# Parameter estimation in an anisotropic expanding spacetime

O. P. de Sá Neto(a) I. G. da Paz(b) P. R. S. Carvalho(b) H. A. S. Costa(b) (a)
Coordenação de Ciência da Computação, Universidade Estadual do Piauí,
64202-220, Parnaíba, PI, Brazil  (b) Universidade Federal do Piauí,
Departamento de Física, 64049-550, Teresina, PI, Brazil

###### Abstract

In this work, we investigate how the anisotropy affects the cosmological
parameters estimation. Here the anisotropy is incorporated as a small
gravitational disturbance. We calculate the Fisher information for both
cosmological parameters $\epsilon$ (expansion volume) and $\rho$ (expansion
rate), and we show that the anisotropy introduces oscillations in the Fisher
information spectrum. This implies that the estimation of the cosmological
parameters is sensible to the direction of the momentum $k$ of particles. In
addition, we observe that for small values of the momentum $k$ there is a
substantial difference between the Fisher information spectrum for the minimum
and conformal couplings.

Fisher Information; Parameter Estimation; Anisotropy; Expanding Spacetime

###### pacs:

03.67.Mn, 03.65.Ud, 04.62.+v

###### Contents

  1. I Introduction
  2. II Physical model
    1. II.1 Isotropic case
    2. II.2 Anisotropic case
  3. III Methods
  4. IV Results
    1. IV.1 Isotropic case
    2. IV.2 Anisotropic case
  5. V Conclusions

##  I Introduction

Quantum metrology exploits quantum properties to improve the estimation of
physical parameters. Roughly speaking, a typical quantum metrology scheme
contains the following steps. First, a quantum probe state is prepared. Then,
the probe state is modified by any evolution process, encoding the set of
parameters to be estimated. The parameter information is then encoded in a
quantum observable. Finally, the estimation of single or multi parameters is
performed. The estimation protocol introduces uncertainty to the measured
values Geremia . The effect of this additional errors can be minimized by
repeating the measurements and averaging the outcomes Helstrom ; Holevo ;
Paris ; Helstrom2 . In addition, it is well known that the mean variance of
the uncertainty for a given measurement of a parameter $\theta$ is lower bound
by the Cramér-Rao inequality, $\mathrm{Var}(\theta)\geq
1/\sqrt{NF_{\mathrm{Q}}(\theta)}$ Braunstein ; Braunstein2 , where $N$ is the
number of identical measurements repeated and $F_{\mathrm{Q}}(\theta)$ is the
quantum Fisher information. It is noteworthy that the quantum Fisher
information is the central quantity that allow us to determine the ultimate
limits of precision in the estimation of a parameter $\theta$.

Recently, the techniques of quantum metrology have been employed in the
context of quantum field theory in curved spacetime Mehdi . In particular,
previous researches have investigated the metrology of a wide range of
relativistic phenomena, including estimation of entanglement Genoni ; Brida ,
gravitational waves Vallisneri ; Aasi ; GW ; Rideout , Unruh-Hawking effect
Aspachs ; Hosler ; Yao ; Wang ; Tian ; Yang , parameters of classical
spacetimes Downes , Schwarzschild spacetime parameters of the Earth Bruschi ,
$\kappa$-deformation of noncommutative spacetime Liu , and Lorentz violation
parameter Costa . In addition, parameter estimation has been studied in the
context of quantum cosmology. As examples of such application of quantum
metrology we can mention the estimation of the expansion parameters of the
universe Wang2 ; Huang  and the influence of the Lorentz invariance violation
on the ultimate limits of precision on the estimation of cosmological
parameters Liu2 . However, these works do not take into account anisotropy
effects on the cosmological parameters estimation. The estimation of
cosmological parameters is one of the central issues in modern cosmology,
thus, it is interesting to investigate quantum metrology protocol in a more
realistic model.

There are two central goals here: First, investigate the effects of an
anisotropic perturbation of the metric on the estimation of the cosmological
parameters $\epsilon$ (expansion volume) and $\rho$ (expansion rate). Second,
examine how the Fisher information spectrum is modified by the type of
coupling between the field and the spacetime curvature. The paper is organized
as follow. In Section 2, we introduce the isotropic and anisotropic models for
the expanding spacetime. In addition, we explicitly compute the Bogoliubov
coefficients for both the models. In Section 3, we present a review of the
basic notions of local quantum estimation theory. In Section 4, we calculate
the Fisher information of the cosmological parameters, and study how the
anisotropy affects the estimation scheme. Finally, Section 5 summarizes the
results of this work and draws our conclusions.

##  II Physical model

In this section we study the isotropic and anisotropic models for the
expanding spacetime. We start with the Friedmann-Robertson-Walker (FRW) model
which describes a homogeneous and isotropic Universe. As it is known, under
such conditions the Einstein field equations $G^{\mu}_{\nu}=\varsigma
T^{\mu}_{\nu}$, where $G^{\mu}_{\nu}$ is the Einstein tensor, $\varsigma=8\pi
G/c^{4}$, and $T^{\mu}_{\nu}$ is the energy-momentum tensor for a perfect
fluid in the comoving frame, can be solved exactly, resulting in the metric

$\displaystyle ds^{2}=c^{2}dt^{2}-a^{2}(t)\left[\frac{dr^{2}}{1-\kappa
r}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})\right].$

Here, $a(t)$ is the scalar factor and $\kappa=-1,0,1$ is the curvature
parameter, respectively for open, flat, and closed geometry.

###  II.1 Isotropic case

Here we study the expanding spacetime in an isotropic medium. Then, we
consider a real and non-interacting scalar field $\phi$ with mass $m$
propagating in (1+1)-dimensional FRW metric with $\kappa=0$. In terms of the
conformal time $\eta=\int\frac{dt}{a(t)}$ the metric of the FRW can be written
as Martinez0

$\displaystyle ds^{2}=a^{2}(\eta)\left(d\eta^{2}-dx^{2}\right).$

From this, it is clear that the FRW spacetime is conformally flat:
$g_{\mu\nu}=a^{2}(\eta)\eta_{\mu\nu}$. Now, we consider the dynamics of the
scalar field $\phi$ which is described by the covariant form of the Klein-
Gordon equation

$\displaystyle\left[\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu})+a^{2}(\eta)m^{2}\right]\phi=0.$
(1)

Since the spatial translation of the spacetime is invariant, we can use the
method of variables to solve the Eq. (1). In the conformal observer frame it
may be written as $\phi_{k}(\eta,x)=a^{-1}(\eta)e^{ikx}\chi_{k}(\eta)$, where
$\chi_{k}(\eta)$ satisfies the following equations of motion

$\displaystyle\left[\partial_{\eta}^{2}+k^{2}+a^{2}(\eta)m^{2}\right]\chi_{k}(\eta)=0.$
(2)

Now, to obtain the solution of Eq. (2) we consider the following form for the
scale factor $a^{2}(\eta)$ Fuentes0

$a^{2}(\eta)=1+\epsilon(1+\tanh(\rho\eta)),$ (3)

where $\epsilon$ and $\rho$ are positive real parameters controlling the total
volume and the rapidity of the expansion, respectively. The scalar factor
$a^{2}(\eta)$ defined above is a smooth function and approaches for a constant
value in the distant past (the so-called in-region, where
$\eta\rightarrow-\infty$) and in the far future (the out-region, where
$\eta\rightarrow\infty$). Consequently, the metric above describes a flat
spacetime in these asymptotic limits. As the spacetime is not stationary but
possesses stationary asymptotic regions we can give a particle interpretation
to the solution of the field equations in both distant past and far future. By
using (3), we can solve the Eq. (2) in terms of hypergeometric functions. The
two set of solutions in the in- and out- regions are given by

$\displaystyle\chi_{k}^{\mathrm{in}}(\eta)$
$\displaystyle=\frac{e^{-i\omega_{+}\eta-\frac{i\omega_{-}}{\rho}\ln[2\cosh(\rho\eta)]}}{\sqrt{4\pi\omega_{\mathrm{in}}}}F_{1}\Bigg{[}1+\frac{i\omega_{-}}{\rho},\frac{i\omega_{-}}{\rho},1-\frac{i\omega_{\mathrm{in}}}{\rho},\frac{1}{2}(1+\tanh(\rho\eta))\Bigg{]},$
$\displaystyle\chi_{k}^{\mathrm{out}}(\eta)$
$\displaystyle=\frac{e^{-i\omega_{+}\eta-\frac{i\omega_{-}}{\rho}\ln[2\cosh(\rho\eta)]}}{\sqrt{4\pi\omega_{\mathrm{out}}}}F_{1}\Bigg{[}1+\frac{i\omega_{-}}{\rho},\frac{i\omega_{-}}{\rho},1+\frac{i\omega_{\mathrm{out}}}{\rho},\frac{1}{2}(1-\tanh(\rho\eta))\Bigg{]},$
(4)

where the $F_{1}$ are hypergeometric functions, and for notational convenience
we have defined

$\displaystyle\omega_{\mathrm{in}}$
$\displaystyle=\sqrt{k^{2}+m^{2}},\quad\omega_{\mathrm{out}}=\sqrt{k^{2}+(1+2\epsilon)m^{2}},$
$\displaystyle\omega_{+}$
$\displaystyle=\frac{1}{2}(\omega_{\mathrm{out}}+\omega_{\mathrm{in}}),\quad\omega_{-}=\frac{1}{2}(\omega_{\mathrm{out}}-\omega_{\mathrm{in}}).$

Notice that in the limit $\eta\rightarrow-\infty$, the first solution in (4)
can be approximated for

$\chi_{k}^{\mathrm{in}}(\eta\rightarrow-\infty)\rightarrow\frac{1}{\sqrt{4\pi\omega_{\mathrm{in}}}}e^{-i\omega_{\mathrm{in}}\eta},$

and by taking the limit $\eta\rightarrow\infty$, the second solution in (4)
becomes

