\,.$ (30)

Acknowledgements: We thank H. Grote for helpful comments and the UK STFC
‘Quantum Technologies for Fundamental Physics’ programme (ST/T006404/1) for
support. H.M acknowledges the support from the Birmingham Institute for
Gravitational Wave Astronomy. H.M has also been supported by UK STFC Ernest
Rutherford Fellowship (Grant No. ST/M005844/11). We also thank the anonymous
referees for comments that helped us improve this Letter.

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algorithm

# Splitting Compounds by Semantic Analogy

Joachim Daiber$*$ Lautaro Quiroz$\dagger$ Roger Wechsler$\dagger$ Stella
Frank$*$ $*$Institute for Logic, Language and Computation $\dagger$Graduate
School of Informatics University of Amsterdam University of Amsterdam Science
Park 107, 1098 XG Amsterdam Science Park 904, 1098 XH Amsterdam
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Compounding is a highly productive word-formation process in some languages
that is often problematic for natural language processing applications. In
this paper, we investigate whether distributional semantics in the form of
word embeddings can enable a deeper, i.e., more knowledge-rich, processing of
compounds than the standard string-based methods. We present an unsupervised
approach that exploits regularities in the semantic vector space (based on
analogies such as “bookshop is to shop as bookshelf is to shelf”) to produce
compound analyses of high quality. A subsequent compound splitting algorithm
based on these analyses is highly effective, particularly for ambiguous
compounds. German to English machine translation experiments show that this
semantic analogy-based compound splitter leads to better translations than a
commonly used frequency-based method.

00footnotetext:  This work is licenced under a Creative Commons Attribution
4.0 International License.

##  1 Introduction

In languages such as German, compound words are a frequent occurrence leading
to difficulties for natural language processing applications, and in
particular machine translation. Several methods for dealing with this
issue—from shallow count-based methods to deeper but more complex neural
network-based processing methods—have been proposed. The recent surge in
practical models for distributional semantics has enabled a multitude of
practical applications in many areas, most recently in morphological analysis
[Soricut and Och (2015]. In this paper, we investigate whether similar methods
can be utilized to perform deeper, i.e. more knowledge-rich, processing of
compounds. A great asset of word embeddings are the regularities that their
multi-dimensional vector space exhibits. ?) showed that regularities such as
“king is to man what queen is to woman” can be expressed and exploited in the
form of basic linear algebra operations on the vectors produced by their
method. This often-cited example can be expressed as follows:
$v(\mathrm{king})-v(\mathrm{man})+v(\mathrm{woman})\approx v(\mathrm{queen})$,
where $v(.)$ maps a word into its word embedding in vector space.

In a very recent approach, ?) exploit these regularities for unsupervised
morphology induction. Their method induces vector representations for basic
morphological transformations in a fully unsupervised manner. String prefix
and suffix replacement rules are induced directly from the data based on the
idea that morphological processes can be modeled on the basis of prototype
transformations, i.e. vectors that are good examples of a morphological
process are applied to a word vector to retrieve its inflected form. A simple
example of this idea is
${\uparrow}d_{\mathrm{\emph{car}\underline{s}}}=v(\mathrm{cars})-v(\mathrm{car})$
and $v(\mathrm{dogs})\approx
v(\mathrm{dog})+{\uparrow}d_{\mathrm{\emph{car}\underline{s}}}$, which
expresses the assumption that the word car is to cars what dog is to dogs. The
direction vector ${\uparrow}d_{\mathrm{\emph{car}\underline{s}}}$ represents
the process of adding the plural morpheme _-s_ to a noun.

