# Quantum probes for universal gravity corrections

Alessandro Candeloro<EMAIL_ADDRESS>Quantum Technology Lab,
Dipartimento di Fisica Aldo Pontremoli, Università degli Studi di Milano, I -
20133, Milano, Italy.  Cristian Degli Esposti Boschi  CNR-IMM, Sezione di
Bologna, Via Gobetti 101, I - 40129 Bologna, Italy.  Matteo G.A. Paris
<EMAIL_ADDRESS>Quantum Technology Lab, Dipartimento di Fisica
Aldo Pontremoli, Università degli Studi di Milano, I - 20133, Milano, Italy.

###### Abstract

We address estimation of the minimum length arising from gravitational
theories. In particular, we provide bounds on precision and assess the use of
quantum probes to enhance the estimation performances. At first, we review the
concept of minimum length and show how it induces a perturbative term
appearing in the Hamiltonian of any quantum system, which is proportional to a
parameter depending on the minimum length. We then systematically study the
effects of this perturbation on different state preparations for several
1-dimensional systems, and we evaluate the Quantum Fisher Information in order
to find the ultimate bounds to the precision of any estimation procedure.
Eventually, we investigate the role of dimensionality by analysing the use of
two-dimensional square well and harmonic oscillator systems to probe the
minimal length. Our results show that quantum probes are convenient resources,
providing potential enhancement in precision. Additionally, our results
provide a set of guidelines to design possible future experiments to detect
minimal length.

##  I  Introduction

In the last decades, various theories of quantum gravity have been proposed,
which tried to jointly describe the quantum world and the gravitational force.
Albeit all of these theories have different postulates on the fundamental
nature of space and time, they all have a common model-independent prediction:
the existence of a minimum length Hossenfelder (2013) commonly associated with
the Planck length ${\hbox{L}}_{\hbox{\tiny P}}$.Thanks to a device-independent
proof Calmet  _et al._ (2004, 2005) the physical reason behind this result is
quite clear: if we want to measure the position of a massive particle, the
more the particle is massive the best precision we can achieve. On the other
hand, if the mass exceeds a specific value (established by the laws of general
relativity), we will run in the black hole regime, thus increasing the
uncertainty on the position. From these considerations, it can be induced that
Maggiore (1993); Calmet  _et al._ (2004, 2005); Markopoulou and Smolin (2004);
Bang and Berger (2006); Das and Vagenas (2008); Hossenfelder (2013)

$\Delta x_{i}\geq{\hbox{L}}_{\hbox{\tiny P}}.$ (1)

Overall, we have that upon assuming minimal compatibility with general
relativity, a momentum-independent lower bound on the precision of any
position measurement should appear, and any length under this lower bound
loses physical meaning. Of course, in standard Quantum Mechanics, we do not
have an independent lower bound on $\Delta x_{i}$, which should just satisfies
the standard uncertainty relations $\Delta x_{i}\Delta
p_{j}\geq\delta_{ij}\hbar/2$. We may ask how reproduce such minimum length
effect in the non-relativistic quantum mechanics. Some solutions have been
suggestedHossenfelder (2013); Das and Vagenas (2008), e.g. by modifying the
particle momentum with an extra ad-hoc-parameter-dependent term Hossenfelder
(2013); Das and Vagenas (2008)

$\vec{p}=\vec{p}_{0}\left(1+\gamma\,\frac{\vec{p}_{0}^{\,2}}{{\hbox{M}}_{\hbox{\tiny
P}}^{2}c^{2}}\right).$ (2)

The parameter $\gamma$ does depend on the minimum length and may be understood
as a self-gravity perturbation 111This kind of interpretation has however some
conceptual problems, see e.g.Hossenfelder (2013) for explanations. As a
result, the standard commutation relations are modifiedHossenfelder (2013);
Das and Vagenas (2008); Rossi  _et al._ (2016); Maggiore (1993), leading to
the so-called Generalized Uncertainty Principle (GUP) holds

$\Delta x_{i}\Delta p_{i}\geq\frac{\hbar}{2}\left(1+\gamma\,\frac{\Delta
p^{2}+\langle p\rangle^{2}}{{\hbox{M}}_{\hbox{\tiny
P}}^{2}c^{2}}+2\gamma\,\frac{\Delta p_{i}^{2}+\langle
p_{i}\rangle^{2}}{{\hbox{M}}_{\hbox{\tiny P}}^{2}c^{2}}\right),$ (3)

which replicates the minimum length effect (1). Furthermore, the momentum
modification (2) affects directly the Hamiltonian of any non-relativistic
system. Indeed, at first order in $\gamma$, we have that
$\mathcal{H}=\mathcal{H}_{0}+\gamma\mathcal{H}_{1}+\mathcal{O}(\gamma^{2})$,
where the extra term

$\mathcal{H}_{1}=\frac{\gamma}{m\,{\hbox{M}}_{\hbox{\tiny
P}}^{2}c^{2}}\,p^{4}$ (4)

is the gravity perturbation and it represents the gravitational effect on a
generic quantum system, due to the modified momentum. This extra term does not
depend on the system under consideration, i.e. on the form of
$\mathcal{H}_{0}$, and it is therefore referred to as the _universal quantum
gravity correction_ term. The consequences of the perturbation on the energy
spectrum have been analysedBrau (1999); Das and Vagenas (2008); Kempf  _et
al._ (1995); Berger and Maziashvili (2011) as well as their effects on
cosmological Ashoorioon  _et al._ (2005); Vakili (2008); Maziashvili (2012)
and inflationary model Kempf and Lorenz (2006); Maziashvili (2012). Other
phenomenological implication had been explored on the set of coherent states
Benczik  _et al._ (2002); Ching  _et al._ (2012); Ching and Ng (2013), and
their superpositions Ching and Ng (2019). Moreover, the concept of GUP can be
applied in the framework of optics, where a formally identical system
describes pulse propagation with higher-order dispersion Braidotti  _et al._
(2017). Finally, a proposal to test such perturbation with a massive
mechanical oscillator was also suggestedPikovski  _et al._ (2012).

In this paper, we address the problem of estimating the parameter $\gamma$ by
exploiting quantum probes, i.e. by performing measurements on a quantum system
subjected to a given potential, and to the gravity corrections. Our goal is to
find the ultimate limits to the precision and to compare different systems in
terms of their ultimate performances. To this aim, we employ tools and ideas
from local quantum estimation theory (LQE) Paris (2009), which allows ones to
quantify the information carried by the state of the system on the parameter
$\gamma$, and to determine the lower bound on the variance of an estimator. In
turn, the paradigm of quantum probing has successfully employed in recent
years to different estimation problems in quantum technology and fundamental
physics and appears as a promising avenue to the search of new physics. As an
example, we cite the approach used by Braun  _et al._ (2017) to find the
minimum intrinsic error on the measurement of the speed of light in a cavity,
which results in restrictions on the probing of quantum gravity fluctuations.
In our work, we assume that the parameter $\gamma$ is small, a fact supported
by the lack of empirical evidence of the perturbation $\mathcal{H}_{1}$, such
that we may use perturbation theory to take into account gravity corrections.
In the perturbative regime, we study different quantum probes, which means
different systems and different state preparations, to find the optimal ones,
i.e. those providing the lowest bound to the precision.

The paper is structured as follows. In Section II we review local quantum
estimation, its main results as well as its geometrical interpretation. In
Section III, using perturbation theory, we study the estimability of the
coupling parameter $\gamma$ of a given perturbation $\mathcal{H}_{1}$. Then,
in IV, we apply these results to the estimation of gravity perturbations (4)
in several 1-dimensional systems to find which one provides better
performance. Eventually, in V, we investigate the relationship between the
dimensionality of the system and the Quantum Fisher Information, to assess a
possible enhancement.

##  II  Quantum Estimation Theory

Estimation theory deals with the problem of estimating the values of a set of
parameters from a data set of empirical values. Differently from a statistical
inference problem, where we do not know the probability distribution of the
empirical values, in an estimation problem this is well known: what it is not
known is the set of the parameters from which the distribution depends on. In
the quantum world, many parameters do not correspond to quantum observable and
they can not be measured directly. Instead, an indirect estimate from a set of
empirical values should be performed. In this procedure, the observer has the
freedom to choose different state preparations and/or different detectors,
i.e. different positive operator-valued measures (POVMs). There are two
different ways to address the problem of quantum estimation. Global Quantum
Estimation Theory pursues the POVM minimizing a suitable cost functional which
must be averaged over all the possible value of the parameter. Thereby it
results in a single POVM which does not depend on the value of the parameter.
Instead, Local Quantum Estimation Theory search for the POVM minimizing the
variance of the parameter estimator at a fixed value of the parameter. Despite
the POVM could depend on the parameter, the minimization concerns only a
specific value of the parameter and we may expect a better estimate.
Hereinafter we will use tools provided by Local QET to find the best
measurements and the best states to achieve the best estimate of $\gamma$ and
in this section we briefly review the ideas behind Local QET Paris (2009).

