#  Complementarity-Entanglement Tradeoff in Quantum Gravity

Yusef Maleki1 and Alireza Maleki2 1Department of Physics and Astronomy, Texas
A&M University, College Station, Texas 77843-4242, USA
2 Department of Physics, Sharif University of Technology, Tehran, Iran

###### Abstract

Quantization of the gravity remains one of the most important, yet extremely
illusive, challenges at the heart of modern physics. Any attempt to resolve
this long-standing problem seems to be doomed, as the route to any direct
empirical evidence (i.e., detecting gravitons) for shedding light on the
quantum aspect of the gravity is far beyond the current capabilities.
Recently, it has been discovered that gravitationally-induced entanglement,
tailored in the interferometric frameworks, can be used to witness the quantum
nature of the gravity. Even though these schemes offer promising tools for
investigating quantum gravity, many fundamental and empirical aspects of the
schemes are yet to be discovered. Considering the fact that, beside quantum
entanglement, quantum uncertainty and complementarity principles are the two
other foundational aspects of quantum physics, the quantum nature of the
gravity needs to manifest all of these features. Here, we lay out an
interferometric platform for testing these three nonclassical aspects of
quantum mechanics in quantum gravity setting, which connects gravity and
quantum physics in a broader and deeper context. As we show in this work, all
of these three fundamental features of quantum gravity can be framed and fully
analyzed in an interferometric scheme.

##  I Introduction

Gravity, as one of the four building block forces of the universe, is the most
sensible of the forces that we deal with in our every day lives. However, it
also is considered as the most controversial one in modern physics. The
general theory of relativity for gravitation, which was formulated by Einstein
in 1915, aside from its thundering triumphs in explaining the universe,
encounters important issues at the high energy levels, such as black holes
information paradox and singularity problem Biswas et al. (2012); Brustein
(2014). These issues arise from the fact that there is no known way to
reconcile gravity with the other three forces of the nature. It seems that to
rectify these issues one needs to integrate quantum physics and gravity in a
unified frame, which remains as one of the most important unsettled issues in
physics.

Therefore, there has been an ongoing attempt to establish a quantum theory of
gravity, leading to development of different approaches such as string theory
and quantum loop gravity Carlip (2001); Kiefer (2007); Rovelli (2008). On the
other hand, considering the aforementioned challenges, the search for
alternative theories such as theories of emergent gravity, instead of quantum
gravity has also attracted a great attention Padmanabhan (2015); Verlinde
(2017). The main hurdle in establishing a quantum theory for gravity is
associated with the weakness of the gravitational interaction. In other words,
the quantum properties of gravity becomes sensible only in the range smaller
than the plank scale, requiring experiments with energies well beyond the
scales that are within the near-future capabilities. This fact makes it
impossible to discern a reliable route to the problem and to discriminate
between various developed models of quantum gravity Alfaro (2005). Therefore,
laying out some empirically feasible methods for studying and testing the
quantum nature of gravity is of utmost importance.

In recent years, there has been an increasing interest in the quantum-
information-theoretic approaches to the study of relativistic and
gravitational systems Peres and Terno (2004); Lunghi et al. (2013); Maleki and
Maleki (2020). Recently, as a novel approach for probing the quantum nature of
the gravity, it has been discovered that gravitationally-induced entanglement
can indeed serve as a witness of quantumness of the gravity Bose et al.
(2017); Marletto and Vedral (2017). As is demonstrated in Bose et al. (2017);
Marletto and Vedral (2017), the detection of entanglement, attained by
gravitational interactions of massive particles, can be considered as a
sufficient criteria for quanta of the gravitational field. As was
demonstrated, such entanglement generations could be tailored in
interferometric schemes, and considering the rapid progress in quantum
information and interferometry technologies, the experimental feasibility of
these approaches are within the sight.

Even though entanglement is central to the Hilbert space structure of
composite quantum systems, the quantum feature of the systems is in no way
restricted to the entanglement. In fact, in quantum physics, Heisenberg’s
uncertainty principle Heisenberg (1985), Bohr’s complementarity principle Bohr
(1928) and quantum correlations in composite quantum systems are the most
fundamental aspects of systems, such that the entire fabric of the quantum
weirdness can be captured by these fundamental characteristics Zubairy (2020).
Therefore, it is imperative to search for all these three building blocks of
quantum mechanics in quantum nature of the gravity, in a feasible experimental
setup.