While this intuition works well for frequently occurring inflectional
morphology, it is not clear whether it extends to more semantically motivated
derivational processes such as compounding. We study this question in the
present paper. Our experiments are based on the German language, in which
compounding is a highly productive phenomenon allowing for a potentially
infinite number of combinations of words into compounds. This fact, coupled
with the issue that many compounds are observed infrequently in data, leads to
a data sparsity problem that hinders the processing of such languages. Our
contributions are as follows: After reviewing related work (Section 2), we
study whether the regularities exhibited by the vector space also apply to
compounds (Section 3). We examine the relationship between the components
within compounds, as illustrated by the analogical relationship “ _Hauptziel_
is to _Ziel_ what _Hauptader_ is to _Ader_.”111In vector algebra:
${\uparrow}d_{\mathrm{\underline{Haupt}\emph{ziel}}}=v(\mathrm{Hauptziel})-v(\mathrm{Ziel})$
and $v(\mathrm{Hauptader})\approx
v(\mathrm{Ader})+{\uparrow}d_{\mathrm{\underline{Haupt}\emph{ziel}}}$. The
compounds translate to main goal (_Hauptziel_) and main artery (_Hauptader_).
As a separate noun, _Haupt_ means head. By leveraging this analogy we can then
analyze the novel compound _Hauptmann_ (captain) by searching for known string
prefixes (e.g.  _Haupt-_) and testing whether the resulting split compound
(_Haupt|mann_) has a similar relation between its components (_haupt_ ,
_mann_) as the prototypical example (_Haupt|ziel_). We induce the compound
components and their prototypes and apply them in a greedy compound splitting
algorithm (Section 4), which we evaluate on a gold standard compound splitting
task (Section 4.3) and as a preprocessing step in a machine translation setup
(Section 5).

##  2 Related work

Our methodology follows from recent work on morphology induction [Soricut and
Och (2015], which combines string edits with distributional semantics to split
words into morphemes. In this model, morphemes are represented as string edits
plus vectors, and are linked into derivation graphs. The authors consider
prefix and suffix morphemes up to six characters in length; in contrast, our
approach to noun compound splitting only considers components at least four
characters long.

###  2.1 Splitting compounds for SMT

Dealing with word compounding in statistical machine translation (SMT) is
essential to mitigate the sparse data problems that productive word generation
causes. There are several issues that need to be addressed: _splitting_
compound words into their correct components (i.e. disambiguating between
split points), _deciding_ whether to split a compound word at all, and, if
translating into a compounding language, _merging_ components into a compound
word (something we do not address, but see ?) and ?) for systems that do). ?)
address German compound splitting using a straightforward approach based on
component frequency. They also present splitting approaches based on word
alignments and POS tag information, but find that while the more resource-
intensive approaches give better splitting performance (measured by gold-
standard segmentations) the frequency-based method results in the best SMT
performance (measured by BLEU). This is attributed to the fact that phrase-
based MT system do not penalize the frequency-based method for over-splitting,
since it can handle components as a phrase.

?), ?) and ?) explore using morphological analyzers for German compound
splitting, with mixed results. Since these approaches use heavy supervision
within the morphological analyzer, they are orthogonal to our unsupervised
approach.

It may be advantageous to split only compositional compounds, and leave
lexicalized compounds whole. ?) investigate this question by using
distributional similarity to split only words that pass a certain threshold
(i.e., where the parts proposed by the morphological analyzer are similar to
the compound). Contrary to their hypothesis, they find no advantage in terms
of SMT, again indicating that oversplitting is not a problem for phrase-based
SMT. The use of distributional similarity as a cue for splitting is similar to
the work presented in this paper. However, the approach we follow in this
paper is fully unsupervised, requiring only word embeddings estimated from a
monolingual corpus. Additionally, it stands out for its simplicity, making it
easy to understand and implement.

###  2.2 Semantic compositionality

Noun compounding has also been treated within the field of distributional
semantics. ?) examine English noun compounds and find that distributional co-
occurrence can capture the relationship between compound parts and whole, as
judged by humans in terms of ‘literalness’. ?) replicate this result for
German, and also show that simple window-based distributional vectors
outperform syntax-based vectors.

##  3 Towards deeper processing of compound words

###  3.1 Unsupervised morphology induction from word embeddings

Our approach is based on the work of ?), who exploit regularities in the
vector space to induce morphological transformations. The authors extract
morphological transformations in the form of prefix and suffix replacement
rules up to a maximum length of 6 characters. The method requires an initial
candidate set which contains all possible prefix and suffix rules that occur
in the monolingual corpus. For English, the candidate set contains rules such
as suffix:ed:ing, which represents the suffix ed replaced by ing (e.g.
walked$\to$walking). This candidate set also contains overgenerated rules that
do not reflect actual morphological transformations; for example
prefix:S:$\epsilon$222$\epsilon$ denotes the empty string. in
scream$\to$cream.