A classical estimation problem consists in a finite set of empirical data
$\\{x_{1},x_{2},\dots,x_{n}\\}$ belonging to the observation space
$\mathcal{S}_{n}$ and following a probability distribution
$\textbf{p}_{\gamma}(x)$ which depends on an unknown parameter
$\gamma\in\mathcal{A}$, whose value we want to estimate. An estimator is a
function $\gamma^{*}$ of the data in the set $\mathcal{A}$ of possible values
of the parameter

$\gamma^{*}:\mathcal{S}_{n}\longrightarrow\mathcal{A}.$ (5)

Among all the possible $\gamma^{*}$, optimal unbiased estimators are those
saturating the Cramer-Rao inequality Van Trees (2004); Lehmann and Casella
(2006)

$\textbf{Var}(\gamma^{*})\geq\frac{1}{n\mathcal{F}_{c}(\gamma)},$ (6)

where $n$ is the number of empirical value and $\mathcal{F}_{c}(\gamma)$ is
the classical Fisher Information

$\mathcal{F}_{c}(\gamma)=\int_{\mathcal{S}}\textbf{p}_{\gamma}(x)\left[\partial_{\gamma}\log{\textbf{p}_{\gamma}(x)}\right]^{2}dx,$
(7)

representing a measure on the amount of information carried by the probability
distribution on the parameter $\gamma$ Petz and Ghinea (2011). This lower
bound on the variance that an estimator $\gamma^{*}$ can achieve is
independent on the estimator used, meaning that it is an universal bound: no
estimator can be more precise than an optimal one. Moving to Quantum
Mechanics, a quantum statistical model consists of a family of quantum states
$\\{\rho_{\gamma}\\}$, depending on a parameter $\gamma$, i.e. a family of
states encoding the information about $\gamma$ Amari and Nagaoka (2007). If we
measure the generalized observable described by the POVM $\mathcal{E}_{m}$
($\mathcal{E}_{m}\geq 0$, $\sum_{m}\mathcal{E}_{m}={\mathbb{I}}$), the
probability distribution is determined both by the state and the POVM
according to the Born rule

$\textbf{p}_{\gamma}(m)=\hbox{Tr}\left[\mathcal{E}_{m}\rho_{\gamma}\right],$
(8)

where $m$ labels a possible outcome of the measurement. The central problem of
Quantum Estimation Theory is to determine the state $\rho_{\gamma}$ and the
POVM $\mathcal{E}_{m}$ that maximizes the $\mathcal{F}_{c}(\gamma)$, i.e.
minimize the lower bound on the variance. Using the Born rule, the classical
lower bound is given by

$\mathcal{F}_{c}[\mathcal{E}_{m}](\gamma)=\int_{\mathcal{S}}\\!dm\,\frac{\left\\{\partial_{\gamma}\hbox{Tr}\left[\mathcal{E}_{m}\rho_{\gamma}\right]\right\\}^{2}}{\hbox{Tr}\left[\mathcal{E}_{m}\rho_{\gamma}\right]}.$
(9)

Using the Schwartz inequality and the completeness property of the POVM one
can see that $\mathcal{F}_{c}[\mathcal{E}_{m}](\gamma)$ has a maximum among
all the possible measurement $\mathcal{E}_{m}$. This maximum is given by the
so-called Quantum Fisher Information (QFI)

$\mathcal{F}_{q}(\gamma)=\hbox{Tr}\left[\Lambda_{\gamma}\rho_{\gamma}^{2}\right]\geq\mathcal{F}_{c}[\mathcal{E}_{m}](\gamma)\quad\forall\mathcal{E}_{m}$
(10)

where $\Lambda_{\gamma}$ is the Symmetric Logarithmic Derivative (SLD) defined
implicitly by $\rho_{\gamma}$ as

$\frac{\Lambda_{\gamma}\rho_{\gamma}+\rho_{\gamma}\Lambda_{\gamma}}{2}=\partial_{\gamma}\rho_{\gamma}.$
(11)

As a result, the quantum counterpart of the Cramer-Rao theorem holds

$\textbf{Var}(\gamma^{*})\geq\frac{1}{n\mathcal{F}_{q}(\gamma)}.$ (12)

The quantum CR bound fixes a lower bound on the precision of any estimator. In
order to saturate the Quantum Cramer Rao Bound, besides using an optimal
estimator $\gamma^{*}$, we need also to implement the optimal measurement,
which is given by the projectors on the eigenspace of $\Lambda_{\gamma}$ Paris
(2009).

The concepts of quantum statistical model and that of Quantum Fisher
Information also has a rather natural geometrical interpretation, related to
the notion of distinguishability Amari and Nagaoka (2007); Facchi  _et al._
(2010). To illustrate this point, let us consider the Bures distance between
two quantum states $\rho$ and $\sigma$

$D_{B}(\rho,\sigma)=\sqrt{2-2\sqrt{F(\rho,\sigma)}}.$ (13)

where $F$ is the fidelity
$F(\rho,\sigma)=\Big{[}\hbox{Tr}\left[\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right]\Big{]}^{2}$
Sommers and Zyczkowski (2003). Using the parameter $\gamma$ as a coordinate,
we may introduce the Bures metric $g_{B}(\gamma)$ in the quantum statistical
model space as

$D^{2}_{B}(\rho_{\gamma},\rho_{\gamma+d\gamma})=g_{B}(\gamma)d\gamma^{2}\,,$
(14)

and it can be proved that it is proportional to the Quantum Fisher Information
$\mathcal{F}_{q}(\gamma)$,

$g_{B}(\gamma)=\frac{\mathcal{F}_{q}(\gamma)}{4}\,.$ (15)

If the distance between two neighbouring states (which differ by an
infinitesimal variation of the parameter $\gamma$) is large, it is easier to
discriminate the states, and consequently to estimate the value of the
parameter $\gamma$.

##  III  QET for a weak perturbation

In this section we apply the results outlined above to the problem of
estimating the coupling parameter $\gamma$, which quantifies the amplitude of
a perturbation $\mathcal{H}_{1}$, to an otherwise unperturbed system governed
by the Hamiltonian $\mathcal{H}_{0}$. Since we know in advance that the
parameter is small, this is a paradigmatic situation where local quantum
estimation theory is providing a consistent approach to the optimization
problem. Assuming that the unperturbed energy spectrum
$\\{E^{(0)}_{n},|\psi_{n}\rangle\\}$ of $\mathcal{H}_{0}$ is discrete, the
corresponding first-order perturbed eigenstates are given by

$|\psi^{\gamma}_{n}\rangle=|\psi_{n}\rangle+\gamma|\psi^{(1)}_{n}\rangle,$
(16)

where

$|\psi^{(1)}_{n}\rangle=\sum_{m\neq
n}^{+\infty}\frac{\langle\psi_{m}|\mathcal{H}_{1}|\psi_{n}\rangle}{E^{(0)}_{n}-E^{(0)}_{m}}|\psi_{m}\rangle\,,$
(17)

is the perturbation ket. The corresponding first-order eigenvalues are
$E^{\gamma}_{n}=E^{(0)}_{n}+\gamma E^{(1)}_{n}$, with the first-order
correction given by
$E^{(1)}_{n}=\langle\psi_{n}|\mathcal{H}_{1}|\psi_{n}\rangle$. For a pure
quantum state $\rho^{\gamma}=|\psi^{\gamma}\rangle\langle\psi^{\gamma}|$, the
QFI is given by

$\mathcal{F}_{q}(\gamma)=4\left[\langle\partial_{\gamma}\psi^{\gamma}|\partial_{\gamma}\psi^{\gamma}\rangle-|\langle\psi^{\gamma}|\partial_{\gamma}\psi^{\gamma}\rangle|^{2}\right]\,,$
(18)

which, for states of the form (16) may be written as (up to first order in
$\gamma$)

$\displaystyle\mathcal{F}_{q}(\gamma)\simeq
4\|\psi^{(1)}_{n}\|^{2}+\mathcal{O}(\gamma^{2})\,,$ (19)

and is independent on $\gamma$ itself. For pure states we may also easily
compute the SLD since for a pure state $\rho^{2}=\rho$ we have

$(\partial_{\gamma}\rho^{\gamma})\rho^{\gamma}+\rho^{\gamma}(\partial_{\gamma}\rho^{\gamma})=\partial_{\gamma}\rho^{\gamma}\,,$
(20)

and in turn, upon comparison to (11),

$\Lambda_{\gamma,n}=2\partial_{\gamma}\rho_{n}^{\gamma}=2\left(|\partial_{\gamma}\psi^{\gamma}_{n}\rangle\langle\psi^{\gamma}_{n}|+|\psi^{\gamma}_{n}\rangle\langle\partial_{\gamma}\psi^{\gamma}_{n}|\right).$
(21)