$\chi_{k}^{\mathrm{out}}(\eta\rightarrow\infty)\rightarrow\frac{1}{\sqrt{4\pi\omega_{\mathrm{out}}}}e^{-i\omega_{\mathrm{out}}\eta}.$

The asymptotic solutions above are connected via a Bogoliubov transformation.
This can be done by using some basic properties of hypergeometric functions.
For example, any hypergeometric function can be written as the linear
combination

$\displaystyle F_{1}[a,b;c;z]$
$\displaystyle=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}F_{1}[a,b;a+b+1-c;1-z]$
$\displaystyle+\frac{\Gamma(c)\Gamma(a+b-c)}{\Gamma(a)\Gamma(b)}(1-z)^{c-a-b}F_{1}[c-a,c-b;1+c-a-b;1-z].$

Another property of hypergeometric functions is the relation

$\displaystyle F_{1}[a,b;c;z]=(1-z)^{c-a-b}F_{1}[c-a,c-b;c;z].$

By using the above properties, we can write $\chi_{k}^{\mathrm{in}}(\eta)$ as

$\displaystyle\chi_{k}^{\mathrm{in}}(\eta)=\alpha_{k}^{\text{iso}}\chi_{k}^{\mathrm{out}}(\eta)+\beta_{k}^{\text{iso}}\chi_{k}^{\mathrm{out}*}(\eta),$
(5)

where

$\displaystyle\alpha_{k}^{\text{iso}}$
$\displaystyle=\sqrt{\frac{\omega_{\mathrm{out}}}{\omega_{\mathrm{in}}}}\frac{\Gamma\Big{(}1-\frac{i\omega_{\mathrm{out}}}{\rho}\Big{)}\Gamma\Big{(}-\frac{i\omega_{\mathrm{in}}}{\rho}\Big{)}}{\Gamma\Big{(}-\frac{i\omega_{+}}{\rho}\Big{)}\Gamma\Big{(}1-\frac{i\omega_{+}}{\rho}\Big{)}},$
(6) $\displaystyle\beta_{k}^{\text{iso}}$
$\displaystyle=\sqrt{\frac{\omega_{\mathrm{out}}}{\omega_{\mathrm{in}}}}\frac{\Gamma\Big{(}1-\frac{i\omega_{\mathrm{out}}}{\rho}\Big{)}\Gamma\Big{(}\frac{i\omega_{\mathrm{in}}}{\rho}\Big{)}}{\Gamma\Big{(}\frac{i\omega_{-}}{\rho}\Big{)}\Gamma\Big{(}1+\frac{i\omega_{-}}{\rho}\Big{)}}.$
(7)

Here $\alpha_{k}^{\text{iso}}$ and $\beta_{k}^{\text{iso}}$ are the Bogoliubov
coefficients for transition from the out-state to the in-state.

###  II.2 Anisotropic case

Here we study the dynamics of a massive scalar field $\phi(x)$ in an expanding
spacetime with anisotropy Pierini . Let’s start with the Lagrangian density
for the scalar field in a general curved spacetime

$\displaystyle\mathcal{L}$
$\displaystyle=\frac{1}{2}\sqrt{-g}[g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-(m^{2}+\xi
R)\phi^{2}],$ (8)

where $g^{\mu\nu}$ is the metric with determiant $g$, $R$ is the Ricci scalar
curvature, and $\xi$ is a dimensionless parameter which characterizes the
coupling. We emphasize two interesting values for the coupling $\xi$, i.e.,
$\xi=0$ and $\xi=1/6$. On the one hand, for $\xi=0$, the field is said to be
minimally coupled to the metric. On the other hand, for $\xi=\frac{1}{6}$, (8)
is conformally invariant in the massless limit. Again, the dynamics of the
field operator $\phi$ is governed by the covariant Klein-Gordon equation (1).
In particular, let us consider a specific model of an anisotropic universe, a
Bianchi Type I spacetime, with line element

$\displaystyle ds^{2}=dt^{2}-\sum_{j=1}^{3}a^{2}_{j}(t)dx^{2}_{j},$ (9)

where $a_{j}(t)$ are arbitrary functions of time. Despite it describes a
spatially homogeneous universe it is a non-isotropic toy model of the
universe. As was done in Refs. Birrell01 ; Birrell02 ; Zeldovich , we consider
the scale factors as $a_{j}(t)=1+h_{j}(t)$ where $h_{j}(t)$ is assumed to be
small and it is treated as a parturbation, i.e., $\mathrm{max}|h_{j}(t)|\ll
1$. In terms of the conformal time coordinate $d\eta=a^{-1}(t)dt$, the metric
(9) reads

$\displaystyle
ds^{2}=a^{2}(\eta)\Bigg{[}d\eta^{2}-\sum_{j=1}^{3}(1+h_{j}(\eta))dx_{j}^{2}\Bigg{]}.$
(10)

As an example, let us now assume $h_{j}(\eta)$ to be

$\displaystyle
h_{j}(\eta)=e^{-\rho\eta^{2}}\cos(\epsilon\eta^{2}+\delta_{j}),$ (11)

with
$\delta_{j}=\frac{\pi}{2},\frac{\pi}{2}+\frac{2\pi}{3},\frac{\pi}{2}+\frac{4\pi}{3}$.
It is worth to mention that this choice satisfies the condition
$\sum_{j=1}^{3}h_{j}(\eta)=0$. Then, the dynamics of the scalar field $\phi$
in the conformal observer frame is given by

$\displaystyle\partial_{\eta}^{2}\phi+2\frac{a^{\prime}(\eta)}{a(\eta)}\partial_{\eta}\phi-\nabla^{2}\phi+a^{2}(\eta)(m^{2}+\xi
R)\phi=0,$ (12)

where the prime denotes derivatives with respect to conformal time. Because of
spatial translation invariance of the spacetime, the solution of the Eq. (12)
may be written as

$\displaystyle\phi_{k}(x)=(2\pi)^{-\frac{3}{2}}a^{-1}(\eta)e^{i\mathbb{k}\cdot\mathbb{x}}f_{k}(\eta),$
(13)

where leading order in $h_{j}$, $f_{k}(\eta)$ satisfies

$\displaystyle\left[\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}+m^{2}\right]f_{k}(\eta)+V(\eta)f_{k}(\eta)=0,$
(14)

with

$\displaystyle V(\eta)$
$\displaystyle=[a^{2}(\eta)-a^{2}(-\infty)]m^{2}+(\xi-\frac{1}{6})a^{2}(\eta)R(\eta)-\sum_{j=1}^{3}h_{j}(\eta)k_{j}^{2}.$

Now, by imposing the following conditions Birrell01 ; Birrell02

$\displaystyle\begin{split}a^{2}(\eta)R(\eta)&\to
0\,\,\text{as}\,\,\eta\to\pm\infty\,\,\text{if}\,\,\xi\neq 1/6,\\\
h_{j}(\eta)&\to 0\,\,\text{as}\,\,\eta\to\pm\infty,\\\ a^{2}(\eta)&\to
a^{2}(\pm\infty)<\infty\,\,\text{as}\,\,\eta\to\pm\infty\,\,\text{if}\,\,m\neq
0,\end{split}$ (15)

one may treat $V(\eta)$ as small and solve the Eq. (14) by iteration to the
lowest order in $V(\eta)$ in terms of the momentum space propagator. In this
case, the integral form of the Eq. (14) becomes

$\displaystyle
f_{k}(\eta)=f^{\mathrm{in}}_{k}(\eta)-\int^{\infty}_{-\infty}G_{r}(\eta,\eta^{\prime})V(\eta^{\prime})f_{k}(\eta^{\prime})d\eta^{\prime},$
(16)

where $f^{\mathrm{in}}_{k}(\eta)$ is the free-wave solution propagating from
the in-region, which is defined by
$f^{\mathrm{in}}_{k}(\eta)=(2\omega)^{-\frac{1}{2}}e^{-i\omega\eta}$ with
$\omega=\sqrt{k^{2}+m^{2}}$. The propagator $G_{r}(\eta,\eta^{\prime})$
satisfies

$\displaystyle\left[\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}+m^{2}\right]G_{r}(\eta,\eta^{\prime})=\delta(\eta-\eta^{\prime})\,,$
(17)

and, in momentum space, it reads

$\displaystyle
G_{r}(\eta,\eta^{\prime})=\frac{1}{2\pi}\int\frac{e^{-ik^{\prime}_{0}(\eta^{\prime}-\eta)}}{k^{\prime
2}_{0}-\omega^{2}_{k}-i\varepsilon}dk^{\prime}_{0}\,.$ (18)

The momentum integral can be performed by closing the integration contour in
the upper-half complex momentum plane. However, for the calculation of the
Bogoliubov coefficients, it suffices to notice that in the limit
$\eta\longrightarrow\infty$, $f_{k}(\eta)$ can be written in terms of the mode
functions $f^{\mathrm{in}}_{k}(\eta)$ as

$\displaystyle
f^{\mathrm{out}}_{k}(\eta)=(2\omega)^{-\frac{1}{2}}\left[\alpha_{k}^{\text{aniso}}e^{-i\omega\eta}+\beta_{k}^{\text{aniso}}e^{i\omega\eta}\right],$
(19)

where the resulting Bogoliubov coefficients are given

$\displaystyle\alpha_{k}^{\text{aniso}}$
$\displaystyle=1+\frac{i}{\sqrt{2\omega}}\int^{\infty}_{-\infty}e^{i\omega\eta^{\prime}}V(\eta^{\prime})f_{k}(\eta^{\prime})d\eta^{\prime},$
$\displaystyle\beta_{k}^{\text{aniso}}$
$\displaystyle=-\frac{i}{\sqrt{2\omega}}\int^{\infty}_{-\infty}e^{-i\omega\eta^{\prime}}V(\eta^{\prime})f_{k}(\eta^{\prime})d\eta^{\prime}.$
(20)

