# Hyperbolic enhancement of a quantum battery

C. A. Downing<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Exeter, Exeter EX4 4QL, United Kingdom  M. S. Ukhtary
Department of Physics and Astronomy, University of Exeter, Exeter EX4 4QL,
United Kingdom  Research Center for Quantum Physics, National Research and
Innovation Agency (BRIN), South Tangerang 15314, Indonesia

###### Abstract

A quantum system which can store energy, and from which one can extract useful
work, is known as a quantum battery. Such a device raises interesting issues
surrounding how quantum physics can provide certain advantages in the
charging, energy storage or discharging of the quantum battery as compared to
their classical equivalents. However, the pernicious effect of dissipation
degrades the performance of any realistic battery. Here we show how one can
circumvent this problem of energy loss by proposing a quantum battery model
which benefits from quantum squeezing. Namely, charging the battery
quadratically with a short temporal pulse induces a hyperbolic enhancement in
the stored energy, such that the dissipation present becomes essentially
negligible in comparison. Furthermore, we show that when the driving is strong
enough the useful work which can be extracted from the quantum battery, that
is the ergotropy, is exactly equal to the stored energy. These impressive
properties imply a highly efficient quantum energetic device with abundant
amounts of ergotropy. Our theoretical results suggest a possible route to
realizing high-performance quantum batteries, which could be realized in a
variety of platforms exploiting quantum continuous variables.

Introduction
The emerging field of quantum energy science seeks to discover unprecedented
improvements in an array of promising energy technologies, from quantum solar
power to nuclear fusion Metzler2023 . Amongst these innovations, the necessity
to improve the world’s existing energy storage technology ensures that quantum
battery research is of utmost importance. Interestingly, as well as quantum
physics and engineering, quantum battery research encompasses both quantum
thermodynamics and quantum information science, such that this subfield poses
appealing fundamental and applied challenges Campaioli2023 .

Early and inventive experiments probing the transfer and storage of energy in
the quantum world have already revealed certain quantum advantages. During the
charging process, superextensive charging rates have been reported in an
organic microcavity Quach2022 , quantum correlations-boosted charging has been
demonstrated in a spin system Joshi2022 , and the manipulation of bright and
dark states has led to stable charging in a superconducting quantum battery
Hu2022 . Meanwhile, careful measurements of the energy transfer between light
and qubits has benchmarked the current capabilities for controlling quantum
energy and information flow Stevens2022 ; Maillette2022 .

However, some problems which harm the expected utility of quantum batteries
remain. Dissipation is intrinsic to open quantum systems, so how to
effectively combat energy loss from a quantum battery? Thermodynamically,
there is a maximum amount of work which can be reasonably extracted from a
quantum storage device (the so-called ergotropy) Allahverdyan2004 ; Alicki2013
, so how to optimize this measure instead of the overall stored energy? A
stream of theoretical proposals for model quantum batteries have sought to
address these important issues, as well as related questions concerning the
speed-up of the charging and discharging processes Andolina2018 ; Friis2018 ;
Farina2019 ; Santos2020 ; NewCrescente2020 ; Santos2021 ; Shaghaghi2002 ;
Gyhm2022 ; Konar2022 ; Mazzoncini2023 ; Centrone2021 ; Catalano2023 ; Song2024
; Ukhtary2024 . Here we suggest a rather different approach based upon quantum
squeezing. We consider a bipartite quantum battery model, where the battery
charger is coupled to the battery holder [cf. the sketch of Fig. 1].
Importantly, we propose that the charger is driven by a quadratic Savona2017 ;
Rota2019 ; Garbe2020 ; Campinas2023  temporal pulse, which induces squeezing
into the system. Consequentially, there is a hyperbolic enhancement in the
energy stored in the battery holder, such that the inevitable energy loss in
the system is made effectively negligible. Previously, continuous (rather than
pulsed) two-photon drives have been studied theoretically within the context
of quantum batteries in Ref. Sassetti2020 ; Downing2023 ; Fan2023 .

Figure 1:  The pulsed quadratic quantum battery. A representation of the
considered bipartite quantum battery model of Eq. (1), associated with the two
bosonic operators $a$ and $b$ [cf. Eq. (2) and Eq. (3)]. The battery charger
is coupled (grey bar) at the rate $g$ to the battery holder [cf. Eq. (4)],
which has the level spacing $\omega_{b}$ (white lines). The charger is driven
by a quadratic pulse (red arrows) of dimensionless strength $\Omega$ [cf. Eq.
(5)], and suffers from dissipation (cyan arrow) at the decay rate $\gamma$
[cf. Eq. (6)]. In this sketch, there is an example stored energy
$E=3\omega_{b}$ in the battery holder [cf. Eq. (7)].

The total Hamiltonian operator $\hat{H}$ of the composite quantum battery
system, as sketched in Fig. 1, can be decomposed into four parts,

$\hat{H}=\hat{H}_{a}+\hat{H}_{b}+\hat{H}_{a-b}+\hat{H}_{d},$ (1)

which account for the battery charger and holder energies, the charger-holding
coupling, and the driving of the battery charger as follows (we take $\hbar=1$
here and throughout)

$\displaystyle\hat{H}_{a}$ $\displaystyle=\omega_{b}\,a^{\dagger}a,$ (2)
$\displaystyle\hat{H}_{b}$ $\displaystyle=\omega_{b}\,b^{\dagger}b,$ (3)
$\displaystyle\hat{H}_{a-b}$
$\displaystyle=g\left(a^{\dagger}b+b^{\dagger}a\right),$ (4)
$\displaystyle\hat{H}_{d}$
$\displaystyle=\tfrac{\Omega}{2}\,\delta(t)\left(a^{\dagger}a^{\dagger}+aa\right).$
(5)

The bosonic creation and annihilation operators $b^{\dagger}$ and $b$ raise
and lower respectively the number of excitations in the battery holder, and
they are subject to the commutation rule $[b,b^{\dagger}]=1$, while
$a^{\dagger}$ and $a$ behave similarly for the battery charger. We consider
the energy level spacing $\omega_{b}$ for the charger and the holder to be the
same [cf. Eq. (2) and Eq. (3)], and there is a coupling rate $g>0$ between
them which mediates the energy transfer [cf. Eq. (4)]. In turn, the battery
charger is driven quadratically with a short temporal pulse of dimensionless
strength $\Omega$, and $\delta(x)$ is the delta function of Dirac [cf. Eq.
(5)]. Experimentally, such kinds of parametric (or two-photon) drives have
been recently realized using a variety of superconducting circuits Leghtas2015
; Wang2019 ; Chien2020 ; Gaikwad2023 , while quantum harmonic oscillators are
known to nicely model realistic quantum batteries Qu2023 .