In particular, for the $n$th first order perturbed ket (16) we have

$\displaystyle\Lambda_{\gamma,n}=\,2\Big{(}\,$
$\displaystyle|\psi^{(1)}_{n}\rangle\langle\psi_{n}|+|\psi_{n}\rangle\langle\psi^{(1)}_{n}|$
$\displaystyle+2\gamma|\psi^{(1)}_{n}\rangle\langle\psi^{(1)}_{n}|\,\Big{)}+\mathcal{O}(\gamma^{2})\,.$
(22)

In order to assess the performance of a given measurement against the optimal
one, one may compute the corresponding FI and compare it with the QFI in Eq.
(19). For energy measurement on the perturbed eigenstates
$|\psi^{\gamma}_{n}\rangle$, i.e. the detection of ${\cal H}_{0}$ on states of
the form (17), we have
$p(k|n,\gamma)=|\langle\psi_{k}|\psi^{\gamma}_{n}\rangle|^{2}=\delta_{kn}+\gamma^{2}|c_{k}|^{2}(1-\delta_{kn})$,
where $c_{k}$ is the perturbation amplitude
$c_{k}=\langle\psi_{k}|\mathcal{H}_{1}|\psi_{n}\rangle/(E^{(0)}_{n}-E^{(0)}_{m})$.
By inserting this expression in Eq. (9) we have
$\mathcal{F}_{q}(\gamma)=\mathcal{F}_{c}[\mathcal{H}_{0}](\gamma)+O(\gamma^{2})$.
In other words, a static energy measurement is optimal (up to second order in
$\gamma$). Other observables may be optimal, however with some constraints on
the form of $\mathcal{H}_{1}$, see appendix A.

Next, we study time-evolving states for the case where the eigenstate of
$\mathcal{H}$, are the same of $\mathcal{H}_{0}$, i.e. the perturbation
commutes with the unperturbed Hamiltonian. A generic initial superposition is
thus given by

$|\psi^{\gamma}(t=0)\rangle=\sum_{n=0}^{N}\psi_{n}(0)|\psi_{n}\rangle\,.$ (23)

The different terms in the superposition acquire a phase proportional to their
energy $E^{\gamma}_{n}$, and this generates an extra dependence on $\gamma$ by
the action of the unitary evolution
$|\psi^{\gamma}(t)\rangle=\exp\\{-i\mathcal{H}t/\hbar\\}|\psi^{\gamma}(0)\rangle$.
From (18) we can compute the QFI, which is given by

$\displaystyle\mathcal{F}_{q}(\gamma,t)=\mathcal{F}_{q}(t)=$
$\displaystyle=4\frac{t^{2}}{\hbar^{2}}\left[\sum_{n=0}^{N}\,|\psi_{n}(0)|^{2}\left[E^{(1)}_{n}\right]^{2}-|\sum_{n=0}^{N}|\psi_{n}(0)|^{2}E^{(1)}_{n}|^{2}\right]\,.$
(24)

The QFI is maximized when the system is initially prepared in a superposition
of only two states: $|\psi_{M}\rangle$ and $|\psi_{m}\rangle$, corresponding
to the maximum and the minimum energy corrections $E^{(1)}_{n}$, respectively
Giovannetti  _et al._ (2006, 2011); Parthasarathy (2001)

$|\psi^{\gamma}(t=0)\rangle=\frac{1}{\sqrt{2}}\left(|\psi_{m}\rangle+|\psi_{M}\rangle\right)\,.$
(25)

The maximized value of the QFI is given by

$\mathcal{F}_{q}(t)=\frac{t^{2}}{\hbar^{2}}\left(\max_{i}{E^{(1)}_{i}}-\min_{j}{E^{(1)}_{j}}\right)^{2}=\left(\frac{t}{\hbar}\Delta
E^{(1)}\right)^{2}.$ (26)

We notice that the QFI is independent on $\gamma$ at any order. Moreover,
since the state is pure, the SLD is of the form (21). For the initial
preparation (25) the SLD rewrites as

$\displaystyle\Lambda_{\gamma}=\frac{it\Delta
E^{(1)}}{\hbar}\left(|\psi_{m}\rangle\langle\psi_{M}|e^{\frac{it\Delta
E^{\gamma}}{\hbar}}-|\psi_{M}\rangle\langle\psi_{m}|e^{-\frac{it\Delta
E^{\gamma}}{\hbar}}\right)$ (27)

where $\Delta E^{\gamma}=E^{\gamma}_{M}-E^{\gamma}_{m}$.

##  IV  QET for gravity perturbation in one dimension

In this section we focus on the perturbation $\mathcal{H}_{1}$ that arises in
the context of the universal gravity corrections, see (4). The section aims to
study different physical systems and compare their performance as potential
quantum probes for the estimation of the gravitational parameter $\gamma$.

###  IV.1 Free Particle

We start our investigation with the most simple physical system, namely the
free particle

$\mathcal{H}_{0}=\frac{p^{2}}{2m}.$ (28)

The momentum eigenstates are both eigenstates of $\mathcal{H}_{0}$ and
$\mathcal{H}_{1}$, thus the full Hamiltonian is diagonalizable. In this case,
eigenstates are not affected by the perturbation and thus superpositions of
eigenstates evolving in time are needed to realize quantum probes. Since we
have a continuous energy spectrum, the superposition is the wave packet
$|\psi^{\gamma}(0)\rangle=\int_{\mathbb{R}}dp\,\psi_{0}(p)\,|p\rangle$ and the
QFI at time $t$ is given by

$\displaystyle\mathcal{F}_{q}(t)=$
$\displaystyle\frac{4\,t^{2}}{\hbar^{2}m^{2}({\hbox{M}}_{\hbox{\tiny
P}}c)^{4}}\times$ $\displaystyle\times$
$\displaystyle\left[\int_{\mathbb{R}}\\!dp\,|\psi_{0}(p)|^{2}\,p^{8}-\left|\int_{\mathbb{R}}\\!dp\,|\psi_{0}(p)|^{2}\,p^{4}\right|^{2}\right]\,,$
(29)

which, in turn, is the continuous counterpart of the discrete results
discussed previously. In order to better understand the meaning of this
result, let us evaluate it for an initial Gaussian wave packet with width
$\sigma$ and mean $p_{0}$. The squared modulus is

$|\psi_{0}(p)|^{2}=\frac{1}{\sqrt{2\pi\sigma^{2}}}\,e^{-(p-p_{0})^{2}/2\sigma^{2}},$
(30)

and the QFI

$\displaystyle\mathcal{F}_{q}(t;\sigma,p_{m})$
$\displaystyle=\frac{32\,t^{2}\,\sigma^{2}}{\hbar^{2}m^{2}({\hbox{M}}_{\hbox{\tiny
P}}c)^{4}}$ $\displaystyle\times$
$\displaystyle\left[2p_{0}^{6}+21p_{0}^{4}\,\sigma^{2}+48p_{0}^{2}\,\sigma^{4}+12\sigma^{6}\right]\,,$
(31)
$\displaystyle=\frac{32\,t^{2}\,\sigma^{2}}{\hbar^{2}m^{2}({\hbox{M}}_{\hbox{\tiny
P}}c)^{4}}$ $\displaystyle\times$
$\displaystyle\left[16h_{0}^{3}m^{3}+60h_{0}^{2}m^{2}\sigma^{2}+24h_{0}m\sigma^{4}-17\sigma^{6}\right]\,,$
(32)

where $h_{0}\equiv\langle{\mathcal{H}}_{0}\rangle=(p_{0}^{2}+\sigma^{2})/2m$
is the energy of the wave packet. Considering that $\sigma\leq\sqrt{2mh_{0}}$,
$\mathcal{F}_{q}(t;\sigma,p_{m})$ is an increasing function of both $y\equiv
h_{0}m$ and $\sigma$, meaning that a free particle may represent an effective
probe if its initial preparation is de-localized and contains high energy
components. For small values of $\sigma$ the QFI is negligible.