The goal is to filter the initial candidate set to remove spurious rules while
keeping useful rules. For all word pairs a rule applies to, word embeddings
are used to calculate a vector representing the transformation. For example,
the direction vector for the rule suffix:ing:ed based on the pair (walking,
walked) would be
${\uparrow}d_{\mathrm{\emph{walk}\underline{ing{\to}ed\vphantom{g}}}}=v(\mathit{walked})-v(\mathit{walking})$.
For each rule there are thus potentially as many direction vectors as word
pairs it applies to. A direction vector is considered to be meaning-preserving
if it successfully predicts the affix replacements of other, similar word
pairs. Specifically, each direction vector is applied to the first word in the
other pair and an ordered list of suggested words is produced. For example,
the direction vector
${\uparrow}d_{\mathrm{\emph{walk}\underline{ing{\to}ed\vphantom{g}}}}$ can be
evaluated against (playing, played) by applying
${\uparrow}d_{\mathrm{\emph{walk}\underline{ing{\to}ed\vphantom{g}}}}$ to
playing to produce the predicted word form:
$v(\mathit{played}*)=v(\mathit{playing})+{\uparrow}d_{\mathrm{\emph{walk}\underline{ing{\to}ed\vphantom{g}}}}$.
This prediction is then compared against the true word embedding
$v(\mathit{played})$ using a generic evaluation function
$E(v(\mathit{played}),v(\mathit{playing})+{\uparrow}d_{\mathrm{\emph{walk}\underline{ing{\to}ed\vphantom{g}}}})$.333We
follow ?) in defining $E$ as either the _cosine_ distance or the _rank_
(position in the predictions). If the evaluation function passes a certain
threshold, we say that the direction vector _explains_ the word pair. Some
direction vectors explain many word pairs while others might explain very few.
To judge the explanatory power of a direction vector, a _hit rate_ metric is
calculated, expressing the percentage of applicable word pairs for which the
vector makes good predictions.444A transformation is considered a _hit_ if the
evaluated score is above a certain threshold for each evaluation method $E$.
Each direction vector has a hit rate and a set of word pairs that it explains
(its evidence set). Apart from their varying explanatory power, morphological
transformation rules are also possibly ambiguous. For example, the rule
suffix:$\epsilon$:s can describe both the pluralization of a noun (one _house_
$\to$two _houses_) and the 3rd person singular form of a verb (I _find_
$\to$she _finds_). Different direction vectors might explain the nouns and
verbs separately.

?) retain only the most explanatory vectors by applying a recursive procedure
to find the minimal set of direction vectors explaining most word pairs. We
call this set of direction vectors _prototypes_ , as they represent a
prototypical transformation for a rule and other words are formed _in analogy
to_ this particular word pair. Finally, ?) show that their prototypes can be
applied successfully in a word similarity task for several languages.

###  3.2 Compound words and the semantic vector space

According to ?), compounds can be classified into several groups based on
whether the semantic head is part of the compound (_endocentric_ compounds; a
doghouse is a also a house) or whether the semantic head is outside of the
compound (_exocentric_ compounds; a skinhead is not a head). In this paper, we
focus on endocentric compounds, which are also the most frequent type in
German. Endocentric compounds consist of a modifier and a semantic head. The
semantic head specifies the basic meaning of the word and the modifier
restricts this meaning. In German, the modifiers come before the semantic
head; hence, the semantic head is always the last component in the compound.
When applying the idea of modeling morphological processes by semantic analogy
to compounds, we can represent either the semantic head or the modifier of the
compound as the transformation (like the morpheme rules above). Since the head
carries the compound’s basic meaning, we add the modifier’s vector
representation to the head word in order to restrict its meaning. We expect
the resulting compound to be in the neighborhood of the head word in the
semantic space (e.g., a doghouse is close to house).

55footnotetext: Gloss for modifiers: (a) main, (b) federal, (c) children, (d)
finance. Heads: (e) piece of work, (f) ministry, (g) man, (h) city.