To the lowest order in $V(\eta)$ we have $f_{k}(\eta)\cong
f^{\mathrm{in}}_{k}(\eta)$, and theBogoliubov coefficients become

$\displaystyle\alpha_{k}^{\text{aniso}}$
$\displaystyle=1+\frac{i}{2\omega}\int^{\infty}_{-\infty}V(\eta^{\prime})d\eta^{\prime},$
$\displaystyle\beta_{k}^{\text{aniso}}$
$\displaystyle=-\frac{i}{2\omega}\int^{\infty}_{-\infty}e^{-2i\omega\eta^{\prime}}V(\eta^{\prime})d\eta^{\prime}.$
(21)

Now, let us consider a scale factor (3) which allows us to calculate the
Bogoliubov coefficients and satisfy the conditions (15). Note that
$a^{2}(\eta)$ represents conveniently a spacetime that undergoes a period of
smooth expansion and becomes flat in the distant past and in the far future.
By inserting in (21) the explicit form of $V(\eta)$ and $a^{2}(\eta)$, we can
write $\alpha_{k}^{\text{aniso}}$ and $\beta_{k}^{\text{aniso}}$ as

$\displaystyle\begin{split}\alpha_{k}^{\text{aniso}}&=1+\alpha_{k}^{(m)}+\alpha_{k}^{(\xi)}+\alpha_{k}^{(h)},\\\
\beta_{k}^{\text{aniso}}&=\beta_{k}^{(m)}+\beta_{k}^{(\xi)}+\beta_{k}^{(h)},\end{split}$
(22)

where to leading order in $\epsilon$ we find

$\displaystyle\alpha_{k}^{(m)}$ $\displaystyle=\frac{i\epsilon
m^{2}}{\omega},$ (23) $\displaystyle\alpha_{k}^{(\xi)}$
$\displaystyle=(\xi-\frac{1}{6})\frac{i\rho}{12\epsilon\omega}[(\epsilon+1)\ln(2\epsilon+1)-2\epsilon],$
(24) $\displaystyle\alpha_{k}^{(h)}$
$\displaystyle=\frac{i\sqrt{\pi}}{2\omega}\sum_{j=1}^{3}k_{j}^{2}\mathrm{Re}\Bigg{(}\frac{e^{-i\delta_{j}}}{\sqrt{\rho+i\epsilon})}\Bigg{)},$
(25) $\displaystyle\beta_{k}^{(m)}$
$\displaystyle=-\frac{m^{2}\epsilon}{2\omega\rho}\frac{\pi}{\sinh(\frac{\pi\omega}{\rho})},$
(26) $\displaystyle\beta_{k}^{(\xi)}$
$\displaystyle=(\xi-\frac{1}{6})\frac{\omega\epsilon}{6\rho}\frac{\pi}{\sinh(\frac{\pi\omega}{\rho})},$
(27) $\displaystyle\beta_{k}^{(h)}$
$\displaystyle=-\frac{\sqrt{\pi}}{2\omega}\sum_{j=1}^{3}k_{j}^{2}\mathrm{Re}\Bigg{(}\frac{e^{-i\delta_{j}+i\frac{\pi}{2}-\frac{\omega^{2}}{\rho+i\epsilon}}}{\sqrt{\rho+i\epsilon})}\Bigg{)}.$
(28)

##  III Methods

In this section, let us calculate the quantum Fisher information matrix (QFIM)
Liu3 ; Safranek . For this, suppose a vector of parameters
$\overrightarrow{x}=(x_{1},x_{2},...,x_{N},...)^{T}$ where $x_{N}$ is the Nth
parameter. The vector of parameters is encoded in the density matrix
$\hat{\rho}=\hat{\rho}(\overrightarrow{x})$. Thus, the QFIM is defined as

$\displaystyle\mathcal{H}_{ab}=\frac{1}{2}\mathrm{Tr}[\hat{\rho}\\{\hat{L}_{a},\hat{L}_{b}\\}],$
(29)

where $\hat{L}_{a}$ and $\hat{L}_{b}$ are the symmetric logarithmic derivative
(SLD) for the parameters $x_{a}$ and $x_{b}$, respectively. $\hat{L}_{a}$ and
$\hat{L}_{b}$ are obtained via the equation

$\displaystyle\partial_{a(b)}\rho=\frac{1}{2}[\hat{L}_{a(b)}\hat{\rho}+\hat{\rho}\hat{L}_{a(b)}].$
(30)

Note that the SLD operators are Hermitian operators with expected value
$\mathrm{Tr}(\hat{L}_{a(b)})=0$. By using the equation above,
$\mathcal{H}_{ab}$ can also be expressed by

$\displaystyle\mathcal{H}_{ab}=\mathrm{Tr}[\hat{L}_{b}\partial_{a}\hat{\rho}]=-\mathrm{Tr}[\hat{\rho}\partial_{a}\hat{L}_{b}].$
(31)

From Eq. (29), the diagonal entries of QFIM are
$\mathcal{H}_{aa}=\mathrm{Tr}[\hat{\rho}\hat{L}_{a}^{2}]$ and
$\mathcal{V}_{bb}=\mathrm{Tr}[\hat{\rho}\hat{L}_{b}^{2}]$. In particular, if
we consider the density matrix with the spectral decomposition
$\hat{\rho}=\sum_{i=0}^{d-1}\lambda_{i}|\lambda_{i}\rangle\langle\lambda_{i}|$,
the QFIM can be evaluated as

$\displaystyle\mathcal{H}_{ab}=\sum_{i,j=0,\lambda_{i}+\lambda_{j}\neq
0}^{d-1}\frac{2\mathrm{Re}[\langle\lambda_{i}|\partial_{a}\hat{\rho}|\lambda_{j}\rangle\langle\lambda_{j}|\partial_{b}\hat{\rho}|\lambda_{i}\rangle]}{\lambda_{i}+\lambda_{j}},$
(32)

where $\lambda_{i}$ and $|\lambda_{i}\rangle$ are the eigenvalues and the
corresponding eigenstates, respectively. It is usually assume that
$\lambda_{i}>0$ and $0\leq i\leq d-1$ where $d:=\mathrm{dim}\rho$ is the
dimension of the density matrix. By substituting the spectral decomposition of
$\hat{\rho}$ into the equation above, it can be rewritten as

$\displaystyle\mathcal{H}_{ab}=\sum_{i=0}^{d-1}\frac{(\partial_{a}\lambda_{i})(\partial_{b}\lambda_{i})}{\lambda_{i}}+\sum_{i\neg
j,\lambda_{i}+\lambda_{j}\neq
0}\frac{2(\lambda_{i}-\lambda_{j})^{2}}{\lambda_{i}+\lambda_{j}}\mathrm{Re}[\langle\lambda_{i}|\partial_{a}\lambda_{j}\rangle\langle\partial_{b}\lambda_{j}|\lambda_{i}\rangle]$
(33)

The first term in the equation above can be viewed as the counterpart of the
classical Fisher information as it only contains the derivatives of the
eigenvalues and can be regarded as the counterpart of the probability
distribution. On the other hand, the second term is purely quantum.

##  IV Results

###  IV.1 Isotropic case

Let us consider that the field is in the vacuum state in the asymptotic past
$|0\rangle_{\mathrm{in}}$, which is annihilated by
$\hat{a}_{\textbf{k}}^{\mathrm{in}}$. We shall restrict our analysis to the
$k$, $-k$ sector of the density matrix for a fixed wavenumber $k$, since the
Lagrangian density we consider here will only mix modes of opposite momenta.
This mixing of positive- and negative-frequency modes of the same wavenumber
by the dynamics of spacetime leads to the excitation of the vacuum
fluctuations, and consequently to particle creation. Thus, we would like to
know the form of the state $|0\rangle_{\mathrm{in}}$ in the asymptotic future.
To achieve this goal, let’s begin with the fact that

$\displaystyle\hat{a}_{\textbf{k}}^{\mathrm{in}}|0\rangle_{\mathrm{in}}=0.$

By using the Bogoliubov transformation
$\hat{a}_{\textbf{k}}^{\mathrm{in}}=\alpha_{k}^{\text{iso}}\hat{a}_{\textbf{k}}^{\mathrm{out}}+\beta_{k}^{\text{iso}}\hat{a}_{-\textbf{k}}^{\mathrm{out}\dagger}$,
we obtain

$\displaystyle(\alpha_{k}^{\text{iso}}\hat{a}_{\textbf{k}}^{\mathrm{out}}+\beta_{k}^{\text{iso}}\hat{a}_{-\textbf{k}}^{\mathrm{out}\dagger})|0\rangle_{\mathrm{in}}=0.$
(34)

Without loss of generality, let us consider the following “ansatz” for the
vacuum state in the out-region

$\displaystyle|0\rangle_{\mathrm{in}}=\sum_{n=0}^{\infty}A_{n}|n_{k},n_{-k}\rangle_{\mathrm{out}}.$
(35)

By substituting the expression above into Eq. (34), we get the recurrence
relation
$A_{n}=\left(\frac{\beta_{k}^{\text{iso}*}}{\alpha_{k}^{\text{iso}*}}\right)A_{0}$
and from the normalization condition we find
$|A_{0}|^{2}=1-\left|\frac{\beta_{k}^{\text{iso}}}{\alpha_{k}^{\text{iso}}}\right|^{2}$.
Therefore, the vacuum state $|0\rangle_{\mathrm{in}}$ in terms of “out” modes
in the far future becomes