###  IV.2 Infinite Square Well

Figure 1: The QFI $\mathcal{F}_{q}$ of the ground state of the finite square
well as a function of the depth and the width of the well. Left panel: the QFI
$\mathcal{F}_{q}(V_{0})$ for different values of $a$ (blue line $a=1$, orange
line $a=1.5$, green line $a=2$). The red-dotted lines denote the points where
there is a discontinuity in $N_{s}$. Below certain values of $V_{0}$,
$\mathcal{F}_{q}(V_{0})$ vanishes, since $N_{s}<2$. Central panel: the QFI
$\mathcal{F}_{q}(a)$ for different value of $V_{0}$ (blue line
$V_{0}=\sqrt{10}$, orange line $V_{0}=\sqrt{75}$, green line
$V_{0}=\sqrt{250}$). Also here the red-dotted lines denote points where there
is a discontinuity in $N_{s}$. For small values of $a$ the QFI vanishes since
$N_{s}(a)<2$. Right panel: the QFI $\mathcal{F}_{q}$ as a function of the
energy of the ground state. The different energies have been obtained by
varying the width $a$ at fixed $V_{0}$ (the blue line for $V_{0}=\sqrt{10}$,
orange line for $V_{0}=\sqrt{75}$, green line for $V_{0}=\sqrt{250}$).
$\mathcal{F}_{q}$ vanishes for energies above a certain threshold.

Let us now consider a particle placed in an infinite square well (ISW) of
width $a$. The unperturbed Hamiltonian is
$\mathcal{H}_{0}=p^{2}/2m+\mathcal{V}$, with potential function given by

$\displaystyle\mathcal{V}(x)=\left\\{\begin{array}[]{cc}0&\quad 0<x<a,\\\
+\infty&\quad\textup{otherwise.}\end{array}\right.$ (35)

The system has a discrete energy spectrum

$E_{n}^{(0)}=\frac{\pi^{2}\hbar^{2}n^{2}}{2ma^{2}}$ (36)

with $n=1,2,3,\dots$. Since $\mathcal{H}_{0}$ is not a bounded operator, we
cannot evaluate the commutator $[\mathcal{H}_{0},\mathcal{H}_{1}]$ to assess
whether the eigenstates of $\mathcal{H}_{0}$ are eigenstates of
$\mathcal{H}_{1}$ too. On the other hand, it is easy to directly check that
the unperturbed energy eigenstates $|n\rangle$ are eigenstates of $p^{4}$,
i.e.

$p^{4}\,|n\rangle=\left(\frac{\pi n}{a}\right)^{4}\hbar^{4}\,|n\rangle,$ (37)

meaning that the full Hamiltonian
$\mathcal{H}=\mathcal{H}_{0}+\gamma\mathcal{H}_{1}$ is diagonal in this basis.
As for the case of the free particle, the perturbation does not affect the
energy eigenstates, but only the spectrum. As a consequence, the QFI for an
energy eigenstate is zero since it does not depend on $\gamma$. However, we
may consider the superpositions of unperturbed energy eigenstates and obtain a
nonzero QFI for the evolved states. Using the results found in III, we have
that the best preparation is given by the superposition of states
corresponding to the maximum and minimum energy corrections $E^{(1)}_{n,\pm}$,
which, for the ISW have the form

$E^{(1)}_{n}=\frac{1}{m({\hbox{M}}_{\hbox{\tiny
P}}c)^{2}}\left(n\frac{\pi\hbar}{a}\right)^{4},$ (38)

The lowest energy correction correspond to the state $|1\rangle$, while we
have no upper bound on the energy correction. Upon setting a constraint on the
overall energy of the superposition, we have that the maximum QFI is obtained
preparing the particle in the state
$|\psi\rangle=(|1\rangle+|N\rangle)/\sqrt{2}$ at $t=0$. The corresponding QFI
value is given by/

$\displaystyle\mathcal{F}_{q}(t;N)=\frac{t^{2}\pi^{8}\hbar^{6}}{m^{2}a^{8}({\hbox{M}}_{\hbox{\tiny
P}}c)^{4}}\left(N^{4}-1\right)^{2}\,.$ (39)

The QFI is thus proportional to $(N/a)^{8}$ and this somehow agrees with the
behaviour observed for the free particle, i.e. an effective probe may be
obtained when the particle has high energy. Moreover, considering the mean
value of the energy

$\bar{E}=\langle\psi|\mathcal{H}_{0}|\psi\rangle=\frac{\pi^{2}\hbar^{2}}{4ma^{2}}\left(1+N^{2}\right),$
(40)

we may rewrite the $\mathcal{F}_{q}(t;N)$ as

$\mathcal{F}_{q}(t;N)=\frac{256t^{2}m^{2}\bar{E}^{4}}{\hbar^{2}\left({\hbox{M}}_{\hbox{\tiny
P}}c\right)^{4}}\frac{\left(N^{4}-1\right)^{2}}{\left(N^{2}+1\right)^{4}},$
(41)

We notice that the $\mathcal{F}_{q}(t;N)$ is proportional to the mean energy
$\bar{E}$ of the state, as observed before. However, it has not a strong
dependence on $N$, since the ratio
$\left(N^{4}-1\right)^{2}/\left(N^{2}+1\right)^{4}\xrightarrow[N\gg 1]{}1$.

###  IV.3 Finite Square Well

A particle in a finite square well is subject to the potential

$\displaystyle\mathcal{V}(x)=\left\\{\begin{array}[]{cc}0&|x|<a\\\
V_{0}&|x|>a\end{array}\right..$ (44)

Given that the potential has a defined parity, the energy eigenstates have
defined parity too. However, the eigenvalue problem is transcendental and we
have no analytical solution. A very good analytic approximation is given by de
Alcantara Bonfim and Griffiths (2006)

$\displaystyle E^{(0)}_{n}\simeq\frac{\hbar^{2}\pi^{2}}{128ma^{2}z_{0}^{2}}$
$\displaystyle\Big{[}4(n-1)z_{0}-\pi+$
$\displaystyle+\left.\sqrt{(4z_{0}+\pi)^{2}-8\pi nz_{0}}\right]^{2},$ (45)

where $z_{0}^{2}=2mV_{0}a^{2}/\hbar^{2}$. Concerning the computation of the
matrix elements of the perturbation

$\displaystyle|\psi^{(1)}_{0}\rangle=\sum_{n\neq 1}^{N_{s}}$
$\displaystyle|n\rangle\frac{\langle
n|\mathcal{H}_{1}|1\rangle}{E_{1}^{(0)}-E_{n}^{(0)}}\hskip 2.84544pt+$
$\displaystyle+\hskip
2.84544pt2\pi\int_{k_{0}}^{+\infty}dk|\phi_{k}^{(\pm)}\rangle\frac{\langle\phi_{k}^{(\pm)}|\mathcal{H}_{1}|1\rangle}{E^{(0)}_{1}-E^{(0)}_{k}}\,,$
(46)

we are forced to use numerical methods, and then evaluate the QFI according
to(19). For the sake of completeness, in Eq. (46) we have also considered the
continuous spectrum. However, we may actually discard it, since it brings
negligible contribution already for moderate values of $V_{0}$. The discrete
sum goes from $n=2$ to $N_{s}$, which is the number of energy levels available
in the well (it depends on both $V_{0}$ and $a$).

The QFI of the ground state, as a function of the different parameters, is
shown in the three panels of Fig. 1 (we set equal to one all the physical
constants, e.g. $\hbar$, ${\hbox{M}}_{\hbox{\tiny P}}$, $c$, and $m$). The
red-dotted lines denote the points where there is a discontinuity in the
number of bound states $N_{s}$. In the left panel, we show $\mathcal{F}_{q}$
as a function of the potential depth $V_{0}$ for different values of the width
$a$ of the well. The QFI shows a maximum, located at a value of $V_{0}$ which
is decreasing for increasing $a$, whereas it vanishes for values of $V_{0}$
below a certain threshold since in this cases we have $N_{s}<2$. We however do
no draw any general conclusion for vanishing value of $V_{0}$ since in our
calculations we have dropped the contribution of the continuous part of the
spectrum. In the central panel of the same figure, we show $\mathcal{F}_{q}$
as a function of the potential width $a$ for different values of the depth
$V_{0}$. At any value of $V_{0}$, the QFI is zero below a certain value of
$a$, since there are no bound states in those cases. The QFI then increases
with $a$ and shows a maximum for a value of the width which decreases for
increasing $V_{0}$. The QFI is then a decreasing function of $a$ for any
$V_{0}$ and vanishes

Figure 2: The QFI obtained for $t=1$ and $\omega=1$ for a harmonic oscillator
initially prepared in the superposition of two perturbed energy eigenstates
with $\omega=1$. The solid lines are for
$|\psi^{\gamma}_{0}\rangle+|\psi^{\gamma}_{n}\rangle$, whereas the dashed
lines denote results for
$|\psi^{\gamma}_{1}\rangle+|\psi^{\gamma}_{n}\rangle$. The blue lines are for
$n=2$, the orange ones for $n=3$, and the green ones for $n=4$. We see that
dashed lines, corresponding to higher excitations in the superpositions, are
above the solid one, thus breaking the hierarchy found for unperturbed
superpositions.

for $a\gg 1$, since in this case the situation is approaching that of a free
particle. In the right panel, we report the QFI as a function of the energy of
the ground state. The different plots have been obtained by varying the width
$a$ at fixed $V_{0}$. The QFI vanishes for vanishing energy and for energies
above a certain threshold. This behaviour may be understood, at least
qualitatively, considering that at fixed $V_{0}$, high energies correspond to
small values of $a$. But if $a$ is smaller than a certain threshold, then
$N_{s}=1$ and therefore we have a null perturbed ket
$|\psi^{(1)}_{0}\rangle=0$ which means a null $\mathcal{F}_{q}$.