We illustrate this intuition by visualizing compound words and their parts in
the vector space. All visualizations are produced by performing principal
component analysis (PCA) to reduce the vector space from 500 to 2 dimensions.
Figure 1 presents the visualization of various compounds with either the same
head or the same modifier. For Figure 1a, we plot all German compounds in our
dataset that have one of the modifiers Haupt-, Super-, Bundes-, Kinder- or
Finanz-. Figure 1b, on the other hand, shows a plot for all German compounds
that have one of the heads -arbeit, -ministerium, -mann or -stadt. Hence, the
two plots illustrate the difference between learning vector representations
for compound modifiers or heads. Words with the same modifier do not
necessarily appear in close proximity in the embedding space. This is
particularly true for modifiers that can be applied liberally to many head
words, such as Super- or Kinder-. On the other hand, compounds with the same
head are close in the embedding space. This observation is crucial to our
method, as we aim to find direction vectors that generalize to as many word
pairs as possible.

(a) Compounds with the same modifier.

(b) Compounds with the same head.

Figure 1: Semantic representations of compounds based on (a) their modifiers
and (b) their heads.

##  4 Compound induction from word embeddings

###  4.1 Compound extraction

#### Candidate extraction

We compile an initial set of modifier candidates by extracting all possible
prefixes with a minimum length of 4 characters.666For efficient computation,
we use a directed acyclic word graph: https://pypi.python.org/pypi/pyDAWG. We
retain a modifier as a candidate if both the modifier and the rest of the
word, i.e. the potential head of the compound, occur in the vocabulary. The
initial candidate set contains 281K modifiers, which are reduced to 165K
candidates by removing the modifiers occurring in only one word. The length of
the average support set (i.e., the set of all compounds the modifier applies
to) is 13.5 words. Table 1a shows the ten candidate modifiers with the biggest
support sets. At this stage, the candidate set contains any modifier-head
split that can be observed in the data, including candidates that do not
reflect real compound splits.777For example, as Para (a river) and dies (this)
occur in the data, an incorrect candidate split occurs for Para|dies
(paradise). Compound splits are not applied recursively here, as we assume
that internal splits can be learned from the occurrences of the heads as
individual words.888For example, for _Haupt|bahn|hof_ (main train station), we
observe both _Haupt|bahnhof_ and _Bahn|hof_.

#### Prototype extraction

To find the prototype vectors that generalize best over the most words in the
support set, we apply the same recursive algorithm as ?). The algorithm
initially computes the direction vector for each _(modifier, compound)_ pair
in the support set by subtracting the embedding of the head from the embedding
of the compound, e.g.
${\uparrow}d_{\mathrm{\underline{dog}\emph{house}}}=v(\mathit{doghouse})-v(\mathit{house})$.
Each direction vector is then evaluated by applying it to all the word pairs
in the support set, for example
${v(\mathit{owner})+{\uparrow}d_{\mathrm{\underline{dog}\emph{house}}}\stackrel{{\scriptstyle?}}{{=}}v(\mathit{dogowner})}$
for the word pair dog|owner. If the resulting vector is close (according to
$E$) to the vector of the actual target compound, we add it to the evidence
set of the vector. The direction vector with the largest evidence set is
selected as a prototype. All pairs this prototype explains are then removed
and the algorithm is applied recursively until no direction vector explains at
least $t_{\text{evd}}$ compounds. As the evaluation function $E$ we use the
rank of the correct word in the list of predictions and experiment with
$t_{\text{evd}}=\\{10,6,4\\}$. Lastly, for efficient computation we sample the
evidence set down to a maximum number of $500$ words.

99footnotetext: Words are related to mouse pointer (_Zeiger_), biological
genus (_Stämme_), mouse costume (_Kostüm_) and control (_Steuerung_).

| Modifier | Support |  |  | Modifier | Support
---|---|---|---|---|---|---
1. | Land- | 8387 |  | 6. | Landes- | 5189
2. | Kinder- | 6249 |  | 7. | Schul- | 5011
3. | Haupt- | 5855 |  | 8. | Jugend- | 4855
4. | Lande- | 5637 |  | 9. | Ober- | 4799
5. | Stadt- | 5327 |  | 10. | Groß- | 4656
(a) Modifiers by size of support set.