The Hamiltonian dynamics of Eq. (1) can be upgraded with the aid of a
Gorini–Kossakowski–Sudarshan–Lindblad quantum master equation in order to
account for the inevitable energy loss from the battery charger into the
external environment. The dissipationless battery holder is assumed to be
isolated from the environment, since it is supposed to be able to store energy
on an exceedingly large timescale. This approximation is indeed close to the
current experimental reality Quach2022 ; Joshi2022 ; Hu2022 ; Stevens2022 ;
Maillette2022 . The density matrix $\rho$ of the overall quantum battery is
then governed by the master equation Breuer2002

$\partial_{t}\rho=\mathrm{i}[\rho,\hat{H}]+\frac{\gamma}{2}\left(2a\rho
a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a\right),$ (6)

where the Hamiltonian operator $\hat{H}$ is defined by Eq. (1), and where
$\gamma\geq 0$ is the decay rate of the battery charger. This completes the
mathematical description of the model, as represented pictorially in Fig. 1.
The average value of the battery holder population $\langle
b^{\dagger}b\rangle$, which can be accessed from Eq. (6) after employing the
trace property $\operatorname{Tr}(\mathcal{O}\rho)=\langle\mathcal{O}\rangle$
with the density matrix $\rho$ (where $\mathcal{O}$ is some operator), then
gives rise to the energy $E$ stored in the quantum battery

$E=\omega_{b}\langle b^{\dagger}b\rangle.$ (7)

Determining this important energetic measure, which takes on the example value
of $E=3\omega_{b}$ in the cartoon battery imagined in Fig. 1, for the
introduced driven-dissipative model of Eq. (6) is a key aim of this
theoretical study, alongside finding the optimal time needed to achieve the
energetic maximum.

After the battery charger is disconnected from the battery holder, the
dissipationless holder is then able to store the energy $E$ indefinitely.
However, thermodynamically not all of this stockpiled energy is useful for
doing work. The useful energy stored in the quantum battery, or more formally
the so-called ergotropy $\mathcal{E}$, is given by the celebrated formula
Allahverdyan2004 ; Alicki2013

$\mathcal{E}=E-E_{\beta},$ (8)

which subtracts off from the stored energy $E$ of Eq. (7) the unuseful energy
$E_{\beta}$ of the passive state $\rho_{\beta}$. Within quantum continuous
variable theory Olivares2012 ; Serafini2017 , the passive state energy
$E_{\beta}$ can be written down exactly in terms of the second moments
$\langle b^{\dagger}b\rangle$ and $\langle bb\rangle$ of the battery holder,
as follows Farina2019 ; Downing2023

$\displaystyle E_{\beta}$
$\displaystyle=\omega_{b}\left(\tfrac{\sqrt{\mathcal{D}}-1}{2}\right),$ (9)
$\displaystyle\mathcal{D}$ $\displaystyle=\left(1+2\langle
b^{\dagger}b\rangle\right)^{2}-4\big{|}\langle bb\rangle\big{|}^{2},$ (10)

since here all first moments are zero (that is $\langle b\rangle=\langle
b^{\dagger}\rangle=0$, as is shown in the Supplemental Material
SupplementalMaterial ). In what follows, we describe analytically the
energetic $E$ and ergtropic $\mathcal{E}$ measures of the presented model,
revealing the high performing nature of the proposed quadratic quantum
battery. We also note that whilst we consider a delta pulse in Eq. (5), our
forthcoming analytic results are also applicable to driving pulses with some
finite temporal width $\tau$, as long as the timescale $\omega_{b}\tau\ll 1$
is observed.

Figure 2:  Energetics of the quantum battery. Panel (a): the ergotropy
$\mathcal{E}$ of the quantum battery in units of
$\omega_{b}\sinh^{2}(\Omega)$, where $\omega_{b}$ is the energy level spacing
and $\Omega$ is the strength of the driving pulse, as a function of time (in
units of the inverse decay rate $1/\gamma$) [cf. the definitions of Eq.
(8)–Eq. (10), along with the results of Eq. (24) and Eq. (25)]). We consider
the case when the coupling strength $g=2\gamma$, and increasing large values
of $\Omega$ are marked with increasingly light (dark green to yellow) lines.
Purple line: the stored energy $E$ [cf. Eq. (17)]. Yellow line: the strong
driving approximation of Eq. (27). Dark green line: the weak driving
approximation of Eq. (26). Panel (b): a log-log plot of the time $t_{E}$ taken
(in units of $1/\gamma$) to reach the maximum in stored energy $E$, as a
function of the coupling-to-dissipation ratio $g/\gamma$ [cf. Eq. (18)].
Dashed cyan line: the weak coupling approximation of Eq. (19). Dashed pink
line: the strong coupling approximation of Eq. (20). Panel (c): a log-log plot
of the maximum stored energy $E(t_{E})$, in units of
$\omega_{b}\sinh^{2}(\Omega)$, as a function of $g/\gamma$ [cf. Eq. (21)].
Dashed cyan line: the weak coupling approximation of Eq. (22). Dashed pink
line: the strong coupling approximation of Eq. (23).

Results
The equation of motion for the second moments of interest, for all times $t>0$
after the action of the driving pulse appearing in Eq. (5), is given by the
Schrödinger-like equation

$\mathrm{i}\partial_{t}\psi=\mathcal{H}\psi,$ (11)

where the mean operators are contained within $\psi$, and where the four-
dimensional dynamical matrix $\mathcal{H}$ are together defined by

$\psi=\begin{pmatrix}\langle a^{\dagger}a\rangle\\\ \langle
b^{\dagger}b\rangle\\\ \langle a^{\dagger}b\rangle\\\ \langle
b^{\dagger}a\rangle\\\
\end{pmatrix},\quad\quad\setstackgap{L}{1.1}\fixTABwidth{T}\mathcal{H}=\parenMatrixstack{-\mathrm{i}\gamma&0g-g\\\
00-gg\\\ g-g-\mathrm{i}\frac{\gamma}{2}0\\\ -gg0-\mathrm{i}\frac{\gamma}{2}}.$
(12)

The formal solution of Eq. (11) leads to the following expressions for the
average populations $\langle a^{\dagger}a\rangle$ and $\langle
b^{\dagger}b\rangle$ of the battery charger and battery holder respectively,