###  IV.4 Harmonic Oscillator

Let us now address a particle trapped in a harmonic potential, i.e. with
Hamiltonian

$\mathcal{H}_{0}=\frac{p_{0}^{2}}{2m}+\frac{1}{2}m\omega^{2}q^{2}.$ (47)

In this system the gravity perturbation takes the form

$\mathcal{H}_{1}\propto(a+a^{\dagger})^{4}\,,$ (48)

and it does not commute with $\mathcal{H}_{0}$. If we choose a perturbed
eigenstate $|\psi^{\gamma}_{n}\rangle$ as a quantum probe, then the QFI is
given by (19), i.e.

$\displaystyle\mathcal{F}_{q}(\omega,n)$ $\displaystyle=\frac{(\hbar
m\omega)^{2}}{32({\hbox{M}}_{\hbox{\tiny P}}c)^{4}}$
$\displaystyle\times\left(65n^{4}+130n^{3}+487n^{2}+422n+156\right)\,,$ (49)

which grows as $n^{4}$ with the energy of the probe. In order to compare the
performance with those of other systems, let us also compute the QFI for
superpositions of unperturbed and perturbed eigenstates, bearing in mind that
the energy correction is

$E^{(1)}_{n}=\frac{3m\hbar^{2}\omega^{2}}{4({\hbox{M}}_{\hbox{\tiny
P}}c)^{2}}\left(1+2n+2n^{2}\right).$ (50)

In the case of unperturbed eigenstates, we know from (26) that the maximum of
QFI is given by

$\mathcal{F}_{q}(t;\omega,n)=\frac{9t^{2}m^{2}\hbar^{2}\omega^{4}n^{2}(1+n)^{2}}{4({\hbox{M}}_{\hbox{\tiny
P}}c)^{4}}\,,$ (51)

corresponding to the QFI of the state evolving in time from the initial
superposition $1/\sqrt{2}(|0\rangle+|n\rangle)$. For superpositions of
perturbed eigenstates, we have no close solution for the probes which maximize
the QFI. However, we can try to evaluate it numerically for different probes
to understand how it behaves. The results are depicted in fig. 2. We see that
the best superposition is not given by the two states with maximum separation
between the corresponding correction $E^{(1)}_{i}$. The underlying reason lies
in the fact that also the state depends itself on the parameter, and the
higher contribution to the $\mathcal{F}_{q}(\gamma)$ comes from the
perturbation ket $|\psi^{(1)}_{n}\rangle$ rather than from the phase that
arises from the time evolution. The plots report results obtained by evolving
the superpositions at second order in $\gamma$. The first order is identically
$0$, with the exception of states containing $n=4$. Also in this last case,
however, the more relevant contribution is coming from the second-order term.
As it is apparent from the plot, the dashed lines, corresponding to higher
excitations in the superpositions, are above the solid one, thus breaking the
hierarchy found for unperturbed superpositions.

###  IV.5 Comparison of the different systems

Using the results from the previous sections, we can compare the different
values of the Quantum Fisher Information to establish which system has the
highest power of estimate for the parameter $\gamma$. To have a faithful
comparison, we choose values of the system’s parameters in a range of real
physical systems and we plot the $\mathcal{F}_{q}$ as a function of the
systems’ energy. For instance, we set the mass $m=10^{-27}$ Kg, which is of
the order of magnitude of the Hydrogen mass Meija  _et al._ (2016). In the
free particle, we set the momentum $p_{m}=1$MeV/c and we vary the width of the
wave packet $\sigma$ in the interval that goes from $0$ to $30$ MeV/c. For the
Infinite Square Well, we vary the width of the well in the range that goes
from $1$ nm to $10$ nm, which is the typical scale of quantum dots Ekimov and
Onushchenko (1981); Wang  _et al._ (2001). Analogously, for the finite Square
well, we choose the same range of $a$ and we fix $V_{0}=50$eV. Finally, for
the Harmonic Oscillator, we vary the frequency $\omega$ from $10^{13}$ to
$10^{14}$, which represents the typical frequencies of a diatomic molecule
Shimanouchi (1977); Shimanouchi and Shimanouchi (1980). The results are shown
in Fig. 3. We see that the most effective probe is provided by the harmonic
oscillator system, whose $\mathcal{F}_{q}$ is larger than the
$\mathcal{F}_{q}$ obtained from other systems by many orders of magnitude.

Figure 3: Logarithmic plot of the Quantum Fisher Information as a function of
the Energy in three different systems: the free particle, the harmonic
oscillator and the Infinite Square Well. For a free particle, we set $p_{m}=1$
MeV/c and we varied the width $\sigma$ of the wave packet; for the Harmonic
oscillator, we considered the Quantum Fisher Information of the time-evolving
state $1/\sqrt{2}|\psi^{\gamma}_{1}\rangle+|\psi^{\gamma}_{4}\rangle$ as well
as for the infinite square well, where our reference state were
$1/\sqrt{2}(|1\rangle+|4\rangle)$. In all the three numerical evaluations,
$t=1$. The grey lines represent the QFI for generic value of the energy, while
the red lines represent the QFI for energy value in the range of real physical
systems as described in section IV.5. Differently from the previous plot, in
this one we used SI values for the fundamental physical constants, i.e. the
Mass Planck ${\hbox{M}}_{\hbox{\tiny P}}=2,176\times 10^{-8}$ Kg, the reduced
Planck constant $\hbar=1,054\times 10^{-34}$ J$\cdot$ s and the speed of light
$c=2,99\times 10^{8}$ m/s. The blue dashed lines show the orders of magnitude
between the minimum of the $\mathcal{F}_{q}$ for the Harmonic Oscillator and
the maximum of the $\mathcal{F}_{q}$ for the Infinite Square Well.

##  V  QET for gravity perturbation in dimension higher than one

In this section, we investigate the role of the dimensionality of the system
in determining the precision in the estimation of the parameter $\gamma$. To
this aim, we study the performance of a quantum probe made of a particle
trapped either in a two-dimensional infinite square well or in a two-
dimensional harmonic potential. This choice is motivated by the result of the
previous Section, indicating that those two potentials are those providing the
best performance in the 1-D case.

###  V.1 2-dimensional Infinite Square Well

The unperturbed 2-dimensional infinite square well of side $a$ is described by

$\mathcal{H}_{0}=\frac{p_{x}^{2}+p_{y}^{2}}{2m}+\mathcal{V}$ (52)

where the potential is

$\mathcal{V}=\begin{cases}0&\text{if }0<x<a\quad\&\quad 0<y<a,\\\
+\infty&\text{Otherwise}.\end{cases}$ (53)

The system is decoupled, meaning that the energy wave functions are factorized
as the solutions of two 1-dimensional ISW and the energies are the sum of the
1-dimensional ISW energies, i.e. employing the boundary conditions we obtain

$\displaystyle\psi_{n_{x},n_{y}}(x,y)=\frac{2}{a}\sin\left(\frac{n_{x}\pi}{a}x\right)\sin\left(\frac{n_{y}\pi}{a}y\right),$
(54) $\displaystyle
E_{n_{x},n_{y}}=\frac{\hbar^{2}\pi^{2}}{2m}\left(\frac{n_{x}^{2}+n_{y}^{2}}{a^{2}}\right).$
(55)

Taking into account the perturbation

$\mathcal{H}_{1}\propto
p^{4}=(\partial_{x}^{4}+2\partial_{x}^{2}\partial_{y}^{2}+\partial_{y}^{4}),$
(56)

we find that the energy eigenstates are eigenstates of $\mathcal{H}_{1}$ too,
since

$p^{4}\psi_{n_{x},n_{y}}(x,y)=\left(n_{x}^{2}+n_{y}^{2}\right)^{2}\frac{\pi^{4}}{a^{4}}\psi_{n_{x},n_{y}}(x,y).$
(57)

It follows that the full Hamiltonian
$\mathcal{H}=\mathcal{H}_{0}+\gamma\mathcal{H}_{1}$ is already diagonal in the
basis of $\mathcal{H}_{0}$. As in the 1-dimensional system, the perturbation
affects only the energy levels, thus to observe the effects of the
perturbation we need to consider superpositions of energy eigenstates evolving
in time. We already know that the superposition maximizing QFI is the
superposition of the state corresponding to the maximum
$E^{(1)}_{n_{x},n_{y}}$ and of the state corresponding to the minimum
$E^{(1)}_{n_{x},n_{y}}$. The energy correction is