Prototype | Evidence words
---|---
$v_{\text{-Zeiger}}$ |  | -Bewegung -Klicks -Klick -Tasten
---
-Zeiger
$v_{\text{-St\"{a}mme}}$ |  | -Mutanten -Gene -Hirnen -Stämme
---
$v_{\text{-Kost\"{u}m}}$ | -Knopf -Hirn -Hirns -Kostüm
$v_{\text{-Steuerung}}$ | -Ersatz -Bedienung -Steuerung
(b) Prototypes and evidence words for _Maus-_.

Table 1: Overall most common modifiers and the prototypes extracted for the
modifier _Maus-_.

###  4.2 Implementation considerations

We now turn to implementation considerations and perform an intrinsic
evaluation of the prototypes.

#### Word embeddings

We use the German data of the News Crawl Corpora
(2007-2014).101010http://www.statmt.org/wmt15/translation-task.html The text
is truecased and tokenized, and all punctuation characters are removed,
resulting in approximately 2B tokens and a vocabulary size of 3M. We use
word2vec to estimate the word
embeddings.111111https://code.google.com/p/word2vec/ We train 500-dimensional
word embeddings using the skip-gram model, a window size of 5 and a minimum
word frequency threshold of 2. The latter ensures that we find word embeddings
for all words that occur at least twice in the corpus, which is useful as long
compounds may occur only very few times.

#### Treatment of interfixes (Fugenelemente)

For mostly phonetic reasons, German allows the insertion of a limited set of
characters between the modifier and the head. As learning this set is not the
aim of our work, we simply allow the fixed set of interfixes $\\{$-s-,
-es-$\\}$ to occur. For any combination of interfix and casing of the head
word, we add the tuple of the two to the support set of the corresponding
modifier.

#### What do the prototypes encode?

An inspection of the prototypes for each modifier shows that the differences
between them are not always clear cut. Often, however, each prototype
expresses one specific sense of the modifier. Table 1b illustrates this on the
example of the German modifier Maus- (Engl. mouse), which can refer to both
the animal and the computer device. Although there are more than two prototype
vectors, it is interesting to observe that the two word senses are almost
fully separated.

#### Calculating the hit rate

To evaluate the quality of the prototypes, we use the hit rate metric defined
by ?). A direction vector’s hit rate is the percentage of relevant word pairs
that can be explained by the vector. A prediction is explainable if the actual
target word is among the top $t_{\text{rank}}$ predictions and, optionally, if
the cosine similarity between the two is at least $t_{\text{sim}}$.

The implementation of this evaluation function $E$ requires the calculation of
the cosine distance between a newly created vector and the word vector of
every item in the vocabulary. Since this score is calculated $N$ times for
every of the $N$ word pairs (i.e., $N^{2}$ times), this is a computationally
extremely expensive process. For more efficient computation, we use an
approximate k-nearest neighbor search
method.121212https://github.com/spotify/annoy While this is not a lossless
search method, it offers an adjustable trade-off between the model’s
prediction accuracy and running time.131313With this fast approx. search
method the total training time would be just below 7 days if run on a single
16 core machine. For a standard setting
($t_{\text{evd}}=6,t_{\text{rank}}=80$), the hit rates using approximate and
exact rank are 85.9% and 60.9% respectively. This shows that the hit rates
obtained with the approximate method are more optimistic, which will affect
how the prototype vectors are extracted. Additionally, restricting both _rank_
and _similarity_ ($t_{\text{rank}}=80,t_{\text{sim}}\geq 0.5$) leads to lower
hit rates (25.9% for approximate and 15% for exact rank).