$\displaystyle\langle a^{\dagger}a\rangle$
$\displaystyle=C\left[\tfrac{1}{2}\left(\tfrac{g}{G}\right)^{2}+\tfrac{8g^{2}-\gamma^{2}}{16G^{2}}\cos\left(2Gt\right)-\tfrac{\gamma}{4G}\sin\left(2Gt\right)\right]\mathrm{e}^{-\frac{\gamma
t}{2}},$ $\displaystyle\langle b^{\dagger}b\rangle$
$\displaystyle=C\left(\tfrac{g}{G}\right)^{2}\sin^{2}\left(Gt\right)\mathrm{e}^{-\frac{\gamma
t}{2}},$ (13)

which gives rise to the renormalized coupling strength $G$, which is defined
as

$G=\sqrt{g^{2}-\left(\tfrac{\gamma}{4}\right)^{2}},\quad\quad\quad\Gamma=\sqrt{\left(\tfrac{\gamma}{4}\right)^{2}-g^{2}},$
(14)

and where we also introduced the related renormalized decay rate $\Gamma$ for
use later on. The solutions of Eq. (Hyperbolic enhancement of a quantum
battery) are given in terms of an unknown constant prefactor $C$, which can be
determined by the boundary conditions of the problem as defined by the
parametric drive. Exactly at $t=0$ the quadratic pulse of Eq. (5) acts, which
couples a wider variety of moments such that Eq. (11) becomes ten-dimensional
(as shown in the Supplemental Material SupplementalMaterial ). Integrating
this generalized equation of motion, using the integrating factor technique
for example, reveals that the $C$ prefactor is simply

$C=\sinh^{2}\left(\Omega\right).$ (15)

The consequence of the quadratic drive of Eq. (5) has been to introduce a
degree of quantum squeezing Walls1983 ; Loudon1987  into the quantum battery
system. This has manifested in the hyperbolic enhancement factor $C$ of Eq.
(15), where the dimensionless drive strength $\Omega$ resembles a kind of
squeezing parameter (as is discussed in the Supplemental Material
SupplementalMaterial ). Importantly, such a striking hyperbolic enhancement as
$C$ is not seen in linearly driven models due to the absence of squeezing in
that more standard case Ukhtary2024 .

Most notably, the mean populations of Eq. (Hyperbolic enhancement of a quantum
battery) also exhibit an exceptional point when $g=g_{\mathrm{EP}}$, where we
introduce the key value [cf. Eq. (14)]

$g_{\mathrm{EP}}=\tfrac{\gamma}{4},$ (16)

which marks where two of the eigenvalues and eigenvectors associated with Eq.
(12) coalesce, in a type of spectral singularity Berry2003 ; Miri2019 ;
Downing2021 ; Fox2023 . The exceptional point of Eq. (16) demarcates the
borderland between oscillatory population cycles and non-oscillatory
population dynamics in the quantum battery. This drastic change in the
dynamics is noticeable from the stored energy $E$, which follows from Eq. (7)
and Eq. (Hyperbolic enhancement of a quantum battery) as

$E=\begin{cases}\omega_{b}\sinh^{2}\left(\Omega\right)\left(\tfrac{g}{\Gamma}\right)^{2}\sinh^{2}\left(\Gamma
t\right)\mathrm{e}^{-\frac{\gamma t}{2}},&g<g_{\mathrm{EP}},\\\
\omega_{b}\sinh^{2}\left(\Omega\right)\left(\frac{\gamma
t}{4}\right)^{2}\mathrm{e}^{-\frac{\gamma t}{2}},&g=g_{\mathrm{EP}},\\\
\omega_{b}\sinh^{2}\left(\Omega\right)\left(\tfrac{g}{G}\right)^{2}\sin^{2}\left(Gt\right)\mathrm{e}^{-\frac{\gamma
t}{2}},&g>g_{\mathrm{EP}}.\end{cases}$ (17)

In particular, there is an unusual quadratic dependence exactly at the
exceptional point, inbetween the more standard hyperbolic and trigonometric
dynamics. There is always an exponential decay of the stored energy $E$ with
the time constant $2/\gamma$ due to the lossy battery charger, as depicted in
Fig. 1. Most importantly, Eq. (17) demonstrates a hyperbolic enhancement in
energy thanks to the $C$ factor of Eq. (15), making the pulsed quadratic
battery an interesting proposition for a quantum storage device exploiting
squeezing (as is further discussed in the Supplemental Material
SupplementalMaterial ). We plot the dynamical energy $E$ of Eq. (17), in units
of $\omega_{b}\sinh^{2}(\Omega)$ and for the example case with the coupling
$g=2\gamma$, as the purple line in Fig. 2 (a). This graph displays some
hallmarks of the proposed quantum battery model in the desirable
$g>g_{\mathrm{EP}}$ regime, including exponentially damped oscillations and
exact energetic zeroes at certain times when the condition $t=n\pi/G$ (where
$n$ is an integer) is fulfilled.

Clearly from the purple line (marking the stored energy $E$) in Fig. 2 (a),
the temporal position of the first energetic peak – which is the global
maximum in the energy $E$ – is of great importance. Indeed, the knowledge of
this energetic maximum allows one to find the optimal charging time $t_{E}$,
where $E(t_{E})=\mathrm{max}_{t}\\{E(t)\\}$. The turning points of Eq. (17)
suggest that the optimal charging times $t_{E}$ are given by

$t_{E}=\begin{cases}\frac{\operatorname{arctanh}\left(\frac{4\Gamma}{\gamma}\right)}{\Gamma},&g<g_{\mathrm{EP}},\\\
\frac{1}{g_{\mathrm{EP}}},&g=g_{\mathrm{EP}},\\\
\frac{\arctan\left(\frac{4G}{\gamma}\right)}{G},&g>g_{\mathrm{EP}},\end{cases}$
(18)

which is plotted as the purple line in Fig. 2 (b), as a function of the
crucial coupling-to-dissipation ratio $g/\gamma$. The asymptotics of Eq. (18)
at small and large ratios of $g/\gamma$ makes the scaling of the optimal time
$t_{E}$ even more explicit as

$\displaystyle\lim_{g\ll g_{\mathrm{EP}}}t_{E}$
$\displaystyle=\frac{4}{\gamma}\ln\left(\tfrac{\gamma}{2g}\right),$ (19)
$\displaystyle\lim_{g\gg g_{\mathrm{EP}}}t_{E}$
$\displaystyle=\frac{\pi}{2g}-\frac{\gamma}{4g^{2}},$ (20)

which are marked with the dashed cyan and dashed pink lines respectively in
Fig. 2 (b). Intuitively, with weaker couplings $g$ the optimal charging time
$t_{E}$ increases – and in fact it eventually becomes logarithmically
divergent [cf. Eq. (19)]. Conversely, stronger couplings $g$ lead to quicker
optimal times $t_{E}$, which rapidly approaches the inverse-$g$
dissipationless result of $t_{E}=\pi/2g$ [which itself comes from the first
maximum of the sinusoidal function, cf. Eq. (20)].