$\displaystyle|0\rangle_{\mathrm{in}}=\sqrt{1-\gamma_{k}}\sum_{n=0}^{\infty}\gamma_{k}^{n}|n_{k},n_{-k}\rangle,$
(36)

where

$\displaystyle\gamma_{k}=\left|\frac{\beta_{k}^{\text{iso}}}{\alpha_{k}^{\text{iso}}}\right|^{2}=\frac{\sinh^{2}(\frac{\pi}{\rho}\omega_{-})}{\sinh^{2}(\frac{\pi}{\rho}\omega_{+})}.$
(37)

Since we are working with a single mode, we will drop the frequency index $k$.
Notice that the state in Eq. (36) is a two mode squeezed state. Let us suppose
that an inertial observer in the out-region has no access to modes $-k$, thus
such observer must trace over the inaccessible modes. This means that there is
an unavoidable loss of information about the state of particles with modes
$-k$, which essentially results in the detection of a mixed state. Therefore,
an inertial observer in the out-region detects a distribution of particles
with modes $k$ according to the marginal state

$\displaystyle\hat{\rho}_{k}$
$\displaystyle=\mathrm{Tr}_{-k}[\hat{\rho}_{k,-k}],$
$\displaystyle=(1-\gamma_{k})\sum_{n=0}^{\infty}\gamma_{k}^{n}|n_{k}\rangle\langle
n_{k}|,$ (38)

where the density matrix of the whole state corresponds to
$\hat{\rho}_{k,-k}=|0\rangle_{\mathrm{in}}\langle 0|$. In the next section, we
would use the state $\hat{\rho}_{k}$ as a probe state, which represents a
class of signals. We are interested in estimating the cosmological parameters
($\epsilon,\rho$) with the minimum variance, which is the ultimate bound of
precision imposed by the quantum theory.

In order to estimate the cosmological parameters $\epsilon$ and $\rho$, we
have evaluated the QFIM of the particle state $\hat{\rho}_{k}$, i.e., we
calculate

$\displaystyle\mathcal{H}_{ab}^{\text{iso}}=\frac{1}{2}\mathrm{Tr}[\hat{\rho}_{k}L_{a}L_{b}],\quad\text{with}\quad
a,b=\epsilon,\rho.$ (39)

Since the density matrix $\hat{\rho}_{k}$ has a spectral decomposition, Eq.
(39) can be rewritten as

$\displaystyle\mathcal{H}_{ab}^{\text{iso}}=\sum_{n=0}^{\infty}\frac{\left(\partial_{a}\lambda_{n}\right)\left(\partial_{b}\lambda_{n}\right)}{\lambda_{n}}+2\sum_{n\neq
m=0}^{\infty}\frac{(\lambda_{m}-\lambda_{n})^{2}}{\lambda_{m}+\lambda_{n}}\mathrm{Re}[\langle
n_{k}|\partial_{a}m_{k}\rangle\langle\partial_{b}m_{k}|n_{k}\rangle],$ (40)

where $\lambda_{n}=(1-\gamma_{k})\gamma_{k}^{n}$. Note that the last term that
contains the truly quantum contribution does not contribute to
$\mathcal{H}_{ab}^{\text{iso}}$ because $|n_{k}\rangle$ and $|m_{k}\rangle$
with $n\neq m$ locate on different subspaces, i.e., $\langle
n_{k}|\partial_{a}m_{k}\rangle=\langle\partial_{b}m_{k}|n_{k}\rangle=0$
$\forall n,m$. In particular, the classical part of the quantum Fisher
information $\mathcal{H}_{ab}^{\text{iso}}$ can be calculated by introducing
the Eq. (37). The analytical expression of
$\mathcal{F}_{\epsilon\epsilon}^{\text{iso}}$ and
$\mathcal{F}_{\rho\rho}^{\text{iso}}$ can be obtained as follow. The
$\mathcal{F}_{\epsilon\epsilon}^{\text{iso}}$ is given by

$\displaystyle\mathcal{F}_{\epsilon\epsilon}^{\text{iso}}$
$\displaystyle=\frac{1}{1-\gamma_{k}}\sum_{n=0}^{\infty}\frac{1}{\gamma_{k}^{n}}\left[\partial_{\epsilon}(1-\gamma_{k})\gamma_{k}^{n}\right]^{2},$
$\displaystyle=\frac{(\partial_{\epsilon}\gamma_{k})^{2}}{1-\gamma_{k}}\sum_{n=0}^{\infty}\gamma_{k}^{n}-2\partial_{\epsilon}\gamma_{k}\sum_{n=0}^{\infty}\partial_{\epsilon}\gamma_{k}^{n}+(1-\gamma_{k})\sum_{n=0}^{\infty}\frac{(\partial_{\epsilon}\gamma_{k}^{n})^{2}}{\gamma_{k}^{n}}.$
(41)

Using the following relations

$\displaystyle\partial_{\epsilon}\gamma_{k}$ $\displaystyle=\frac{\pi
m^{2}}{\omega^{\mathrm{out}}_{k}\rho}\left[\frac{1}{\tanh(\frac{\pi\omega_{-}}{\rho})}-\frac{1}{\tanh(\frac{\pi\omega_{+}}{\rho})}\right]\gamma_{k},$
$\displaystyle\partial_{\epsilon}\gamma_{k}^{n}$ $\displaystyle=\frac{\pi
m^{2}}{\omega^{\mathrm{out}}_{k}\rho}\left[\frac{1}{\tanh(\frac{\pi\omega_{-}}{\rho})}-\frac{1}{\tanh(\frac{\pi\omega_{+}}{\rho})}\right]n\gamma_{k}^{n},$
(42)

one can easily obtain the following result

$\displaystyle\mathcal{F}_{\epsilon\epsilon}^{\text{iso}}=\frac{\gamma_{k}}{(1-\gamma_{k})^{2}}\frac{\pi^{2}m^{4}}{\omega^{\mathrm{out}2}_{k}\rho^{2}}\left[\frac{\tanh(\frac{\pi\omega_{+}}{\rho})-\tanh(\frac{\pi\omega_{-}}{\rho})}{\tanh(\frac{\pi\omega_{-}}{\rho})\tanh(\frac{\pi\omega_{+}}{\rho})}\right]^{2}.$
(43)

Similarly, the Fisher information $\mathcal{F}_{\rho\rho}^{\text{iso}}$ is
given by

$\displaystyle\mathcal{F}_{\rho\rho}^{\text{iso}}$
$\displaystyle=\frac{1}{1-\gamma_{k}}\sum_{n=0}^{\infty}\frac{1}{\gamma_{k}^{n}}\left[\partial_{\rho}(1-\gamma_{k})\gamma_{k}^{n}\right]^{2}$
$\displaystyle=\frac{(\partial_{\rho}\gamma_{k})^{2}}{1-\gamma_{k}}\sum_{n=0}^{\infty}\gamma_{k}^{n}-2\partial_{\rho}\gamma_{k}\sum_{n=0}^{\infty}\partial_{\rho}\gamma_{k}^{n}+(1-\gamma_{k})\sum_{n=0}^{\infty}\frac{(\partial_{\rho}\gamma_{k}^{n})^{2}}{\gamma_{k}^{n}}.$
(44)

Now, by using the relations

$\displaystyle\partial_{\rho}\gamma_{k}$
$\displaystyle=-\frac{\pi}{\rho^{2}}\left[\frac{\omega_{-}}{\tanh(\frac{\pi\omega_{-}}{\rho})}-\frac{\omega_{+}}{\tanh(\frac{\pi\omega_{+}}{\rho})}\right]\gamma_{k},$
$\displaystyle\partial_{\rho}\gamma_{k}^{n}$
$\displaystyle=-\frac{\pi}{\rho^{2}}\left[\frac{\omega_{-}}{\tanh(\frac{\pi\omega_{-}}{\rho})}-\frac{\omega_{+}}{\tanh(\frac{\pi\omega_{+}}{\rho})}\right]n\gamma_{k}^{n},$
(45)

we find

$\displaystyle\mathcal{F}_{\rho\rho}^{\text{iso}}=\frac{\gamma_{k}}{(1-\gamma_{k})^{2}}\frac{\pi^{2}}{\rho^{4}}\left[\frac{\omega_{-}\tanh(\frac{\pi\omega_{+}}{\rho})-\omega_{+}\tanh(\frac{\pi\omega_{+}}{\rho})}{\tanh(\frac{\pi\omega_{-}}{\rho})\tanh(\frac{\pi\omega_{+}}{\rho})}\right]^{2}.$
(46)

Note that the Fisher information $\mathcal{F}_{\epsilon\epsilon}^{\text{iso}}$
and $\mathcal{F}_{\rho\rho}^{\text{iso}}$ quantify the error in the estimation
of the individual cosmological parameters. Fig. 1 (a) shows the Fisher
information $\mathcal{F}_{\epsilon\epsilon}^{\text{iso}}$ and
$\mathcal{F}_{\rho\rho}^{\text{iso}}$ as a function of the momentum of the
mode $k$. We can see that $\mathcal{F}_{\epsilon\epsilon}^{\text{iso}}$ and
$\mathcal{F}_{\rho\rho}^{\text{iso}}$ are monotonic decreasing function of the
momentum of the mode $k$. In this case, notice that the spectrum of the Fisher
information $\mathcal{F}_{\epsilon\epsilon}^{\text{iso}}$ and
$\mathcal{F}_{\rho\rho}^{\text{iso}}$ are similar. Moreover, notice that the
highest precision in the estimation of both the parametrs $\epsilon$ and
$\rho$ can be achieved for small momentum $k$.

Figure 1:  (a) Fisher information
$\mathcal{F}_{\epsilon\epsilon}^{\text{iso}}$ and
$\mathcal{F}_{\rho\rho}^{\text{iso}}$ as a function of momentum $k$, with
$\rho=15$, $\epsilon=0.1$ and $m=1$. (b) Fisher information
$\mathcal{F}_{\epsilon\epsilon}^{\text{iso}}$ and
$\mathcal{F}_{\rho\rho}^{\text{iso}}$ as a function of mass $m$, with
$\rho=15$, $\epsilon=0.1$ and $k=1$.