#### Influence of thresholds

Table 2 compares the parameters of our model based on (a) the mean hit rate,
(b) cosine similarity, (c) the percentage of candidate modifiers with at least
one prototype and (d) the mean number of prototypes per rule. Higher values of
$t_{\text{evd}}$ (minimum evidence set size) lead to better quality in terms
of hit rate and cosine similarity as prototypes have to be able to cover a
larger number of word pairs in order to be retained. The rank threshold
$t_{\text{rank}}$ also behaves as expected. Reducing $t_{\text{rank}}$ to 80
means that the predicted vectors are of higher quality as they need to be
closer to the true compound embeddings. Tables (c) and (d) illustrate that the
more restrictive parameter settings reduce the amount of modifiers for which
prototypes can be extracted. From a total of 165399 candidate prefixes, only
3%-10% are retained in the end for our settings. Similarly, the average number
of prototypes per modifier also decreases with more restrictive settings.
Interestingly, however, for the most restrictive setting ($t_{\text{evd}}=10$,
$t_{\text{rank}}=80$), this number is still a relatively high 2 prototypes per
vector.

| (a) Mean hit rate |  | (b) Mean cosine sim. |  | (c) % with prototypes |  | (d) Mean # of prototypes
---|---|---|---|---|---|---|---
$t_{\text{rank}}=$ |  $80$ |  $100$ |  |  $80$ |  $100$ |  |  $80$ |  $100$ |  |  $80$ |  $100$
$t_{\text{evd}}=4$ |  26% |  22% |  |  0.39 |  0.39 |  |  8.93% |  9.52% |  |  4.20 |  4.16
$t_{\text{evd}}=6$ |  31% |  26% |  |  0.43 |  0.43 |  |  5.13% |  5.47% |  |  3.29 |  3.30
$t_{\text{evd}}=10$ |  36% |  31% |  |  0.45 |  0.45 |  |  2.91% |  3.14% |  |  2.25 |  2.29
Table 2: Overview of the influence of the hyperparameters on prototype
extraction.

###  4.3 Compound splitting

To obtain a clearer view of the quality of the extracted compound
representations, we apply the prototypes to a compound splitting task.

#### Splitting compounds by semantic analogy

The extracted compound modifiers and their prototypes can be employed directly
to split a compound into its components. Algorithm 4.3 presents the greedy
algorithm applied to every word in the text. $\mathit{V}$ is the word
embedding vocabulary, $\mathit{M}$ is the set of extracted modifiers with
their prototypes, and Prefixes(.) is a function returning all string prefixes.

[h]

1:procedure Decompound($\mathit{word}$, $\mathit{V}$, $\mathit{M}$)

2: $\mathit{modifiers}\leftarrow\left\\{m\mid
p\leftarrow\textsc{Prefixes}(\mathit{word})\text{ if }p\in\mathit{M}\right\\}$

3: if $\mathit{modifiers}=\emptyset$ OR $\mathit{word}\notin\mathit{V}$ then

4: return $\mathit{word}$

5: $\mathit{bestModifier}\leftarrow\emptyset$

6: for $\mathit{modifier}\in\mathit{modifiers}$ do

7: $\mathit{head}\leftarrow\mathit{word}$ without $\mathit{modifier}$
$\triangleright$ e.g. house $\leftarrow$ doghouse without dog-

8: if $\mathit{head}\in\mathit{V}$ then

9: for
$(\mathit{head}_{\mathit{proto}},\mathit{word}_{\mathit{proto}})\in\mathit{modifier}$
do

10: Evaluate “$\mathit{word}$ is to $\mathit{head}$ what
$\mathit{word}_{\mathit{proto}}$ is to $\mathit{head}_{\mathit{proto}}$”

11:$\triangleright$ e.g. doghouse is to house what dogowner is to owner

12: Update $\mathit{bestModifier}$ if this is the best match so far

13: return $\mathit{word}$ split based on $\mathit{bestModifier}$

Greedy compound splitting algorithm.

Compounds may only be split if (a) the full compound word is in the vocabulary
$\mathit{V}$, i.e. it has been observed at least twice in the training data
(Line 3), (b) it has a string prefix in the modifier set and this modifier has
at least one prototype (Line 3), (c) the potential head word resulting from
splitting the compound based on the modifier is also in our vocabulary (Line
8). The last case, namely that the compound head candidate is not in the
vocabulary can occur for two reasons: either this potential head is a valid
word that has not been observed frequently enough or, the more common reason,
the substring is not a valid word in the language.141414For example, when
applying the algorithm to _Herrengarderobe_ (male cloak room), two possible
prefixes apply: _Herr_ and _Herren_. In the first case, the remaining slice is
_engarderobe_ , which is not a valid word and thus the candidate prefix is
discarded. The algorithm’s coverage can be increased by backing off to a
frequency-based method if conditions (a) or (c) are violated. The core of the
algorithm is the evaluation of meaning preservation in Line 10. This
evaluation is performed using the _rank_ -based and _cosine similarity_ -based
evaluation functions. Modifiers that do not pass the thresholds defined for
these functions are discarded as weak splits. To split compounds with more
than two components, the algorithm is applied recursively.