The energetic maximums $E(t_{E})$ themselves are determinable directly from
Eq. (17) with Eq. (18), leading to the compact results

$E\left(t_{E}\right)=\begin{cases}\omega_{b}\sinh^{2}\left(\Omega\right)\mathrm{e}^{-\frac{\gamma}{2\Gamma}\operatorname{arccoth}\left(\frac{\gamma}{4\Gamma}\right)},&g<g_{\mathrm{EP}},\\\
\omega_{b}\sinh^{2}\left(\Omega\right)\mathrm{e}^{-2},&g=g_{\mathrm{EP}},\\\
\omega_{b}\sinh^{2}\left(\Omega\right)\mathrm{e}^{-\frac{\gamma}{2G}\operatorname{arccot}\left(\frac{\gamma}{4G}\right)},&g>g_{\mathrm{EP}},\end{cases}$
(21)

where the case exactly at the exceptional point ($g=g_{\mathrm{EP}}$) sees the
dimensionless number $1/\mathrm{e}^{2}\simeq 0.135$ arise. This optimal energy
$E(t_{E})$ is plotted as the the purple line in Fig. 2 (c) as a function of
the reduced coupling strength $g/\gamma$. The limiting behaviour of this
optimal energy follows from Eq. (21) as

$\displaystyle\lim_{g\ll g_{\mathrm{EP}}}E\left(t_{E}\right)$
$\displaystyle=\omega_{b}\sinh^{2}\left(\Omega\right)\left(\tfrac{2g}{\gamma}\right)^{2},$
(22) $\displaystyle\lim_{g\gg g_{\mathrm{EP}}}E\left(t_{E}\right)$
$\displaystyle=\omega_{b}\sinh^{2}\left(\Omega\right)\left(1-\tfrac{\pi\gamma}{4g}\right),$
(23)

which captures the quadratic in $g$ decrease in optimal energy at very small
couplings where $g\ll\gamma$ [cf. Eq. (22)], and defines the energetic upper
bound of $\omega_{b}\sinh^{2}(\Omega)$, which is approached inverse-linearly
in $g$ at large couplings where $g\gg\gamma$ [cf. Eq. (23)]. These asymptotics
are represented with the dashed cyan and dashed pink lines respectively in
Fig. 2 (c), completing the basic energetic analysis of the proposed pulsed
quantum battery.

In the discharging phase, when the battery charger is disconnected from the
battery holder, the maximal amount of work which can be extracted from the
quantum battery (via unitary operations) is governed by the ergotopy
$\mathcal{E}$ Allahverdyan2004 ; Alicki2013 . For the case of the pulsed
quantum battery, this important measure follows directly from the definitions
of Eq. (8)–Eq. (10). Pleasingly, the results of Eq. (Hyperbolic enhancement of
a quantum battery) and Eq. (15), as well as auxiliary results presented in the
Supplemental Material SupplementalMaterial , allow for an explicit expression
for the influential quantity $\mathcal{D}$ to be determined [cf. Eq. (10)],
via

$\displaystyle\mathcal{D}$
$\displaystyle=1+4\sinh^{2}\left(\Omega\right)\mathcal{P}\left(1-\mathcal{P}\right),$
(24) $\displaystyle\mathcal{P}$
$\displaystyle=\left(\tfrac{g}{G}\right)^{2}\sin^{2}\left(Gt\right)\mathrm{e}^{-\frac{\gamma
t}{2}}.$ (25)

These brief formulae allow for the dynamical ergotropy $\mathcal{E}$ to
calculated analytically, the results of which are shown in Fig. 2 (a), where
increasing large values of the drive $\Omega$ are marked with increasingly
light (from dark green to light green to yellow) lines. The functional
behaviour of the ergotopy $\mathcal{E}$ is highly reminiscent of the stored
energy $E$ (purple line in the panel), essentially being damped sinusoidal-
like oscillations. In the two limiting cases of very weak and very strong
drives, the ergotropy $\mathcal{E}$ is captured by the exact asymptotics

$\displaystyle\lim_{\Omega\ll 1}\mathcal{E}$
$\displaystyle=\omega_{b}\sinh^{2}\left(\Omega\right)\mathcal{P}^{2},$ (26)
$\displaystyle\lim_{\Omega\gg 1}\mathcal{E}$
$\displaystyle=\omega_{b}\sinh^{2}\left(\Omega\right)\mathcal{P},$ (27)

where the dynamic function $\mathcal{P}$ was just introduced in Eq. (25). When
$\Omega\ll 1$, there is a significant difference between the stored energy $E$
(purple line) and the ergotropy $\mathcal{E}$ (dark green line) as shown in
Fig. 2 (a), signifying the inefficiency of this weak driving regime [cf. Eq.
(26)]. However, when $\Omega\gg 1$ the ergotropy $\mathcal{E}$ (yellow line)
swiftly approaches the stored energy $E$ (purple line) exactly [cf. Eq. (17)
and Eq. (27)]. Hence the pulsed quadratic battery exhibits a remarkable strong
driving regime with two highly desirable measures of performance: (i) it is
essentially perfectly efficient since $\mathcal{E}\simeq E$, and (ii) it is
able to store an abundance of energy thanks to the hyperbolic enhancement
carried by the all-important $C$ factor [cf. Eq. (15)].

Discussion
In conclusion, we have suggested a quantum continuous variable model of a
quantum battery, which features a hyperbolic enhancement of its stored energy
(due to its quadratic pulsed drive inducing quantum squeezing). This
remarkable feature acts to combat the unavoidable energy loss suffered by the
battery charger. Importantly, the ergotropy is essentially equal to the stored
energy for strong enough driving pulses, revealing the efficient nature of the
proposed quantum battery in the discharging phase. Our analytic theory also
allows for compact expressions for the optimal charging times and optimal
energetics of this high-performing quantum battery. The underlying idea of
exploiting squeezing in order to boost the achievements of the quantum battery
may also have some significance for wider efforts surrounding quantum
technological advancements in the generation, transfer and storage of energy
Deutsch2020 ; Auffeves2022 .

Acknowledgments
Funding: CAD is supported by the Royal Society via a University Research
Fellowship (URF/R1/201158) and by Royal Society Enhanced Research Expenses
which support MSU. CAD also gratefully acknowledges an Exeter-FAPESP SPRINT
grant with the Universidade Federal de São Carlos (São Paulo, Brazil).
Discussions: CAD thanks R. Bachelard, A. Cidrim, A. C. Santos and C. J.
Villas-Boas for fruitful discussions and for their hospitality during his
visits to UFSCar.