$\displaystyle E^{(1)}_{n_{x},n_{y}}$ $\displaystyle=\langle
n_{x},n_{y}|\mathcal{H}_{1}|n_{x},n_{y}\rangle$
$\displaystyle=\frac{\hbar^{4}}{m\left({\hbox{M}}_{\hbox{\tiny
P}}c\right)^{2}}\frac{\pi^{4}}{a^{4}}\left(n_{x}^{2}+n_{y}^{2}\right)^{2},$
(58)

and the minimum is realized for $\\{n_{x}=1,n_{y}=1\\}$, while the maximum is
not fixed, depending on the bound we choose. The corresponding maximum QFI is

$\mathcal{F}_{q}(t;n_{x},n_{y})=\frac{t^{2}\hbar^{6}\pi^{8}}{a^{8}m^{2}({\hbox{M}}_{\hbox{\tiny
P}}c)^{4}}\left((n_{x}^{2}+n_{y}^{2})^{2}-4\right)^{2}$ (59)

and it is realized by the time evolution of the state

$|\psi(0)\rangle=\frac{1}{\sqrt{2}}\left(|1_{x},1_{y}\rangle+|n_{x},n_{y}\rangle\right).$
(60)

Figure 4: Plot of the weighted ratio $\mathcal{R}(n_{x},n_{y})$ (63) between
the maximum QFI for the 2D and the 1D Infinite Square Well for a
superposition. The ratio is a function of $n_{x}$ and $n_{y}$. We clearly
notice that the maximum is realized when the state has equal excitation in
both $x$ and $y$ directions, i.e. the state
$1/\sqrt{2}(|n,n\rangle+|1,1\rangle)$. The value of the maximum ratio is $8$,
independently of the value of $n$, which means that the enhancement does not
depend on the energy we are. However, the absolute value of the QFI depends on
$n$.

The analogous 1-dimensional states for the comparison are the normalized
superpositions $|1_{x}\rangle+|n_{x}\rangle$ and
$|1_{y}\rangle+|n_{y}\rangle$, whose QFI, after a time evolution, are
respectively

$\displaystyle\mathcal{F}_{q}(t;n_{x})=\frac{t^{2}\hbar^{6}\pi^{8}}{a^{8}m^{2}({\hbox{M}}_{\hbox{\tiny
P}}c)^{4}}\left(n_{x}^{4}-1\right)^{2},$ (61)
$\displaystyle\mathcal{F}_{q}(t;n_{y})=\frac{t^{2}\hbar^{6}\pi^{8}}{a^{8}m^{2}({\hbox{M}}_{\hbox{\tiny
P}}c)^{4}}\left(n_{y}^{4}-1\right)^{2}.$ (62)

Then the weighted ratio $\mathcal{R}(n_{x},n_{y})$ between the 2-dimensional
and 1-dimensional systems is

$\displaystyle\mathcal{R}(n_{x},n_{y})$
$\displaystyle=\frac{\mathcal{F}_{q}(t;n_{x},n_{y})}{\mathcal{F}_{q}(t;n_{x})+\mathcal{F}_{q}(t;n_{y})}=$
$\displaystyle=\frac{((n_{x}^{2}+n_{y}^{2})^{2}-4)^{2}}{(n_{x}^{4}-1)^{2}+(n_{y}^{4}-1)^{2}}$
(63)

$\begin{array}[]{||c|c|c||c|c||c||c||}\hline\cr\vrule\vrule\lx@intercol\hfil\hbox{\multirowsetup{States}}\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\textup{QFI
in 1D HO}\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\textup{QFI in
2D HO}&\hbox{\multirowsetup{Weigthed Ratio}}\\\
\vrule\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\lx@intercol\hfil\textit{in
units of }{\displaystyle\frac{(\hbar\omega m)^{2}}{({\hbox{M}}_{\hbox{\tiny
P}}c)^{4}}}\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&\textit{in
units of }{\displaystyle\frac{(\hbar\omega m)^{2}}{({\hbox{M}}_{\hbox{\tiny
P}}c)^{4}}}&\\\
\hline\cr\vrule\vrule\lx@intercol\hfil|0\rangle\hfil\lx@intercol\vrule\lx@intercol&|0,0\rangle&\lx@intercol\hfil\displaystyle\frac{39}{8}\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&17&\frac{68}{39}\simeq
1.74\\\ \hline\cr\hskip 19.0633pt|0\rangle&|1\rangle&|1,0\rangle&\hskip
12.80365pt{\displaystyle\frac{39}{8}}&{\displaystyle\frac{315}{8}}&75&\frac{100}{59}\simeq
1.69\\\
\hline\cr|0\rangle&\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)&\frac{1}{\sqrt{2}}(|0,0\rangle+|0,1\rangle)&\hskip
12.80365pt{\displaystyle\frac{39}{8}}&{\displaystyle\frac{177}{8}}&46&\frac{46}{27}\simeq
1.70\\\
\hline\cr\vrule\vrule\lx@intercol\hfil\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\hfil\lx@intercol\vrule\lx@intercol&\frac{1}{\sqrt{2}}(|1,0\rangle+|0,1\rangle)&\lx@intercol\hfil\displaystyle\frac{177}{8}\hfil\lx@intercol\vrule\lx@intercol\vrule\lx@intercol&75&\frac{100}{59}\simeq
1.69\\\ \hline\cr\end{array}$ Table 1: We sum up the comparison for the
Harmonic Oscillator. In the first column are listed the analogous
1-dimensional states of the 2-dimensional ones, which are in the second column
instead. In the third column we evaluate the QFI for the corresponding
1-dimensional states, while in the fourth columns there are the QFI for the
2-dimensional states. Finally, in the last column we evaluated the weighted
ratio.

which is depicted in figure 4. We see that the maximum is realized for
$n_{x}=n_{y}=n$, for which the weighted ratio is

$\frac{\mathcal{F}_{q}(t;n,n)}{2\mathcal{F}_{q}(t;n)}=\frac{((2n^{2})^{2}-4)^{2}}{2(n^{4}-1)^{2}}=8=2^{3}\,,$
(64)

i.e. the QFI shows a super-additive behaviour in terms of dimensionality,
which in turn represents a metrological resource. Doing the same comparison
between 3D and 1D Infinite Square Well, we find that the maximum is realized
when $n_{x}=n_{y}=n_{z}=n$ and the QFI is

$\frac{\mathcal{F}_{q}(t;n,n,n)}{3\mathcal{F}_{q}(t;n)}=\frac{81(n^{4}-1)^{2}}{3(n^{4}-1)^{2}}=27=3^{3}.$
(65)

We see that the maximum of the QFI scales as the third power of the dimension
of the system. Since the states are not affected by the perturbation, the
enhancement does not originate from any possible entangling power of
$\mathcal{H}_{1}$. Instead, it is the larger correction in the higher
dimensional systems that generates the gain.

###  V.2 2-dimensional Harmonic Oscillator

In this system, the unperturbed Hamiltonian is given by the sum of two
independent (but, for the sake of simplicity, with the same frequency
$\omega$) 1-dimensional Harmonic Oscillator

$\mathcal{H}_{0}=\frac{p_{0x}^{2}}{2m}+\frac{p_{0y}^{2}}{2m}+\frac{1}{2}m\omega^{2}q_{x}^{2}+\frac{1}{2}m\omega^{2}q_{y}^{2},$
(66)

which is easily diagonalized as $\mathcal{H}_{0}=\hbar\omega(N_{x}+N_{y}+1)$.
The main difference with the 1-dimensional case is that the energy spectrum is
always degenerate, with the exception of the ground state. In general, the
degree of degeneracy is $g_{n}=n+1$. If we express the perturbation
$\mathcal{H}_{1}\propto p^{4}$ in terms of the ladder operators, we obtain
that

$\displaystyle\mathcal{H}_{1}\propto$
$\displaystyle\left(a_{x}+a_{x}\right)^{4}+\left(a_{y}+a^{\dagger}_{y}\right)^{4}+$
(67)
$\displaystyle+2\left(a_{x}+a_{x}^{\dagger}\right)^{2}\left(a_{y}+a_{y}^{\dagger}\right)^{2}.$
(68)

We clearly see that a coupling term appears, which causes the two independent
harmonic oscillators not to be independent anymore. The main consequence of
this extra coupling is the appearance of entanglement between the two degrees
of freedom of the system. However, as we will see in the following,
entanglement does not represent a resource for the estimation of $\gamma$, at
least in our perturbative regime. As mentioned above, the ground state is non-
degenerate, and we may use standard non-degenerate perturbation theory to
compute the state $|\psi^{(1)}_{0,0}\rangle$ and then evaluating its norm,
which is equal to the QFI at first order in $\gamma$