In Fig. 1 (b) we plot the Fisher information
$\mathcal{F}_{\epsilon\epsilon}^{\text{iso}}$ and
$\mathcal{F}_{\rho\rho}^{\text{iso}}$ with respect to the mass $m$. Fig. 1 (b)
shows that the Fisher information
$\mathcal{F}_{\epsilon\epsilon}^{\text{iso}}$ reaches the maximum at the
optimal value of mass $m\approx 3.0$. This implies that the optimal precision
in the estimation of the expansion volume can be achieved at certain mass of
particles in the measuring process. On the other hand, notice that
$\mathcal{F}_{\rho\rho}^{\text{iso}}$ is a monotonic decreasing function of
mass, which means that when $m\rightarrow 0$, the estimation of the expansion
rate reaches the highest precision.

###  IV.2 Anisotropic case

In this section, we investigate the influence of the anisotropic perturbation
on the estimation of the cosmological parameters $\epsilon$ and $\rho$. In
addition, we consider the Fisher information for two values of $\xi$, namely,
the weak coupling ($\xi=0$) and the conformal coupling ($\xi=1/6$). For this
purpose, let us calculate the general analytical expression for the entries of
$\mathcal{F}_{ab}^{\text{aniso}}$ associated with $\hat{\rho}_{k}$:

$\displaystyle\mathcal{F}_{ab}^{\text{aniso}}$
$\displaystyle=\frac{1}{1-\Lambda_{k}}\sum_{n=0}^{\infty}\frac{1}{\Lambda_{k}^{n}}\left[\partial_{a}(1-\Lambda_{k})\Lambda_{k}^{n}\right]\left[\partial_{b}(1-\Lambda_{k})\Lambda_{k}^{n}\right]$
$\displaystyle=\frac{(\partial_{a}\Lambda_{k})(\partial_{b}\Lambda_{k})}{1-\Lambda_{k}}\sum_{n=0}^{\infty}\Lambda_{k}^{n}-\partial_{a}\Lambda_{k}\sum_{n=0}^{\infty}\partial_{b}\Lambda_{k}^{n}-\partial_{b}\Lambda_{k}\sum_{n=0}^{\infty}\partial_{a}\Lambda_{k}^{n}+(1-\Lambda_{k})\sum_{n=0}^{\infty}\frac{(\partial_{a}\Lambda_{k}^{n})(\partial_{b}\Lambda_{k}^{n})}{\Lambda_{k}^{n}},$

where
$\Lambda_{k}=\left|\frac{\beta_{k}^{\text{aniso}}}{\alpha_{k}^{\text{aniso}}}\right|^{2}$.
By replacing the following relations in the expression above

$\displaystyle\partial_{a(b)}\Lambda_{k}$
$\displaystyle=\Lambda_{k}\left[\frac{\partial_{a(b)}\beta_{k}^{\text{aniso}}}{\beta_{k}^{\text{aniso}}}+\frac{\partial_{a(b)}\beta_{k}^{\text{aniso}*}}{\beta_{k}^{\text{aniso}*}}-\frac{\partial_{a(b)}\alpha_{k}^{\text{aniso}}}{\alpha_{k}^{\text{aniso}}}-\frac{\partial_{a(b)}\alpha_{k}^{\text{aniso}*}}{\alpha_{k}^{\text{aniso}*}}\right],$
$\displaystyle=\Lambda_{k}\partial_{a(b)}[\ln\Lambda_{k}],$
$\displaystyle\partial_{a(b)}\Lambda_{k}^{n}$
$\displaystyle=n\Lambda_{k}^{n}\left[\frac{\partial_{a(b)}\beta_{k}^{\text{aniso}}}{\beta_{k}^{\text{aniso}}}+\frac{\partial_{a(b)}\beta_{k}^{\text{aniso}*}}{\beta_{k}^{\text{aniso}*}}-\frac{\partial_{a(b)}\alpha_{k}^{\text{aniso}}}{\alpha_{k}^{\text{aniso}}}-\frac{\partial_{a(b)}\alpha_{k}^{\text{aniso}*}}{\alpha_{k}^{\text{aniso}*}}\right],$
$\displaystyle=n\Lambda_{k}^{n}\partial_{a(b)}[\ln\Lambda_{k}],$

we obtain

$\displaystyle\mathcal{F}_{ab}^{\text{aniso}}$
$\displaystyle=\frac{\Lambda_{k}}{(1-\Lambda_{k})^{2}}[\partial_{a}\ln\Lambda_{k}\partial_{b}\ln\Lambda_{k}],$
$\displaystyle=\frac{4\Lambda_{k}}{(1-\Lambda_{k})^{2}}\left[\partial_{a}\ln\left|\frac{\beta_{k}^{\text{aniso}}}{\alpha_{k}^{\text{aniso}}}\right|\partial_{b}\ln\left|\frac{\beta_{k}^{\text{aniso}}}{\alpha_{k}^{\text{aniso}}}\right|\right],$
$\displaystyle=\frac{4\Lambda_{k}}{(1-\Lambda_{k})^{2}}\left[\partial_{a}\ln\left|\frac{\beta_{k}^{(m)}+\beta_{k}^{(\xi)}+\beta_{k}^{(h)}}{1+\alpha_{k}^{(m)}+\alpha_{k}^{(\xi)}+\alpha_{k}^{(h)}}\right|\partial_{b}\ln\left|\frac{\beta_{k}^{(m)}+\beta_{k}^{(\xi)}+\beta_{k}^{(h)}}{1+\alpha_{k}^{(m)}+\alpha_{k}^{(\xi)}+\alpha_{k}^{(h)}}\right|\right].$
(47)

This result, together with (22), allows us to determine the effects of the
anisotropic perturbation on the estimation of the cosmological parameters. In
order to perform the plots, let us assume that the wavevector
$\overrightarrow{k}=(k_{1},k_{2},k_{3})$ has a general direction specified by
the spherical coordinates $(k,\theta,\phi)$ as

$\displaystyle\overrightarrow{k}=k\sin\theta\cos\phi\hat{x}+k\sin\theta\sin\phi\hat{y}+\cos\theta\hat{z},$

where $k^{2}=k_{1}^{2}+k_{2}^{2}+k_{3}^{2}$. Notice that the effects of the
anisotropy is expected to depend on the direction of the particle momentum.
Thus, the influence of the anisotropy on the Fisher information can be
quantified by the azimuthal angle $\theta$ and/or the polar angle $\phi$.

Note that the Fisher information $\mathcal{F}_{\epsilon\epsilon}^{aniso}$ and
$\mathcal{F}_{\rho\rho}^{aniso}$ can be analytically evaluated, but it’s not a
convenient way. Of course, this can be analytically tackled under a particular
approximation. However, let’s do a numerical analysis of
$\mathcal{F}_{\epsilon\epsilon}^{aniso}$ and $\mathcal{F}_{\rho\rho}^{aniso}$
without approximation. In Fig. 2 we plot
$\mathcal{F}_{\epsilon\epsilon}^{aniso}$ and $\mathcal{F}_{\rho\rho}^{aniso}$
as a function of the momentum of the mode $k$ for different values of
expansion rate $\rho$. The plots show that
$\mathcal{F}_{\epsilon\epsilon}^{aniso}$ and $\mathcal{F}_{\rho\rho}^{aniso}$
are oscillating functions of the momentum $k$. In particular, in Fig. 2 (a) we
observe that the oscillation of $\mathcal{F}_{\epsilon\epsilon}^{aniso}$
increases when $\rho$ grows. On the other hand, the Fig. 2 (b) shows that the
oscillation of $\mathcal{F}_{\rho\rho}^{aniso}$ decreases when the expansion
rate grows. In addition, it is worth mentioning that, unlike of the isotropic
case, the Fisher information spectrum for the estimation of the expansion rate
$\rho$ is different of that for the estimation of the expansion volume
$\epsilon$.

Figure 2:  Fisher information $\mathcal{F}_{\epsilon\epsilon}^{\text{aniso}}$
and $\mathcal{F}_{\rho\rho}^{\text{aniso}}$ as function of the momentum $k$
for different values of $\rho$. Here we have fixed $m=0.001$, $\epsilon=0.1$,
$\xi=\frac{1}{6}$, $\theta=\phi=\frac{\pi}{2}$.

We are also interested in knowing how the coupling between the field and the
spacetime curvature affects the estimation of the cosmological parameters. In
Fig. 3 we plot $\mathcal{F}_{\epsilon\epsilon}^{aniso}$ and
$\mathcal{F}_{\rho\rho}^{aniso}$ as a function of the momentum of the mode $k$
for different values of the paraneter $\xi$. Fig. 3 shows that, in low energy
regime (for smaller values of momentum $k$), there is a difference in the
behavior of the Fisher information $\mathcal{F}_{\epsilon\epsilon}^{aniso}$
and $\mathcal{F}_{\rho\rho}^{aniso}$ as the coupling $\xi$ changes from
minimal ($\xi=0$) to conformal ($\xi=1/6$).

Figure 3:  Fisher information $\mathcal{F}_{\epsilon\epsilon}^{\text{aniso}}$
and $\mathcal{F}_{\rho\rho}^{\text{aniso}}$ as function of the momentum $k$
for different values of $\xi$. Here we have fixed $m=0.001$, $\epsilon=0.1$,
$\rho=15$, $\theta=\phi=\frac{\pi}{2}$.