#### General evaluation

We use the test set from ?), which contains a list of 54569 compounds
annotated with binary splits. As we only consider prefixes with a minimal
length of 4 characters, we filter the test set accordingly, leaving 50651
compounds. Moses [Koehn et al. (2007] offers a compound splitter that splits a
word if the geometric average of the frequencies of its components is higher
than the frequency of the compound. We trained two instances of this compound
splitter to use as references: one using the German monolingual dataset used
to train the Word2Vec models and a second using a subset of the previous
dataset.151515Subset: _News Crawl 2007-2009_ (275M tokens, 2.09M types). Full
set: _News Crawl 2007-2014_ (2B tokens, 3M types). Unlike our method, the two
baseline systems do not consider the meaning preservation criteria of the
compound splitting rules that are applied. Results for the full test set
(accuracy and coverage, i.e. $\frac{|\text{correct
splits}|}{|\text{compounds}|}$ and $\frac{|\text{compounds
split}|}{|\text{compounds}|}$) are presented in the first row of Table 3b.

#### Evaluation of highly ambiguous compounds

The strength of our method resides in the capacity to discriminate good
candidate splits from bad ones. By capturing the meaning relation between
compounds and their components, we are able to decide for a given word which
splitting rule is the most appropriate. With this in mind, our approach should
stand out in contexts where multiple split points may apply to a compound. We
simulate different ambiguity scenarios based on Henrich and Hinrich’s gold
standard dataset: We extract compounds for which we find 2, 3, 4, and 5
potential split points.161616Each string prefix which occurs as a separate
word produces a potential split point (indicated by  ). The potential split
points may not be linguistically motivated and can lead to correct
(_general|stabs_) or incorrect splits (_gene rals  tabs_). Examples include
Einkauf  s  wagen, Eis  en  bahn  unternehmen, Wissen  s  chaft  s  park and
Gene  ra  l  s  tab  s.  The resulting test sets consists of 18571, 1815, 842
and 104 compounds, respectively. For all compound splitting experiments, we
use the prototype vectors extracted with the parameters $t_{\text{evd}}=6$ and
$t_{\text{rank}}=100$.

$2$$3$$4$$5$$0$$20$$40$$60$AmbiguityAccuracy and Coverage (%)Acc. This
workAcc. Moses p.Acc. Moses f.Cov. This workCov. Moses p.Cov. Moses f.

(a) Evaluation of highly ambiguous compounds.

This work Moses (partial) Moses (full) Scenario Acc. Cov. Acc. Cov. Acc. Cov.
Full test set 27.43 58.45 18.04 31.41 6.57 13.75 2 splits 24.94 56.75 13.13
20.13 1.79 3.11 3 splits 21.10 68.37 8.04 18.35 1.21 2.92 4 splits 22.09 62.11
9.98 15.91 1.19 1.90 5 splits 24.04 69.23 9.62 11.54 0.96 1.92

(b) Evaluation of all compounds and highly ambiguous compounds only.

Table 3: Gold standard evaluation of compound splitting.

Table 3b presents accuracy and coverage for the compounds within the different
ambiguity scenarios. To better visualize the trends for highly ambiguous
compounds, we plot the accuracy and coverage scores in relation to the
ambiguity of the compounds in Table 3a. The analogy-based method outperforms
the frequency-based baselines in both coverage and accuracy. While for the
Moses splitter, the coverage decreases with increasing ambiguity, the opposite
behavior is shown by our approach, as having more possible splits results in a
higher number of direction vectors increasing the likelihood of obtaining
meaning-preserving splits. This experiment shows that the analogy-based
compound splitter is advantageous for words that can potentially be explained
by several candidate splits.