$\mathcal{F}_{q}(\omega;0,0)=17\frac{(\hbar
m\omega)^{2}}{({\hbox{M}}_{\hbox{\tiny P}}c)^{4}}.$ (69)

We can compare this value with the corresponding QFI of the ground state of
the 1-dimensional Harmonic Oscillator. We multiply the latter by two, to match
the dimensionality of the systems. Eventually, we have

$\frac{\mathcal{F}_{q}(\omega;0,0)}{2\mathcal{F}_{q}(\omega;0)}=\frac{68}{39}\simeq
1.74.$ (70)

We see that we have an enhancement of a factor approximately equal to $7/4$.
Likewise, we can evaluate the QFI for the state $|1,0\rangle$. We obtain that

$\mathcal{F}_{q}(\gamma;1,0)=75\frac{(m\omega\hbar)^{2}}{({\hbox{M}}_{\hbox{\tiny
P}}c)^{4}}.$ (71)

To have a meaningful comparison we use a weighted ratio and we obtain

$\frac{\mathcal{F}_{q}(\gamma;1,0)}{\mathcal{F}_{q}(\gamma;0)+\mathcal{F}_{q}(\gamma;1)}=\frac{100}{59}\simeq
1.69,$ (72)

which is slightly lower than the one obtained for the ground state, but still
larger than unity, i.e. the QFI is superadditive also in this case.

We summarize results in table 1. We observe that the highest ratio is given by
the ground state, while the others are slightly lower but still around this
value. Moreover, the weighted ratio for the state $|1,0\rangle$ is exactly the
same of the state $1/\sqrt{2}(|1,0\rangle+|0,1\rangle)$. It thus follows that
the enhancement is not given by the fact that the probe state is entangled.
Rather, it depends only on the norm of perturbation ket
$|\psi^{(1)}_{n_{x},n_{y}}\rangle$. In particular, since the 2D oscillator has
a higher number of superposed states than the 1D counterpart, it has a higher
norm, ensuring that the ratio is always larger than $1$. Moreover, the states
$|0,1\rangle$ and $|1,0\rangle$ give the same contributions. Overall, this
explains why the weighted ratio gives the same result for both $|0,1\rangle$
and $1/\sqrt{2}(|1,0\rangle+|0,1\rangle)$.

##  VI Conclusion

We have addressed the problem of estimating the minimum length parameter,
possibly arising from quantum gravity theories in low energy physical systems.
Upon exploiting tools from quantum estimation theory, we found general bounds
on precision and have assessed the use of different quantum probes to enhance
the estimation performances. In particular, we have systematically studied the
effects of gravity-like perturbations on different state preparations for
several 1-dimensional systems, and have evaluated the Quantum Fisher
Information in order to find the ultimate bounds to the precision of any
estimation procedure. Our results indicate that the largest values of QFI are
obtained with a quantum probe subject to a harmonic potential and initially
prepared in a superposition of perturbed energy eigenstates (see fig. 3).

We have also investigated the role of dimensionality by analysing the use of
two-dimensional square well and harmonic oscillator systems to probe the
minimal length. We have shown that QFI is super-additive with the dimension of
the system, which therefore represents a metrological resource. The gain in
precision is not due to the appearance of entanglement of the state, but
rather to the increasing number of superposed states generated by the
perturbation or to the larger energy corrections. We evaluated analytically
the QFI ratio $R$, showing that it scales as $R\propto d^{3}$ for the infinite
square well and at most as $R\simeq 1.71$ for the harmonic oscillator, at
least for low-lying energy states.

Our results show that quantum probes are convenient resources, providing a
potential enhancement in precision, provide a set of guidelines to design
possible future experiments to detect minimal length.

###### Acknowledgements.

MGAP is member of GNFM-IndAM and thanks Marco Genoni, Sholeh Razavian, Andrea
Caprotti, Hakim Gharbi, Hamza Adnane and Sid Ali Mohammdi for useful
discussions. AC thanks Stefano Biagi for useful discussions.

##  Appendix A Optimal Observables

We consider a general pure state $|\psi^{\gamma}\rangle$ depending on a
parameter $\gamma$ and a generic projective measurement with projectors
$|x\rangle\langle x|$. The corresponding probability distribution function is
given by the Born rule

$\textbf{p}(x;\gamma)=\textup{Tr}\left[|\psi^{\gamma}\rangle\langle\psi^{\gamma}|x\rangle\langle
x|\right]=|\langle\psi^{\gamma}|x\rangle|^{2}=|\psi^{\gamma}(x)|^{2},$ (73)

and as a result, the Quantum Fisher Information is

$\displaystyle\mathcal{F}_{q}(\gamma)=4\Bigg{[}\int
dx\left[\partial_{\gamma}\left(\psi^{\gamma}(x)^{*}\right)\partial_{\gamma}\left(\psi^{\gamma}(x)\right)\right]+$
$\displaystyle-\iint
dxdy\psi^{\gamma}(x)^{*}\partial_{\gamma}\left(\psi^{\gamma}(x)\right)\partial_{\gamma}\left(\psi^{\gamma}(y)^{*}\right)\psi^{\gamma}(y)\Bigg{]}.$
(74)

We can rewrite the wave function $\psi^{\gamma}(x)$ in terms of its complex
phase

$\theta^{\gamma}_{\psi}=\textup{arctan}\left[\frac{\psi^{\gamma}_{\Im}(x)}{\psi^{\gamma}_{\Re}(x)}\right]$
(75)

and its radius

$r^{\gamma}_{\psi}(x)=\sqrt{\psi^{\gamma}_{\Re}(x)^{2}+\psi^{\gamma}_{\Im}(x)^{2}}\\\
$ (76)

as

$\displaystyle\psi^{\gamma}(x)=\exp\\{i\theta^{\gamma}_{\psi}(x)\\}r^{\gamma}_{\psi}(x).$
(77)

In this representation, the normalization takes the following form

$\displaystyle\int dx\psi^{\gamma}(x)^{*}\psi^{\gamma}(x)=\int
dx\left(r^{\gamma}_{\psi}(x)\right)^{2}=1.$ (78)

If we derive both sides we have that

$\displaystyle\int
dx\left(\partial_{\gamma}r^{\gamma}_{\psi}(x)r^{\gamma}_{\psi}(x)+r^{\gamma}_{\psi}(x)\partial_{\gamma}r^{\gamma}_{\psi}(x)\right)=0,$
$\displaystyle\int
dx\partial_{\gamma}r^{\gamma}_{\psi}(x)r^{\gamma}_{\psi}(x)=0.$ (79)

that will be useful in the following.

If we expand the integrals in (74) in terms of $\theta^{\gamma}_{\psi}$ and
$r^{\gamma}_{\psi}$, considering that

$\partial_{\gamma}\psi^{\gamma}(x)=\exp\\{i\theta^{\gamma}_{\psi}(x)\\}\left(\partial_{\gamma}r^{\gamma}_{\psi}(x)+i\partial_{\gamma}\\{\theta^{\gamma}_{\psi}(x)\\}r^{\gamma}_{\psi}(x)\right),$
(80)

and that (79) holds, we eventually obtain

$\displaystyle\mathcal{F}_{q}(\gamma)=4\bigg{[}\|\partial_{\gamma}r^{\gamma}_{\psi}\|^{2}+\|\partial_{\gamma}\\{\theta^{\gamma}_{\psi}\\}r^{\gamma}_{\psi}\|^{2}+$
$\displaystyle-\left(\int
dx\partial_{\gamma}\theta^{\gamma}_{\psi}(x)\left(r^{\gamma}_{\psi}(x)\right)^{2}\right)^{2}\bigg{]}.$
(81)

Instead, we find that the classical Fisher information
$\mathcal{F}_{c}(\gamma)$, using again (79), is

$\displaystyle\mathcal{F}_{c}(\gamma)=\int^{+\infty}_{-\infty}dx\frac{1}{|\psi_{\gamma}(x)|^{2}}\left[\partial_{\gamma}|\psi_{\gamma}(x)|^{2}\right]^{2}=$
$\displaystyle=4\|\partial_{\gamma}r^{\gamma}_{\psi}\|^{2}.$ (82)

In this representation, the Quantum Cramer Rao inequality
$\mathcal{F}_{q}(\gamma)\geq\mathcal{F}_{c}(\gamma)$ reads as

$\|\partial_{\gamma}\theta^{\gamma}_{\psi}r^{\gamma}_{\psi}\|^{2}\geq\left(\int
dx\partial_{\gamma}\theta^{\gamma}_{\psi}(x)\left(r^{\gamma}_{\psi}(x)\right)^{2}\right)^{2}.$
(83)

A sufficient but not necessary conditions for the equality is that the phase
does not depend on $\gamma$, $\partial_{\gamma}\theta^{\gamma}_{\psi}(x)=0$,
which includes the case of a real wave function.