In Fig. 4 we plot $\mathcal{F}_{\epsilon\epsilon}^{aniso}$ and
$\mathcal{F}_{\rho\rho}^{aniso}$ as a function of azimuthal angle $\theta$ for
different values of $\phi$. Fig. 4 shows that the estimation of the expansion
rate $\rho$ and the expansion volume $\epsilon$ is sensible to the direction
of the momentum $k$. For $\xi=0$ the highest precision in the estimation of
the expansion rate and the expansion volume can be achieved at a certain value
of the azimuthal angle $\theta$. On the other hand, when $\xi=1/6$ we observe
that $\mathcal{F}_{\epsilon\epsilon}^{aniso}$ and
$\mathcal{F}_{\rho\rho}^{aniso}$ are oscillating functions of the azimuthal
angle $\theta$. It is interesting to note that the oscillations change as
$\phi$ varies from $\pi/6$ to $\pi/2$.

Figure 4:  Fisher information $\mathcal{F}_{\epsilon\epsilon}^{\text{aniso}}$
and $\mathcal{F}_{\rho\rho}^{\text{aniso}}$ as function of azimuthal angle
$\theta$ for different values of $\phi$. Here we have fixed $k=m=0.01$,
$\epsilon=0.1$, $\rho=1$.

The above results suggest that a small gravitational disturbance (anisotropy)
changes the behavior of the Fisher information, introducing oscillations, when
compared with the isotropic case. The estimation of the expansion rate $\rho$
and the expansion volume $\epsilon$ is sensible to the direction of the
momentum $k$. Other interesting feature to note is that there is a substantial
difference between the Fisher information spectrum for the minimum and
conformal couplings. This differerence is significant only for small values of
momentum $k$.

##  V Conclusions

Starting from expanding spacetime models, we presented the estimate of the
cosmological parameters for the isotropic and anisotropic models. We have
observed that, unlike of the isotropic case, a small gravitational disturbance
(anisotropy) introduces oscillations in the Fisher information spectrum. The
estimation of the expansion rate $\rho$ and the expansion volume $\epsilon$ is
sensible to the direction of the momentum $k$ of particles created during the
cosmic expansion. Moreover, we have seen that there is a substantial
difference between the Fisher information spectrum for the minimum and
conformal couplings. Our analysis suggests that both anisotropy and coupling
between the field and the spacetime curvature significantly affect the
estimation of the cosmological parameters.

## Acknowledgments

OPSN wishes to acknowledge support by the EDITAL FAPEPI/MCT/CNPq No. 007/2018:
Programa de Infraestrutura para Jovens Pesquisadores/Programa Primeiro
Projetos (PPP). I.G.P. acknowledges Grant No. 307942/2019-8 from CNPq. PRSC
would like to thank CNPq for grants: Universal-431727/2018 and Produtividade
307982/2019-0. HASC would like to thank the Brazilian funding agency CAPES for
financial support.

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# Privacy-preserving Linear Computations in Spiking Neural P Systems

Mihail-Iulian Plesa University of Bucharest
Bucharest, RomaniaDepartment of Computer Science mihail-
<EMAIL_ADDRESS>University of Bradford
Bradford, UKSchool of Electrical Engineering and Computer ScienceUniversity of
Bucharest
Bucharest, RomaniaDepartment of Computer Science Marian Gheorghe University of
Bradford
Bradford, UKSchool of Electrical Engineering and Computer Science
<EMAIL_ADDRESS>University of Bucharest
Bucharest, RomaniaDepartment of Computer Science Florentin Ipate University of
Bucharest
Bucharest, RomaniaDepartment of Computer Science<EMAIL_ADDRESS>
###### Abstract

Spiking Neural P systems are a class of membrane computing models inspired
directly by biological neurons. Besides the theoretical progress made in this
new computational model, there are also numerous applications of P systems in
fields like formal verification, artificial intelligence, or cryptography.
Motivated by all the use cases of SN P systems, in this paper, we present a
new privacy-preserving protocol that enables a client to compute a linear
function using an SN P system hosted on a remote server. Our protocol allows
the client to use the server to evaluate functions of the form $t_{1}k+t_{2}$
without revealing $t_{1},t_{2}$ or $k$ and without the server knowing the
result. We also present an SN P system to implement any linear function over
natural numbers and some security considerations of our protocol in the
honest-but-curious security model.

##  1 Introduction

Membrane computing (or P systems) is a new model of computation inspired by
how membranes work and interact in living cells [18]. There are several
variants of the model e.g. neural P systems, cell P systems, tissue P systems,
etc., [30, 16, 13]. P systems have generated new perspectives on the P vs NP
problem, being used to efficiently solve hard problems [28, 4, 8, 29]. There
are also multiple applications of P systems in various fields like formal
verification, artificial intelligence, or cryptography [31].

In this work we used a special type of P systems called Spiking Neural P
systems (SN P systems for short) [13]. SN P systems are inspired by biological
neurons. There are also numerous variants of SN P systems: SN P systems with
astrocytes, SN P systems with communication on request, SN P systems with
polarization, SN P systems with colored spikes, etc., [17, 15, 26, 24].

Although there are many theoretical aspects and simulations in the literature,
to gain the maximum efficiency of these systems, they must be implemented on
dedicated hardware [2]. If these systems are implemented at a large scale,
they will have to be accessed remotely in the cloud. This raises privacy
concerns about data uploaded to the server that hosts the P system. This paper
approaches the problem of confidentiality in SN P systems by describing a
protocol that allows a client to perform a simple linear computation using an
SN P system that is served remotely without revealing private information.

###  1.1 Related work

Besides the theoretical work, there are also many applications of P systems.
In [23] the authors propose a new key agreement protocol based on SN P
systems. In [9, 11, 12] the authors describe how to implement the RSA
algorithm in the framework of membrane computing. One ingenious way of
applying P systems is shown in [25] which presents an algorithm to break the
RSA encryption. There are also applications in artificial intelligence. In [3]
the authors present a survey of the learning aspects in SN P systems.
Clustering algorithms have also been developed in the framework of membrane
computing [22, 21, 20]. Image processing is another common application of P
systems [5, 27, 6].

###  1.2 Our contribution

In this paper, we present a protocol that allows a client to perform a linear
computation using an SN P system hosted on a server without revealing any
private data. The SN P system computes functions of the form $t_{1}k+t_{2}$
over natural numbers. The client must retrieve from the server the result of
the computation without the server knowing $t_{1},t_{2}$ or $k$. Also, the
server must not learn the value $t_{1}k+t_{2}$. To enable privacy-preserving
computations on the server side, we use the ElGamal cryptosystem and its
homomorphic properties [7]. We also provide an SN P system that computes any
linear function over natural numbers and some security considerations of our
protocol. The paper is organized as follows: in Section 2 we present the
background on the SN P system and homomorphic encryption. In Section 3, we
show an SN P system that computes linear functions over the natural numbers.
In Section 4, we introduce our protocol and some security considerations.
Section 5 is left for the conclusions and further directions.

##  2 Preliminaries

In this section, we briefly present the Spiking Neural P systems (SN P
systems) and the cryptographic algorithm used in our protocol. We stress some
useful properties of the encryption scheme.

###  2.1 Spiking Neural P systems

A Spiking Neural P system (SN P system) of degree $m\geq 1$ is defined as the
following construct:

######  Definition 2.1.

$\Pi=\left(O,\sigma_{1},\sigma_{2},\dots,\sigma_{m},syn,i_{0}\right)$ where:

  * •

$O=\\{a\\}$ is the alphabet. The symbol $a$ denotes a spike.

  * •

$\sigma_{i},\ 1\leq i\leq m$ represents a neuron. Each neuron is characterized
by the initial number of spikes denoted by $n_{i}\geq 0$ and the finite set of
rules denoted by $R_{i}$: $\sigma_{i}=\left(n_{i},R_{i}\right)$.

  * •

Each rule can be of the following two forms:

    1. 1.

$E/a^{r}\rightarrow a;t$ where $E$ is a regular expression over the alphabet
$O$, $r\in\mathbf{N^{*}}$ represents the current number of spikes in the
neuron and $t\geq 0$ is the refractory period. This type of rule is called a
firing rule.

    2. 2.

$a^{s}\rightarrow\lambda$ where $s\geq 1$ is the current number of spikes in
the neuron and $\lambda$ is a special symbol that denotes an empty set of
spikes. This type of rule is called a forgetting rule.

  * •

$syn\subseteq\\{1,2,\dots,m\\}\times\\{1,2,\dots,m\\}$ is the set of synapses
between neurons. No neuron can have a synapse to it i.e.
$\left(i,i\right)\notin syn$ $\forall\ 1\leq i\leq m$.

  * •

$i_{0}$ represents the output neuron.

A neuron can fire using the firing rule $E/a^{r}\rightarrow a;t$ only if it
contains $n$ spikes such that $a^{n}\in L\left(E\right)$ and $n\geq r$ where
$L\left(E\right)$ is a language defined in the following way:

  * •

$L\left(\lambda\right)=\\{\lambda\\}$

  * •

$L\left(a\right)=\\{a\\}\ \forall a\in O$

  * •

$L\left(\left(E_{1}\right)\cup\left(E_{2}\right)\right)=L\left(E_{1}\right)\cup
L\left(E_{2}\right)$

  * •

$L\left(\left(E_{1}\right)\left(E_{2}\right)\right)=L\left(E_{1}\right)L\left(E_{2}\right)$

  * •

$L\left(\left(E_{1}\right)^{+}\right)=L\left(E_{1}\right)^{+}$

for all regular expressions over the alphabet $O$.

After firing, $r$ spikes are consumed. A firing rule that is applied when the
neuron contains exactly $r$ spikes i.e. $L\left(E\right)=\\{a^{r}\\}$, is
simply denoted as $a^{r}\rightarrow a;t$.

At the neuron level, all rules are applied sequentially, but the system as a
whole evolves with maximum parallelism i.e. if a rule can be applied in a
neuron then that rule will be applied.