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2023), co-located with FOIS 2023, 19-20 July, 2023, Sherbrooke, Québec,
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# OPPO: An Ontology for Describing Fine-Grained Data Practices in Privacy
Policies of Online Social Networks

Sanonda Datta Gupta  Torsten Hahmann  School of Computing and Information
Science, University of Maine, ME, USA

(2023)

###### Abstract

Privacy policies outline the data practices of Online Social Networks (OSN) to
comply with privacy regulations such as the EU-GDPR and CCPA. Several
ontologies for modeling privacy regulations, policies, and compliance have
emerged in recent years. However, they are limited in various ways: (1) they
specifically model what is required of privacy policies according to one
specific privacy regulations – GDPR; (2) they provide taxonomies of concepts
but are not sufficiently axiomatized to afford automated reasoning with them;
and (3) they do not model data practices of privacy policies in sufficient
detail to allow assessing the transparency of policies. This paper presents an
OWL Ontology for Privacy Policies of OSNs – OPPO – that aims to fill these
gaps by formalizing detailed data practices from OSNS’ privacy policies. OPPO
is grounded in BFO, IAO, OMRSE, and OBI, and its design is guided by the use
case of representing and reasoning over the content of OSNs’ privacy policies
and evaluating policies’ transparency in a greater detailed.

###### keywords:

Privacy Policy Web Ontology Language (OWL) Conceptual Modeling Data Practices

##  1 Introduction

The widespread use of Online Social Network (OSN) raises many privacy
concerns, such as storing users’ personally identifiable information (PII)
longer than required or without following proper security mechanisms. To
mitigate such concerns, various privacy regulations (eg., GDPR [1] and
CCPA[2]) have been introduced, which require organizations to be transparent
about their data practices (collection, processing, or storage of users’
data). An essential aspect to comply with these regulations are the privacy
policies that describe the data practices of OSNs. However, privacy policies
are usually long and complex and often do not explain the data practices in
sufficient detail [3, 4]. For instance, a policy may mention that it follows
best practices in security without actually specifying what mechanisms it uses
and to which types of data (personal vs. non-personal) they apply. Such
generic or overly vague descriptions are indicators of an OSN’s lack of
transparency in informing the users of its actual data practices. In this
paper, we present the Ontology for Privacy Policies of OSNs (OPPO) that is
designed to encode the data practices from the privacy policies of OSNs (and
other companies) in as much detail as possible. The ontology is intended to
let the OSN, users, and others (e.g. privacy advocacy groups) query the formal
ontological representations of OSNs’ data practices for specific details or
the general level of detail they provide.

Prior research on formal representations of online privacy has focused on
extracting rights and obligations to check compliance with regulations [5, 6,
7] and developed privacy related vocabularies and taxonomies [8, 9, 10, 11,
12]. However, most of these works miss more detailed yet important concepts
and relations, such as response time for data rectification or erasure
requests, retention duration (limited or unlimited) for different data types
(personal vs. non-personal), or security mechanisms that are applied to
different storage entities (device vs. data center). These details are not
omitted just for a lack of methods that are able to extract such details but,
more fundamentally, the ontologies (e.g., Pronto [8] and PrivOnto [10]) lack
important concepts for modeling data practices at such levels of details in
the first place.

Towards the goal of filling this gap, we develop OPPO as an upper-level
ontological model for the domain of privacy policies and data practices. It is
grounded in top-level ontological distinctions and capable of describing data
practices at more granular levels. As a proof-of-concept of OPPO’s ability to
describe fine-grained data practices, we focus on introducing classes and
properties for capturing data retention, storage, and security practices in
detail, including aspects such as how users can request data rectification or
what types of security mechanisms are applied to various data types. OPPO is
developed following the methodology METHONTOLOGY [13] and is guided by a set
of 45 competency questions, manual analysis of the GDPR (Art. 5, 13–17, and
32), CCPA (Art. 1798.100, 1798.125, 1798.135, 1798.140, 1798.81.5), and the
privacy policies of ten major OSNs111WhatsApp [14], Signal [15], Telegram
[16], Twitter [17], Tiktok [18], WeChat [19], SnapChat [20], Reddit [21],
Pinterest [22], and Instagram [23]. OPPO is grounded in the Basic Formal
Ontology (BFO) [24] as well as the Information Artifacts Ontology (IAO) [25],
the Ontology for Biomedical Investigations (OBI) [26], and the Ontology for
Medically Related Social Entities (OMRSE) [27] as widely-used specializations
of BFO. OPPO further reuses parts of the Data Privacy Vocabulary (DPV) [9] and
the OWL Time ontology [28]. The ontology is shared via GitHub at
https://github.com/SanondaDattaGupta/OPPO-Ontology; it currently contains 60
classes and relations and is formalized in OWL2
[29]222OPPO_Ontology_Full_Import.ttl fully imports the ontologies BFO, IAO,
OBI, OMRSE, DPV and OWL-Time from which concepts are reused or extended.
Because only a small fraction of these concepts are relevant to OPPO, we also
provide a version of OPPO that imports only small subsets of IAO, OBI, OMRSE
and DPV that are limited to the classes (and their superclasses) and relations
that are reused or refined. This version is called
OPPO_Ontology_Minimal_Import.ttl and gives a better impression of the concepts
introduced by OPPO when loaded into an ontology editor like Protégé.. To
demonstrate its use and to verify its consistency, we populate the ontology
with data from the privacy policy of Telegram as an example. Its logical
consistency has been verified using the OWL Reasoner Hermit [30]. It has been
validated on a subset of the guiding competency questions, expressing and
executing them as SPARQL queries over the Telegram data.

##  2 Use Case

OPPO is designed as a tool for OSN users, privacy researchers, policy makers,
regulatory bodies, and organizations to help analyze how OSNs’ privacy
practices relate to different privacy regulations, identify inconsistencies
between practices, or better judge their transparency – just to name a few
examples. Generally, OPPO intends to serve as a tool to formulate and answer
questions about the data practices described in privacy policies and, thus,
achieve greater transparency while dealing with often long and complex
policies. For the first released version of OPPO described here, we focus on
capturing the practices that describe storage and retention of data, including
the security mechanisms an OSN may employ.

To guide OPPO’s development, we have defined 45 competency questions (CQ)
[31]333The complete set of competency questions are available from the GitHub
repository.. The competency questions help define what terms should be
included in the ontology and, later on, they serve as questions to measure
whether the ontology can express and answer the questions. These questions
include simple queries such as Where does the OSN store my information? to
more complex questions such as Which of my personal information will still be
available after I delete my account? Note that our competency questions are
not limited to questions about data storage and retention practices but also
cover other data practices such as collection or notification mechanisms as
well. However, in this paper, we focus on the 27 CQs that pertain to data
storage, retention, and security practices. In the following, we present three
of those competency questions to illustrate their coverage and the kind of
answers we would expect.

  1. 1.

Where does the social media site/company store my message?
An OSN may store users’ personal information in several places which differ
both in the type of storage (on a device vs. a data center) and the storage
location (Europe vs. United States). Hence, this CQ enables us to capture the
location and physical devices where an OSN may store users’ personal
information.