##  5 Compound splitting for machine translation

#### Translation setup

We use the Moses decoder [Koehn et al. (2007] to train a phrase-based MT
system on the English–German _Common crawl_ parallel corpus and _WMT news
test_ 2010 (tuning). Word alignment is performed with Giza++ [Och and Ney
(2003]. We use a 3rd order language model estimated using IRSTLM [Federico et
al. (2008], as well as lexicalized reordering. The test data set is _WMT news
test_ 2015,171717http://www.statmt.org/wmt15/translation-task.html which
contains approx. 2100 de-en sentence pairs and 10000 tokens (with one
reference translation). We compare our method against a baseline translation
system with no compound splitter, and the same system implementing Moses’
default compound splitting tool. The test set contains 2111 out-of-vocabulary
word types (natural OOV words), which yields a total of 2765 unknown tokens,
consisting mostly of compounds, brand names, and city names. This implies that
22.16$\%$ (word types) resp. 7.15$\%$ (tokens) of the test corpus are unknown
to the baseline system.

#### Translation experiments

|  | (a) No comp. splitting |  | (b) OOV only |  | (c) Rare: $\text{c(w)}<20$ |  | (d) All words
---|---|---|---|---|---|---|---|---
|  |  Splits |  BLEU |  MTR |  |  Splits |  BLEU |  MTR |  |  Splits |  BLEU |  MTR |  |  Splits |  BLEU |  MTR
Moses splitter |  |  0 |  17.6 |  25.5 |  |  226 |  17.6 |  25.7A |  |  231 |  17.6 |  25.7 |  |  244 |  17.9 |  25.8A
This work |  |  |  |  |  |  317 |  17.6 |  25.8A |  |  744 |  18.2ABC |  26.1ABC |  |  1616 |  17.7 |  26.3A

A Stat. sign. against (a) at $p<0.05$ B Stat. sign. against Moses splitter at
same c(w) at $p<0.05$ C Stat. sign. against best Moses splitter (d) at
$p<0.05$

Table 4: Translation results for various integration methods.

To test the analogy-based compound splitter on a realistic setting, we perform
a standard machine translation task. We translate a German text using a
translation baseline system with no compound handling (a), a translation
system integrating the standard Moses compound splitter tool trained using the
best-performing settings, and a translation system using our analogy-based
compound splitter. We test the following basic methods of integration:
Splitting only words that are OOV to the translation model (b), splitting all
words that occur less than 20 times in the training corpus (c),and applying
the compound splitters to every word in the datasets (d). Table 4 shows the
results of these translation experiments. For each experiment, we report BLEU
[Papineni et al. (2002], METEOR [Denkowski and Lavie (2014], and the number of
compound splits performed on the test set. Statistical significance tests are
performed using bootstrap resampling [Koehn (2004].

#### Discussion

The results show that when applied without restrictions, our method splits a
large number of words and leads to minor improvements. When applied only to
rare words the splitter produces statistically significant improvements in
both BLEU and METEOR over the best frequency-based compound splitter. This
difference indicates that a better method for deciding which words the
splitter should be applied to could lead to further improvements. Overall, the
output of the analogy-based compound splitter is more beneficial to the
machine translation system than the baseline splitter.

##  6 Conclusion

In this paper, we have studied whether regularities in the semantic word
embedding space can be exploited to model the composition of compound words
based on analogy. To approach this question, we made the following
contributions: First, we evaluated whether properties of compounds can be
found in the semantic vector space. We found that this space lends itself to
modeling compounds based on their semantic head. Based on this finding, we
discussed how to extract compound transformations and prototypes following the
method of ?) and proposed an algorithm for applying these structures to
compound splitting. Our experiments show that the analogy-based compound
splitter outperforms a commonly used compound splitter on a gold standard
task. Our novel compound splitter is particularly adept at splitting highly
ambiguous compounds. Finally, we applied the analogy-based compound splitter
in a machine translation task and found that it compares favorably to the
commonly used shallow frequency-based method.

#### Acknowledgements

Joachim Daiber is supported by the EXPERT (EXPloiting Empirical appRoaches to
Translation) Initial Training Network (ITN) of the European Union’s Seventh
Framework Programme. Stella Frank is supported by funding from the European
Union’s Horizon 2020 research and innovation programme under grant agreement
Nr. 645452.

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