At a certain point, a neuron can be firing, spiking, or closed. If a neuron
applies the firing rule $E/a^{r}\rightarrow a;t$ at moment $q$ then the neuron
will send a spike to all the neurons to which it is connected by synapses at
moment $q+t$. At times $q+1,q+2,\dots q+t-1$ the neuron will be in the
refractory period i.e. the neuron will not receive or send any spikes. When
neuron $\sigma_{i}$ is spiking, the spikes are replicated in such a way that
each neuron $\sigma_{j}$ with $\left(i,j\right)\in syn$ receives one spike
although the number of spikes consumed by $\sigma_{i}$ is exactly $r$.

When a forgetting rule $a^{s}\rightarrow\lambda$ is applied in a neuron, $s$
spikes are removed from that neuron. A neuron can apply a forgetting rule only
if the number of spikes is exactly $s$.

There are several ways in which we can record the output of an SN P system:

  * •

The moments of time at which the output neuron $i_{0}$ sends a spike i.e. if
the neuron $i_{0}$ releases spikes at the moments $q_{1},q_{2}\dots$ then the
output of $\Pi$ is the sequence $q_{1},q_{2}\dots$.

  * •

The interval between the moments at which the output neuron $i_{0}$ sends a
spike i.e. if the neuron $i_{0}$ releases spikes at the moments
$q_{1},q_{2}\dots$ then the output of $\Pi$ is the sequence
$q_{2}-q_{1},q_{3}-q_{2},\dots$

An SN P system is constructed using the principle of minimal determinism i.e.
at a certain moment in time, either a firing or a forgetting rule is applied
without being able to choose which of the two types of rules is applied [13].

###  2.2 Homomorphic encryption

In this work, we use the ElGamal cryptosystem [7]. The security of the
encryption scheme is based on the computational Diffie-Hellman assumption
(CDH). Moreover, the scheme achieves semantic security based on the decisional
Diffie-Hellman assumption (DDH) i.e. the scheme is randomized. Randomization
implies that when encrypting the same message multiple times, each resulting
ciphertext will be different. A consequence of this property is the fact that
an attacker cannot distinguish two plaintexts by analyzing the corresponding
ciphertext with non-negligible probability. The scheme works over a group $G$
of order $q$ with a generator $g$. We now proceed to the description of the
cryptosystem:

  * •

The key generation algorithm denoted by KeyGen generates a key pair i.e. a
private key and the corresponding public key. The algorithm takes the
following steps:

    1. 1.

Generate a random integer $x\in\\{1,2,\dots q-1\\}$.

    2. 2.

Compute $h:=g^{x}$.

    3. 3.

Output the public key $h$ and the corresponding private key $x$.

  * •

The encryption algorithm denoted by $\textbf{Enc}^{y}_{h}$ encrypts a
plaintext $m\in G$ using the public key $h$ and a random number
$y\in\\{1,2,\dots,q-1\\}$. The algorithm performs the following steps:

    1. 1.

Computes $s:=h^{y}$.

    2. 2.

Computes $c_{1}:=g^{y}$ and $c_{2}:=m\cdot s$.

    3. 3.

Outputs the ciphertext $c:=\left(c_{1},c_{2}\right)$.

  * •

The decryption algorithm denoted by $\textbf{Dec}_{x}$ takes as input a
ciphertext $c=\left(c_{1},c_{2}\right)$ and decrypts it under the private key
$x$. The algorithm is composed of the following steps:

    1. 1.

Computes $s:=c_{1}^{x}$.

    2. 2.

Computes $s^{-1}$, the inverse of $s$ in the group $G$.

    3. 3.

Computes the plaintext $m:=c_{2}\cdot s^{-1}$.

    4. 4.

Outputs the plaintext $m$.

The proof of correctness is straightforward:

$c_{2}\cdot s^{-1}=c_{2}\cdot c_{1}^{-x}=m\cdot h^{y}\cdot g^{-xy}=m\cdot
g^{-xy}\cdot g^{xy}=m$ (1)

The encryption of a message $m\in G$ can be summarized by the following two
equations:

$c_{1}=g^{y}$ (2) $c_{2}=m\cdot h^{y}$ (3)

The scheme is homomorphic with respect to multiplication. Let
$c=\left(c_{1},c_{2}\right)$ and
$c^{\prime}=\left(c_{1}^{\prime},c_{2}^{\prime}\right)$ be the encryptions of
two plaintexts $m$ and $m^{\prime}$ under the same public key $h$ i.e.
$c=\textbf{Enc}^{y}_{h}\left(m\right)$ and
$c^{\prime}=\textbf{Enc}^{y}_{h}\left(m^{\prime}\right)$. We define the
following two operations:

  1. 1.

Let $c\odot c^{\prime}=\left(c_{1}\cdot c_{1}^{\prime},c_{2}\cdot
c_{2}^{\prime}\right)$ be the multiplication of two ciphertexts. The result of
this operation is another ciphertext that encrypts the sum between $m$ and
$m^{\prime}$:

$c_{1}\cdot c_{1}^{\prime}=g^{y}\cdot g^{y^{\prime}}=g^{y+y^{\prime}}$ (4)
$c_{2}\cdot c_{2}^{\prime}=m\cdot s\cdot m^{\prime}\cdot s^{\prime}=m\cdot
m^{\prime}\cdot h^{y}\cdot h^{y^{\prime}}=m\cdot m^{\prime}\cdot
h^{y+y^{\prime}}$ (5)

From 4 and 5 we can see that $c\odot c^{\prime}=\left(c_{1}\cdot
c_{1}^{\prime},c_{2}\cdot c_{2}^{\prime}\right)$ is a ciphertext that encrypts
$m\cdot m^{\prime}$ under the public key $h$.

  2. 2.

Let $c\otimes k=\left(c_{1},c_{2}\cdot k\right)$ be the multiplication between
a ciphertext and a constant $k$. The result of this operation is a ciphertext
that encrypts the product between $m$ and $k$ as it can be seen from 6 and 7:

$c_{1}=g^{y}$ (6) $c_{2}\cdot k=m\cdot s\cdot k=\left(m\cdot k\right)\cdot
s=\left(m\cdot k\right)h^{y}$ (7)

  3. 3.

When $c_{1}=c_{1}^{\prime}$ i.e. $y=y^{\prime}$, we can also define the
addition between two ciphertexts as follows: $c\oplus
c^{\prime}=\left(c_{1},c_{2}+c_{2}^{\prime}\right)$. The result of this
operation is a ciphertext that encrypts the sum between $m$ and $m^{\prime}$:

$c_{1}=g^{y}$ (8) $c_{2}+c_{2}^{\prime}=m\cdot s+m^{\prime}\cdot
s=\left(m+m^{\prime}\right)\cdot s=\left(m+m^{\prime}\right)\cdot h^{y}$ (9)

It is important to notice that when $y=y^{\prime}$ the scheme is no longer
semantically secure. Although an attacker who observes the two ciphertexts $c$
and $c^{\prime}$ could not recover any of the plaintexts, it could determine
additional information about them e.g. whether they are different. In many
scenarios, this is not acceptable but in this work, we will use the $\oplus$
operation. All the messages encrypted with the same random $y$ are not
critical i.e. the impact of the lack of semantic security does not affect the
security of the protocol in which the encryption scheme is used.

##  3 SN P system to compute linear functions

In this section, we describe an SN P system that computes functions of the
form $t_{1}k+t_{2}$ over natural numbers.

$a^{2k-1}$$a^{+}/a\rightarrow a;t_{1}-1$$\sigma_{1}$$a$$a\rightarrow
a;0$$\sigma_{2}$$a^{k}\rightarrow a;t_{2}-1$$\sigma_{3}$ Figure 1:  SN P
system to compute linear functions

Let
$\Pi_{Add}\left(t_{1},t_{2},k\right)=\\{\\{a\\},\sigma_{1},\sigma_{2},\sigma_{3},syn_{add},\sigma_{2}\\}$
be the SN P system that computes the linear function $t_{1}k+t_{2}$,
$t_{1},t_{2}\in\mathbf{N}$ with the following components:

  * •

The alphabet is made from a single symbol $\\{a\\}$ that denotes a spike.

  * •

There are three neurons: $\sigma_{1},\sigma_{2}$ and $\sigma_{3}$ with the
following firing rules:

    1. 1.

For $\sigma_{1}$ the firing rule is $a^{+}/a\rightarrow a;t_{1}-1$.

    2. 2.

For $\sigma_{2}$ the firing rule is $a\rightarrow a;0$.

    3. 3.

For $\sigma_{3}$ the firing rule is $a^{k}\rightarrow a;t_{2}-1$.

  * •

The set of synapses $syn_{add}$ is the set
$\\{\left(1,3\right),\left(3,2\right)\\}$.

  * •

The output neuron is $\sigma_{2}$.