  2. 2.

What contents may be stored for a maximum of 12 months?
Both GDPR and CCPA require organizations to explicitly mention the retention
period of the collected personal information. While analyzing the ten privacy
policies, we found that some privacy policies (such as Telegram), specify
different duration description (such as maximum of X months or as long as they
need) for different information types. Thus, this CQ can help us capture and
query such details about retention practices and also identify where OSNs lack
specificity in such details.

  3. 3.

What types of security mechanism are applied to my photos and private chats?
Privacy regulations require organizations to describe how they ensure the
security of the retained information. While most organizations briefly
describe the security mechanisms, some organizations (such as Signal,
WhatsApp, and Telegram) do explicitly mention different security mechanisms
(hashing mechanism vs. encryption mechanism) that they apply to different
information types (public chat vs. media). Moreover, certain information types
are more sensitive (such as bio-metric or health information), and thus may
pose higher privacy risks, if not stored following sufficiently secure
mechanisms. Hence, this CQ exemplifies the kind of competency questions that
help evaluate how transparent and concrete OSNs are about their security and
data practices.

##  3 Related Work

In recent years, several vocabularies, ontologies, and conceptual models for
modeling privacy policies, regulations, and compliance have been introduced
[32, 33, 34, 11, 35] but are primarily concerned with modeling rights and
compliance issues with respect to regulations (e.g., BPR4GDPR [11] and SPECIAL
[35]) and permissions and prohibitions (e.g., CDMM [12], ODRL [36]). Most
closely related to our work are PrivOnto [10], PrOnto [8], and DPV [9], which
are vocabularies and conceptual models that also cover – to some extent – data
practices. However, they do not capture more granular aspects of data
practices such as the duration for which data is stored, the location it is
stored in, the specific security protocols that are employed, or the specific
process to request to rectify user data. While outlining such detailed
practices is not explicitly required by most existing privacy regulations, it
can help organizations be more transparent. In the following, we briefly
describe the differences in scope, representation, and other limitations of
PrivOnto, PrOnto and DPV that distinguish them from OPPO.

The purpose of PrivOnto [10] is quite different from OPPO: it is designed to
annotate paragraphs from policies with concepts from their vocabulary as a
kind of semantic tagging but PrivOnto does not allow encoding the content of
the policies in a formal representation that it can be automatically reasoned
with later on. Moreover, the development of PrivOnto predates the release of
GDPR and other privacy regulations; hence the concepts do not tie well to data
practices mentioned in regulations.

PrOnto [8] overlaps with OPPO in aspects such as the modeling of different
privacy-related data types but lacks a full coverage of types of personal
data, such as financial, identity or activity data that we add. A more
fundamental difference between PrOnto and OPPO is that Pronto is encoded in
the LegalRuleML [37] language, a defeasible logic, which restricts the ability
to automatically reason with PrOnto444Reasoning with OWL2 ontologies is
supported by a wide range of off-the-shelf reasoners but we are not aware of
any tools that can reason directly with LegalRuleML. Instead, specifications
in LegalRuleML first need to be converted to another format to enable
reasoning as described in [38]. or to integrate it with other non-legal
ontologies.

DPV [9] provides a comprehensive set of privacy-related terms, including for
different data types, purposes, legal entities, and data processing operations
(e.g. transmit or store) and encodes it in OWL2. However, it still is more of
a taxonomy that does not describe or restrict how these concepts are related
to one another axiomatically. Moreover, DPV does not tie the concepts to top-
level concepts and does not use ontological analysis tools to structure its
taxonomies. For example, while DPV loosely defines a group of _legal roles_ ,
data subjects or regulation authorities are not treated as such. Likewise,
multiple distinctions between different types of personal data are made, but
these distinctions are not integrated into a single coherent taxonomy.
However, we reuse some of DPV’s terms where appropriate. But OPPO still
introduces an additional 60 concepts such as specific security mechanisms
(encryption and hashing), different data types (personal and non-personal),
duration descriptions (definite and indefinite) and location descriptions
(storage type and spatial location). OPPO further introduces concepts to model
specific practices, such as for describing how an OSN deals with requests to
rectify or delete data, and the associated properties (e.g., request and
response type and response delay). These allow capturing data practices of
OSNs more fully and in greater detail, thus enabling querying and evaluating
the transparency and level of detail across OSNs.

##  4 The Conceptual Model of OPPO

Figure 1: OPPO’s high-level concepts and how they extend concepts from BFO,
IAO, OBI, and OMRSE. Concepts in yellow are introduced by OPPO, while all
other concepts are being reused.

OPPO builds and reuses a number of existing ontologies. We directly reuse the
classification of purposes from DPV [9] and time-related concepts (e.g. to
capture storage duration) from OWL-Time [39] as discussed in more detail later
on. In addition, we use top- and mid-level ontologies to ground OPPO concepts.
Because privacy policies and data practices are primarily about information,
we rely on the Information Artifact Ontology (IAO)555Available from
https://github.com/information-artifact-ontology/IAO [25] as the main source
for upper-level reference concepts while OMRSE [27] and OBI [26] provide
additional high-level role classifications of interest to OPPO. All of IAO,
OMRSE, and OBI build on the top-level ontology BFO [24]. Thus, to maximize
compatibility we also reuse BFO as top-level ontology. Figure 1 demonstrates
how OPPO’s most generic concepts relate to BFO and the other ontologies. In
the remainder of this section we will provide overview of OPPO’s key concepts
and their relationship to top- and mid-level ontologies.

#### Information content entities:

Privacy policies describe the data practices an OSN employs to collect, store
and process users’ data in compliance with privacy regulations. We have
identified the four underlined concepts as central to this endeavor, which all
are modeled as subclasses of IAO:InformationContentEntity: (i) a
PrivacyRegulation such as GDPR or CCPA that regulate the storage, collection,
and processing of user’s personal information; (ii) a IAO:DataItem is a piece
of personal information directly or indirectly associated with a user (e.g.,
name or age); (iii) a DataPractice describes a way that an OSN processes user
data (e.g., how long or where it retains data); and (iv) a PrivacyPolicy
describes the entirety of the data practices of an OSN or, more generally, any
kind of organization. Note that by modeling both PrivacyPolicy and
DataPractice as subclasses of InformationContentEntity we make the intentional
choice to treat them as the content of a privacy policy (PrivacyPolicy) or a
portion of that content that describes a specific practice (DataPractice).
They are distinct from the text itself and from what the OSN actually does in
practice with the data. Data practices may mention specific techniques or
tools employed to protect user data, which are modeled as
SecurityMechanisms666For brevity, we omit the OPPO: namespace; concepts and
properties without a namespace are implicitly assumed to be within that
namespace..