In the case of a first order perturbed state

$\displaystyle\psi^{\gamma}_{n}(x)=\psi^{(0)}_{n}(x)+\gamma\psi^{(1)}_{n}(x)+\mathcal{O}(\gamma^{2})$
(84)

we can separate the real and the imaginary part as

$\displaystyle\psi^{\gamma}_{n}(x)=\left[\psi^{(0)}_{n}(x)^{\Re}+\gamma\psi^{(1)}_{n}(x)^{\Re}\right]+$
$\displaystyle+i\left[\psi^{(0)}_{n}(x)^{\Im}+\gamma\psi^{(1)}_{n}(x)^{\Im}\right].$
(85)

As a result, the phase is

$\theta^{\gamma}_{\psi}(x)=\arctan\left[\frac{\psi^{(0)}_{n}(x)^{\Im}+\gamma\psi^{(1)}_{n}(x)^{\Im}}{\psi^{(0)}_{n}(x)^{\Re}+\gamma\psi^{(1)}_{n}(x)^{\Re}}\right].$
(86)

We see that it does not depend on $\gamma$ in only two cases. In the first
scenario it must be

$\displaystyle\psi_{n}^{(0)}(x)^{\Im}=0\quad\&\quad\psi_{n}^{(0)}(x)^{\Re}=0,$
(87)

but these conditions can not be satisfied since the unperturbed wave function
$\psi^{(0)}(x)$ must be different from $0$.

Instead, in the second scenario it must be

$\displaystyle\psi_{n}^{(1)}(x)^{\Im}=0\quad\&\quad\psi_{n}^{(1)}(x)^{\Re}=0.$
(88)

These conditions may be satisfied if the perturbation $\mathcal{H}_{1}$ is
diagonal on the same basis as the unperturbed Hamiltonian. However, the QFI is
null since the state $\psi^{\gamma}_{n}(x)$ does not depend on $\gamma$. In
this case we already know that the time-evolving states are the necessary
probes. However, due to the unitary evolution, the state acquires a complex
phase depending on $\gamma$ and the condition
$\partial_{\gamma}\theta^{\gamma}_{\psi}(x)=0$ can not be satisfied.

From these considerations we induce that the condition
$\partial_{\gamma}\theta^{\gamma}_{\psi}(x)=0$ is too restrictive for
perturbed state of the form (84) and no useful constraints on the wave
function may be found from it. Moreover, the condition
$\partial_{\gamma}\theta^{\gamma}_{\psi}(x)=0$ is not a necessary one, meaning
that it does not exclude the possibility of saturating (83), a condition that
can be checked by directly evaluating the two sides of (83) in any specific
case.

Eventually, comparing the QFI (81) and the Classical Fisher (82), we note that
the acquired phase depending on $\gamma$ may be considered as the quantum
enhancement since it is the term that makes the Quantum Fisher Information
larger than the classical one.

## References

  * Hossenfelder (2013) S. Hossenfelder, Living Reviews in Relativity 16, 2 (2013).
  * Calmet  _et al._ (2004) X. Calmet, M. Graesser, and S. D. H. Hsu, Phys. Rev. Lett. 93, 211101 (2004).
  * Calmet  _et al._ (2005) X. Calmet, M. Graesser, and S. D. H. Hsu, International Journal of Modern Physics D 14, 2195 (2005), https://doi.org/10.1142/S0218271805008005 .
  * Maggiore (1993) M. Maggiore, Physics Letters B 319, 83 (1993).
  * Markopoulou and Smolin (2004) F. Markopoulou and L. Smolin, Physical Review D 70, 124029 (2004).
  * Bang and Berger (2006) J. Y. Bang and M. S. Berger, Physical Review D 74, 125012 (2006).
  * Das and Vagenas (2008) S. Das and E. C. Vagenas, Physical review letters 101, 221301 (2008).
  * Note (1) This kind of interpretation has however some conceptual problems, see e.g.Hossenfelder (2013) for explanations.
  * Rossi  _et al._ (2016) M. A. Rossi, T. Giani, and M. G. Paris, Physical Review D 94, 024014 (2016).
  * Brau (1999) F. Brau, Journal of Physics A: Mathematical and General 32, 7691 (1999).
  * Kempf  _et al._ (1995) A. Kempf, G. Mangano, and R. B. Mann, Physical Review D 52, 1108 (1995).
  * Berger and Maziashvili (2011) M. S. Berger and M. Maziashvili, Physical Review D 84, 044043 (2011).
  * Ashoorioon  _et al._ (2005) A. Ashoorioon, A. Kempf, and R. B. Mann, Physical Review D 71, 023503 (2005).
  * Vakili (2008) B. Vakili, Physical Review D 77, 044023 (2008).
  * Maziashvili (2012) M. Maziashvili, Physical Review D 85, 125026 (2012).
  * Kempf and Lorenz (2006) A. Kempf and L. Lorenz, Physical Review D 74, 103517 (2006).
  * Benczik  _et al._ (2002) S. Benczik, L. N. Chang, D. Minic, N. Okamura, S. Rayyan, and T. Takeuchi, Physical Review D 66, 026003 (2002).
  * Ching  _et al._ (2012) C.-L. Ching, R. R. Parwani, and K. Singh, Physical Review D 86, 084053 (2012).
  * Ching and Ng (2013) C. L. Ching and W. K. Ng, Physical Review D 88, 084009 (2013).
  * Ching and Ng (2019) C. L. Ching and W. K. Ng, Physical Review D 100, 085018 (2019).
  * Braidotti  _et al._ (2017) M. C. Braidotti, Z. H. Musslimani, and C. Conti, Physica D: Nonlinear Phenomena 338, 34 (2017).
  * Pikovski  _et al._ (2012) I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. Kim, and Č. Brukner, Nature Physics 8, 393 (2012).
  * Paris (2009) M. G. Paris, International Journal of Quantum Information 7, 125 (2009).
  * Braun  _et al._ (2017) D. Braun, F. Schneiter, and U. R. Fischer, Classical and Quantum Gravity 34, 175009 (2017).
  * Van Trees (2004) H. L. Van Trees,  _Detection, estimation, and modulation theory, part I: detection, estimation, and linear modulation theory_ (John Wiley & Sons, 2004).
  * Lehmann and Casella (2006) E. L. Lehmann and G. Casella,  _Theory of point estimation_ (Springer Science & Business Media, 2006).
  * Petz and Ghinea (2011) D. Petz and C. Ghinea, in  _Quantum probability and related topics_ (World Scientific, 2011) pp. 261–281.
  * Amari and Nagaoka (2007) S.-i. Amari and H. Nagaoka,  _Methods of information geometry_ , Vol. 191 (American Mathematical Soc., 2007).
  * Facchi  _et al._ (2010) P. Facchi, R. Kulkarni, V. Man’ko, G. Marmo, E. Sudarshan, and F. Ventriglia, Physics Letters A 374, 4801 (2010).
  * Sommers and Zyczkowski (2003) H.-J. Sommers and K. Zyczkowski, Journal of Physics A: Mathematical and General 36, 10083 (2003).
  * Giovannetti  _et al._ (2006) V. Giovannetti, S. Lloyd, and L. Maccone, Physical review letters 96, 010401 (2006).
  * Giovannetti  _et al._ (2011) V. Giovannetti, S. Lloyd, and L. Maccone, Nature photonics 5, 222 (2011).
  * Parthasarathy (2001) K. Parthasarathy, in  _Stochastics in finite and infinite dimensions_ (Springer, 2001) pp. 361–377.
  * de Alcantara Bonfim and Griffiths (2006) O. de Alcantara Bonfim and D. J. Griffiths, American journal of physics 74, 43 (2006).
  * Meija  _et al._ (2016) J. Meija, T. B. Coplen, M. Berglund, W. A. Brand, P. De Bièvre, M. Gröning, N. E. Holden, J. Irrgeher, R. D. Loss, T. Walczyk, _et al._ , Pure and Applied Chemistry 88, 265 (2016).
  * Ekimov and Onushchenko (1981) A. I. Ekimov and A. A. Onushchenko, Jetp Lett 34, 345 (1981).
  * Wang  _et al._ (2001) C. Wang, M. Shim, and P. Guyot-Sionnest, Science 291, 2390 (2001).
  * Shimanouchi (1977) T. Shimanouchi, Journal of physical and chemical reference data 6, 993 (1977).
  * Shimanouchi and Shimanouchi (1980) T. Shimanouchi and T. Shimanouchi,  _Tables of molecular vibrational frequencies_ (National Bureau of Standards Washington, DC, 1980).