Initially $\sigma_{1}$ has $2k-1$ spikes, $\sigma_{2}$ has $1$ spike and
$\sigma_{3}$ has no spikes. Since $\sigma_{2}$ has no refractory time and one
spike, it will release it in the first step of the computation. During the
first step, $\sigma_{1}$ will be firing. Since its refractory period is
$t_{1}-1$ during the time steps $2,3,\dots t_{1}-1$ the neuron will be closed
i.e. it will not receive or send any spikes. In step $t_{1}$ the neuron will
send one spike to $\sigma_{3}$ and fire again. Thus $\sigma_{1}$ fires every
$t_{1}$ steps consuming one spike and sending one spike to $\sigma_{3}$.
$\sigma_{3}$ will fire when it acumulates $k$ spikes from neuron $\sigma_{1}$.
Since $\sigma_{1}$ fires one spikes every $t_{1}$ steps, at time step, $k\cdot
t_{1}$ $\sigma_{3}$ will receive the $k^{th}$ spike. At the moment $t_{1}\cdot
k+1$, $\sigma_{3}$ will fire. The refractory period of this neuron is
$t_{2}-1$ thus at the steps $t_{1}\cdot k+2,t_{1}\cdot k+3,\dots,t_{1}\cdot
k+t_{2}-1$ it will be closed and it will release one spike to $\sigma_{2}$ at
$t_{1}\cdot k+t_{2}$. Since $\sigma_{3}$ has no refractory period, it will
release the spike at the moment $t_{1}\cdot k+t_{2}+1$. At this point, the
number of spikes left in $\sigma_{1}$ is $k-1$ because it already sent $k$
spikes to neuron $\sigma_{3}$ and the initial number of spikes was $2k-1$. The
neuron will continue to send spikes to $\sigma_{3}$ at the appropriate time
steps until the spikes run out. The neuron $\sigma_{3}$ will never fire again
since it can no longer accumulate $k$ spikes. $\sigma_{2}$ will never fire
again either since it will no longer receive the spike from $\sigma_{3}$.
Thus, after $\sigma_{1}$ exhausts all the spikes, the computation will stop.
There are two moments when the output neuron $\sigma_{2}$ fires: $1$ and
$t_{1}\cdot k+t_{2}+1$. Thus, the result of the computation i.e. the
difference between the time points at which the output neuron fires, is
$t_{1}\cdot k+t_{2}$. The $\Pi_{Add}\left(t_{1},t_{2},k\right)$ system is
depicted in Figure 1.

##  4 Privacy-preserving computations in SN P systems

In this section, we describe our protocol which enables the running of an SN P
system to compute linear functions in a privacy-preserving way. We also make
some remarks regarding security.

###  4.1 The protocol

There are two actors in the protocol:

  1. 1.

The Server: it can instantiate and run an SN P system of the form
$\Pi_{Add}\left(t_{1},t_{2},k\right)$ for any integers $t_{1},t_{2}$ and $k$.

  2. 2.

The Client: it wants to evaluate the linear function $t_{1}\cdot k+t_{2}$
using the system hosted by the Server without revealing any of inputs
$t_{1},t_{2}$ or $k$.

The protocol uses the holomorphic properties of the ElGamal cryptosystem to
allow the client to use the server without revealing the inputs of the SN P
system. There are $7$ steps:

  1. 1.

The Client will use the KeyGen algorithm to generate a key pair: $h$ and $x$.

  2. 2.

The Client will use the encryption algorithm $\textbf{Enc}^{y}_{h}$ to encrypt
$t_{1},t_{2}$ and $k$. We denote by
$c^{t_{1}}=\left(c^{t_{1}}_{1},c^{t_{1}}_{2}\right),c^{t_{2}}=\left(c^{t_{2}}_{1},c^{t_{2}}_{2}\right)$
and $c^{k}=\left(c^{t_{k}}_{1},c^{k}_{2}\right)$ the encryptions of
$t_{1},t_{2}$ and $k$:

     * •

$c^{t_{1}}=\left(c^{t_{1}}_{1},c^{t_{1}}_{2}\right)=\textbf{Enc}^{y_{1}}_{h}\left(t_{1}\right)$

     * •

$c^{k}=\left(c^{k}_{1},c^{k}_{2}\right)=\textbf{Enc}^{y_{2}}_{h}\left(k\right)$

     * •

$c^{t_{2}}=\left(c^{t_{2}}_{1},c^{t_{2}}_{2}\right)=\textbf{Enc}^{y_{1}+y_{2}}_{h}\left(t_{2}\right)$

  3. 3.

The Client will store locally $c^{t_{1}}_{1},c^{t_{2}}_{1}$ and $c^{k}_{1}$
and send to the server $c^{t_{1}}_{2},c^{t_{2}}_{2}$ and $c^{k}_{2}$.

  4. 4.

The Server will instantiate and run an SN P system of the form
$\Pi_{Add}\left(c^{t_{1}}_{2},c^{t_{2}}_{2},c^{k}_{2}\right)$.

  5. 5.

After the computation stops, the Server will return to the client the result
of the computation i.e. $c_{2}=c^{t_{1}}_{2}\cdot c^{k}_{2}+c^{t_{2}}_{2}$.

  6. 6.

The Client will compose a new ciphertext $c=\left(c_{1},c_{2}\right)$ where
$c_{1}=c^{t_{1}}_{1}\cdot c^{k}_{1}$.

  7. 7.

The Client will decrypt the ciphertext $c$ using the algorithm
$\textbf{Dec}_{x}$ and retrived the result of the computation i.e. $t_{1}\cdot
k+t_{2}$.

We now prove that the ciphertext computed in step $6$ of the protocol is a
valid ElGamal ciphertext that correctly decrypts to the final result of the
computation i.e. $t_{1}\cdot k+t_{2}$. Let $c^{\prime}$ be the following
ciphertext:

$c^{\prime}=\left(c_{1},c^{t_{1}}_{2}\cdot c^{k}_{2}\right)$ (10)

Since $c_{1}=c^{t_{1}}_{1}\cdot c^{k}_{1}$ we can write $c^{\prime}$ as:

$c^{\prime}=\left(c^{t_{1}}_{1}\cdot c^{k}_{1},c^{t_{1}}_{2}\cdot
c^{k}_{2}\right)$ (11)

The ciphertext $c^{\prime}$ represents the multiplication between the
ciphertexts $c^{t_{1}}$ and $c^{k}$:

$c^{\prime}=c^{t_{1}}\odot c^{k}$ (12)

Since $c$ and $c^{\prime}$ use the same randomess i.e. $c_{1}$, we can write
$c$ as the sum between the ciphertext $c^{\prime}$ and $c^{t_{2}}$:

$c=c^{\prime}\oplus c^{t_{2}}$ (13)

In conclusion, we can express $c$ as a composition of valid ElGamal
ciphertexts which is also a valid ElGamal ciphertext:

$c=\left(c^{t_{1}}\odot c^{k}\right)\oplus c^{t_{2}}$ (14)

The ciphertext $\left(c^{t_{1}}\odot c^{k}\right)$ represents the encryption
of $t_{1}\cdot k$. When we add this ciphertext with $c^{t_{2}}$ using the
$\oplus$ operation, the resulting ciphertext will be the encryption of
$t_{1}\cdot k+t_{2}$ which is the result of the computation performed over
plaintext data. Figure 2 depicts our protocol.

Since the SN P system works over natural numbers, we can use
$G=\mathbf{Z}_{q}$ for a large prime $q$.

Public
parameters:$G,g,q$ClientKeyGen$c^{t_{1}},c^{t_{2}},c^{k}$$c=\left(c_{1},c_{2}\right)$$\textbf{Dec}_{x}\left(c\right)$Server$\Pi_{Add}\left(c^{t_{1}}_{2},c^{t_{2}}_{2},c^{k}_{2}\right)$$c_{2}=c^{t_{1}}_{2}\cdot
c^{k}_{2}+c^{t_{2}}_{2}$${c^{t_{1}}_{2},c^{t_{2}}_{2},c^{k}_{2}}$$c_{2}$
Figure 2:  Privacy-preserving linear function computation using SN P systems

###  4.2 Security considerations

We analyze our protocol in the honest-but-curious security model [19]. In this
model, we assume that the adversary is the Server. There are two properties of
the adversary:

  1. 1.

Curious: the Server will try to find information about the underlying
plaintexts. In our case, these are the parameters of the SN P system:
$t_{1},t_{2}$, and $k$.

  2. 2.

Honest: the Server will respect the protocol and it will complete every step.
It will not modify in any way the messages received or sent to the Client.

The underlying encryption scheme i.e. the ElGamal cryptosystem is semantically
secure given the DDH assumption as long as each ciphertext is generated using
different randomness [10]. The security of the scheme can be illustrated using
the following game. We suppose that the attacker runs in probabilistic
polynomial time and has access to an encryption oracle that receives as input
a plaintext and returns the corresponding ciphertext:

  * •

The attacker chooses as many plaintexts as it wants and encrypts them using
the oracle. For each ciphertext, the attacker knows the corresponding
plaintext.

  * •

The attacker chooses two plaintext $m_{0}$ and $m_{1}$ and send them to the
oracle.

  * •

The oracle will generate a random bit $b$ and return the encryption of
$m_{b}$.

  * •

The attacker outputs the bit $b$.

The scheme is secure as long as the attacker cannot output the correct bit
with non-negligible probability.

Given the fact that the Client encrypts each parameter of the SN P system
using the ElGamal cryptosystem with different randomness, the Server cannot
learn any information about them with non-negligible probability.

##  5 Conclusions and further directions of research

In this paper, we presented a protocol for performing privacy-preserving
computation over SN P systems. There are two actors involved: the client and
the server. The server hosts an SN P system that computes linear function over
natural numbers i.e. $t_{1}k+t_{2}$. The client uses the protocol to retrieve
the result of the computation without revealing $t_{1},t_{2}$ or $k$ and
without the server knowing the result of the calculation. We presented an SN P
system that computes any linear function over natural numbers and also some
security considerations about our protocol which is based on the ElGamal
cryptosystem.

There are several directions of research. The first one is to give formal
proof of the security of the protocol. Although the protocol is secure at
first sight, we must prove it by reducing the security of it to the security
of the underlying cryptosystem. The second direction of research is to enable
complex computation on SN P systems in a privacy-preserving way. The third
direction is to use dedicated cryptosystems e.g. fully homomorphic encryption
schemes which are created to enable privacy-preserving computations [14]. The
challenge here is to map the operations performed over encrypted data to the
operations performed by the SN P system. We should also consider using other
privacy-enhancing technologies e.g. secure multi-party computation or
differential privacy to enable private computations over SN P systems. Another
direction of research is to implement the protocol and perform the appropriate
benchmarking to study its efficiency and communication overhead in practice.

####  5.0.1 Acknowledgements

This research was supported by the European Regional Development Fund,
Competitiveness Operational Program 2014-2020 through project IDBC (code SMIS
2014+: 121512).

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