#### Roles:

We have identified three distinct roles relevant to privacy policies and the
data practice described therein, which align well with roles distinguished in
the OMRSE ontology. (i) DataSubjectRole is a role played by a person whom the
collected data is about (e.g. a user’s date of birth or a message they
posted). It specializes OMRSE:LegalPersonRole. (ii) DataRecipientRole is a
role played by an organization (OBI:Organization in OBI’s terms) that receives
information either directly from a person or from a third party. It
specializes OMRSE:OrganizationRole. (iii) DataProviderRole is a role played by
either a person or an organization that shares data with others. To
distinguish whether the data is shared by a person or an organization, we
introduce the subclasses LegalPersonDataProviderRole and
OrganizationalDataProviderRole that also specialize OMRSE:OrganizationRole and
OMRSE:LegalPersonRole, respectively. Note also that users and organizations
can play multiple roles for a particular piece of data, for example, a user
can share data about themselves, in which case the user acts in both a
DataSubjectRole and a DataProviderRole. Likewise, one organization can act in
a DataRecipientRole when it receives some data from a third party and in a
DataProviderRole when sharing data with the same or other third parties.

These roles can be further refined as shown in the right half of Figure 1. One
additional refinement is motivated by regulations, such as GDPR and CCPA,
imposing stricter conditions on handling data from minors, that is, users
under a certain age (which may be defined differently by different
regulations). Thus, we introduce MinorDataSubjectRole as a subclass of
DataSubjectRole. Another distinction is between two kinds of
DataRecipientRoles. An organization may act in a FirstPartyDataRecipientRole
role when it receives data directly from a person in which case the person
acts in both a DataSubjectRole and a DataProviderRole. An organization may
also share the collected data with other organizations according to its own
data practices. Any other organization that receives such data then acts in
the ThirdPartyDataRecipientRole and is bound not only by its own privacy
policies but also by the policy of the organization that it receives the data
from. For example, if an organization that is a FirstPartyDataRecipientRole
for some piece of data says that it shares data only for specific purposes
with third parties, then these third parties are expected not to share that
information with others for any other purposes either.

Figure 2: The core concepts in OPPO and the relations between them.

#### Purposes:

Many privacy regulations, such as GDPR, require privacy policies to outline
for what purpose an organization collects or processes data. For instance,
GDPR Art. 5 [1] states that an organization may collect different information
types for different purposes such as archiving statistical or research
purpose. To model these different purposes we adopt the generic concept
DPVO:Purpose from the DPV ontology [9], which is a subclass of BFO:Disposition
in BFO. Thus, all specialized purposes from DPV can be reused as well.

#### Relations among OPPO’s core concepts:

Figure 2 shows the key relations among the core concepts of OPPO, forming its
central Ontology Design Pattern (ODP). It consists of the following relations.
A PrivacyPolicy of an OSN (modeled as an OBI:Organization) contains data
practices ((hasDataPractice) relation to DataPractice) that describe how the
OSN collects, stores, and processes users’ data. These data practices may be
explicitly allowed or disallowed (allows and disallows) by different
PrivacyRegulation. DataPractices apply to (actsOn) specific kinds of data
(IAO:DataItem), such as specific subclasses of data (e.g.
DemographicPersonalData or AnonymizedData as elaborated in Figure 3) or data
that is constrained in other ways, for example, what kind of legal person the
data is about (a minor or not), what purpose it is collected for, or how it
was received. More specifically, IAO:DataItems can be linked to specific data
subjects, such as who the data is about (isAbout linking to a
DataSubjectRole), data providers (isProvidedBy linking to a DataProviderRole
describing who provides the data), and data recipients (isReceivedBy linking
to a DataRecipientRole describing who receives the data). Because an OSN may
describe different practices for storing or collecting data for distinct
purposes (DPVO:Purpose), such as archiving or marketing purpose, a practice is
related to purposes via the hasPurpose relation. By specifying both purposes
and specific types, instances of the DataPractice class can capture that
certain DataItems (e.g. personal data or statistical data) are stored or
processed only for certain purposes. Similarly, an OSN may employ different
security mechanism in different DataPractice and thus, implicitly, for
different kinds of DataItems. For example, an organization may apply end-to-
end encryption mechanism to biometric data while using a pseudonymization
mechanism for personal technical data such as IP addresses. We capture this by
relating SecurityMechanisms to IAO:DataItem using the appliesTo relation. In
the next subsections, we will discuss further refinements of IAO:DataItem,
DataPractice, and SecurityMechanism.

###  4.1 Data Item Module

To describe the different kinds of data collected by an OSN, we reuse and
refine IAO:DataItem [25] as shown in Figure 3. At the highest level, we
distinguish data related to an individual (IndividualData), such as a name,
age, or credit card information, from data that is an aggregated across
multiple individuals (AggregatedData). IndividualData can be further
distinguished based on whether it is anonymous, that is, data that cannot be
used to personally identify any specific individual person (AnonymizedData).
Data which may – directly or indirectly – reveal an individual’s identity
falls into the complementary class of PersonalData. It includes, for example,
photos, fingerprints, posts, reviews, location, or credit card information.
Based on the analysis of GDPR, CCPA, privacy policies, and prior works [40,
9], we have identified thirteen subclasses of PersonalData as shown in Figure
3. Their full definitions are provided in the ontology using the
skos:definition relation. One noteworthy concept is that of
PseudonymizedPersonalData, which is widely considered to be still
PersonalData, though any personal identifiers have been replaced by a pseudo-
identity. But it is different from AnonymizedData in that the data can be
still ascribed to an individual by anyone who knows the mapping (or mapping
algorithm) between pseudonyms and personal identifiers. We also include
dpvo:InferredPersonalData from DPV as subclass of PersonalData. It includes
any new data that is derived from existing data (e.g., demographic information
from the browsing history) and which may, directly or indirectly, identify an
individual.

Finally, two distinct subclasses of AggregatedData are included in OPPO: (i)
statistical aggregations of user data (StatisticalData), such as the number of
views on a product page or the number of likes of a post, and (ii) artificial
data produced to mimick real user data (dpvo:SyntheticData).

###  4.2 Data Practice Module

As mentioned in Section 1, the upper level of OPPO is currently only refined
to the extent needed to model fine-grained data storage and retention
practices (DataStoragePractice) and security practices (DataSecurityPractice).
For each of them, we distinguish three subclasses as shown in Figure 4 and
explained next.

Figure 3: OPPO’s taxonomy of subclasses of IAO:DataItem. The yellow concepts
are introduced by OPPO; all other concepts are reused from DPVO and IAO.

#### Data Storage Practices:

DataStoragePractices can specify restrictions on the duration of the storage,
the location of the storage, and how to get stored data corrected or deleted.
While all storage practices can give such details, those that do fall into a
of three subclasses. A DataStorageDurationPractice must specify the duration
of the stored data, which may be definite or indefinite. A
DataStorageLocationPractice includes restrictions on where or how the data is
stored. For instance, it may apply to data stored in specific geographic
locations (e.g., EU-GDPR imposes restrictions on data practices while the data
is stored outside of the EU) or sites of a specific company. We use the new
concept SpatialEntity that generalizes both BFO:SpatialRegion and BFO:Site (as
shown in Figure 1) as location to remain flexible and compatible with how
different BFO-based ontologies may specify locations. Alternatively, a
DataStorageLocationPractice may specify the kind of physical infrastructure
(StorageEntity) the data is stored in, such as a data center or the user’s
device.

Figure 4: OPPO’s taxonomy of DataPractices with indicative relations for the
DataStoragePractices.

DataStorageModificationPractice is a subclass of DataStoragePractice that
specifically captures practices that allow the modification (i.e., correction
or deletion) of stored data. It provides relations to specify how users can
request to rectify inaccurate data (a data property RequestType), how the OSN
may respond to such requests (a data property ResponseType), or how fast long
the OSN may take to process such data rectification or erasure request (using
hasResponseDelay). To capture different data retention practices as well as
response delays, we reuse the TIME:GeneralizedDurationDescription concept and
refine it by introducing DefiniteDurationDescription and
IndefiniteDurationDescription as our own specializations. We refer to the
ontology for details.

#### Data Security Practices:

DataSecurityPractices are organizational data practices that are followed to
maintain security to the collected/stored data. OPPO distinguishes three
classes of security practices (DataSecurityPractice). A
DataSecurityAuditingPractice is a practice that inspects whether and how the
organization maintains proper safeguard mechanisms while collecting, storing,
or processing personal data. A DataSecurityRestorationPractice is a practice
that discusses how data will be recovered if the data has been lost, stolen,
or compromised in other ways. A DataSecurityAccessPractice is a practice that
limits access to the data, thus preventing unauthorized access.

###  4.3 Security Mechanism Module

Both GDPR and CCPA require organizations to apply suitable techniques or tools
(referred to as SecurityMechanism in OPPO) to ensure the security of the
collected data. The regulations themselves distinguish two types of security
mechanisms: (i) PseudonymizationMechanisms that replace personal identifiers
with a pseudo identity; and (ii) EncryptionMechanisms that make personal data
unintelligible without the necessary keys for decryption. Our analysis of the
ten OSN privacy policies, however, identified additional security mechanisms
that are employed by these social networks. For instance, Signal’s privacy
policy [15] states that it applies cryptographic hashing mechanisms to
collected data before transmitting it to their server. As another example,
Telegram’s policy [16] explicitly states that it employs (if the user enables
it) two-factor authentication mechanisms to limit unauthorized access to their
data. As a result, OPPO also distinguishes HashingMechanism and
AuthenticationMechanism as two additional subclasses of security mechanisms,
with further subclasses for AuthenticationMechanism. The hierarchy of
subclasses of SecurityMechanism is shown in Figure 5.

Figure 5: OPPO’s taxonomy of SecurityMechanism and its subclasses.

##  5 Formalization and Evaluation

The ontology is encoded using the Web Ontology Language (OWL2) as computer-
interpretable format. The axiomatization currently contains 60 new classes, 17
object properties, and 271 logical axioms777The numbers are based on a core
version of the ontology that only replicates some upper level BFO, IAO and
OMRSE concepts, but those have been excluded in the concept and property
counts. The syntactic correctness of the ontology has been verified with a
simple RDFS validator888http://rdfvalidator.mybluemix.net/. We further checked
for common pitfalls in the ontology such as missing domain and range
restrictions using the OntOlogy Pitfall Scanner (OOPS!) [41].

#### Verification:

We used the HermiT [30] OWL2 reasoner that is provided with Protégé to check
the ontology for logical consistency. In the current version, no
inconsistencies are found nor are any classes inferred as being equivalent in
any problematic ways, such as being equivalent to owl:Thing. Additionally, we
manually created instances of the classes and properties from Telegram’s real
privacy policy999The dataset can be found in our GitHub repository.. When re-
checked the consistency of this dataset together with the ontology using the
HermiT reasoner.

#### Validation:

To test the expressivity of OPPO, we were able to express 15 out of a total of
27 competency questions as SPARQL queries101010The SPARQL queries are provided
in our GitHub Repository.. The remaining questions will guide the refinement
and further development of the ontology. We loaded the ontology and the sample
dataset into a GraphDB instance [42] that supports RDF and OWL reasoning. We
selected the OWL-RL (Optimized) ruleset provided by GraphDB that implements
OWL2 reasoning with the limitations described by the OWL2 RL profile. We
executed each SPARQL query and analyzed the results to ensure that they match
what we expect for our sample data. This validated that the ontology is
sufficiently expressive to adequately encode and answer these competency
questions. For instance, Figure 6 shows one example of a CQ that focuses on
capturing the specific data types that are being stored by Telegram for a
maximum of 12 months. The output indicates that Telegram stores four types of
data for a maximum of 12 months. A more in-depth evaluation of the CQs on a
larger dataset will be completed in the near future.

Figure 6: Encoding of a sample competency question as a SPARQL Query and the
answer when executed within GraphDB over the ontology and test dataset.

##  6 Conclusion and Future Work

In this paper, we presented OOPO as an extensible ontology for the privacy
domain that is designed to model and formally encode detailed data practices
as described in OSNs’ privacy policies. As a proof-of-concept, the ontology
provides a core pattern that connects privacy policies and their contained
data practices to data items that are described by their data types and roles.
The pattern identifies refined data practices based on the kinds of constrains
they impose (e.g. the duration, location, or type of practice) in order to
allow formally representing and reasoning over data practices of different
privacy policies of online social networks and similar companies. The ontology
leverages and connects to a number of existing ontologies based on the Basic
Formal Ontology (BFO). OPPO is encoded in OWL2 and provided as an open source
resource to the community. We evaluated the ontology’s logical consistency
with some small but real dataset and using a set of competency questions. The
competency questions were encoded in SPARQL, with the answers validating that
the ontology can indeed express and answer queries about the details of data
practices from privacy policies.

In the future, we plan to improve OPPO by (i) extending the ontology with
other practices, such as those related to data collection and sharing; by (ii)
constructing larger datasets from multiple privacy policies in order to
evaluate and compare their degrees of transparencies while also further
evaluating the ontology; and by (iii) expanding the ontology to also model
different privacy regulation in order to pose and answer questions about
compliance and non-compliance between individual privacy policies and
different privacy regulations, which is becoming increasingly important as
more regulations emerge.

#### Acknowledgments

We thanks the anonymous reviewers for the helpful suggestions to improve the
final version of the paper.

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