As an other fundamental aspect of quantum physics, Bohr’s complementarity
principle Bohr (1928) provides one of the most fundamental aspects of the
nature, which explains that the two mutually exclusive attributes, such as the
waviness and the particleness, can both be imprinted in quantum systems, such
that measuring one feature prohibits its dual feature to be exhibited Bohr
(1928); Zubairy (2020). As an example, in the case of a single photon passing
through an interferometer, the particleness of the photon is embedded in the
path predictability of the photon, while the waviness is encoded in the
visibility of the interference pattern on screen Englert et al. (1994). The
quantitative notion of such a discipline was first introduced by Wootters and
Zurek in 1979 Wootters and Zurek (1979), which lead to a mathematical
description in form of an inequality as $\mathcal{P}^{2}+\mathcal{V}^{2}\leq
1$, where $\mathcal{P}$ represents the predictability of a quantum system,
which contains the path information and indicates a quantity for particleness,
and $\mathcal{V}$ stands for the visibility of interference pattern, measuring
the waviness of the system Glauber (1986); Mandel (1991); Jaeger et al.
(1993); Englert (1996).

In an interesting attempt, it has been shown that the wave-particle duality
has a relationship with the entanglement in the system Jakob and Bergou
(2007); de Melo et al. (2007); Qian et al. (2018). This unification of the
duality and entanglement in double slit (two-qubit) analyses provided an
important relation as $\mathcal{P}^{2}+\mathcal{V}^{2}+\mathcal{C}^{2}=1$
Jakob and Bergou (2007); de Melo et al. (2007); Qian et al. (2018), in which
$\mathcal{C}$ is a measure of entanglement known as concurrence Wootters
(1998). More recently, an interesting geometrical correspondence between this
relation and stereographic projection of $S^{7}$ geometry was also formulated
Maleki (2019), which gives a full geometrical proof for the duality-
entanglement relation Maleki (2019).

In this paper, we put forward an experimentally feasible platform that enables
testing Heisenberg’s uncertainty principle, Bohr’s complementarity principle
and quantum entanglement as the nonclassical aspects of quantum mechanics in
quantum gravity framework, connecting gravity and quantum physics in a broader
and deeper context. As one important aspect of our study, we show that all of
these three fundamental charactristics of quantum gravity can be framed and
tested in an interferometric scheme.

The paper is organized as follows. In Section II, we introduce the quantum
gravitational interaction potential and discuss its fundamental implications.
In Section III, we briefly address the entanglement-complementarity relation
and its main features. In Section IV, we put forward the gravitaionally
induced complementarity and entanglement analyses in an interferometric
quantum superposition scenario. In Section V, we briefly analyze Bell
inequality which provides a practical way for the detection of entanglement in
quantum systems. In Section VI, we lay out the discuss of the uncertainty
principle in the context of graviationally induced quantum phases. In Section
VII, we discuss the experimental feasibility of the gravitaionally induced
entanglement and its analyses provided in this work. We finally provide a
short summary and conclude in VIII.

##  II Main aspects of quantum gravity

Here, we briefly consider some important aspects of the gravitationally
induced phase, and the significance of potential that results into such an
entanglement. To this end, we start with the action of the gravitational field
as Hobson et al. (2006)

$S=\int d^{4}x\sqrt{-g}[R+\mathcal{L}_{m}],$ (1)

where $g$ is the determinant of metric $g_{\mu\nu}$, and $R$ is Ricci scalar
which depends on the derivations of the metric, and is related to the Ricci
tensor by $R=g^{\mu\nu}R_{\mu\nu}$. Also, $\mathcal{L}_{m}$ represents the
matter part of the action.

Now, taking the variation over metric one can reach to the Einstein field
equation Hobson et al. (2006)

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi GT_{\mu\nu},$ (2)

where $T_{\mu\nu}$ is the stress-momentum tensor. To make the quantum nature
of the gravity manifest, we study the perturbation $h_{\mu\nu}$ on the
background metric of Minkowski space-time $\eta_{\mu\nu}$. Therefore, the
entire metric can be written as $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$.
Considering the weak field limit, after expanding the metric and omitting the
orders higher than quadratic terms and choosing the harmonic gauge we obtain
Zee (2010)

$\displaystyle S_{WF}$ $\displaystyle=\frac{1}{2}\int d^{4}x((-\frac{1}{32\pi
G}$
$\displaystyle\times(\partial_{\lambda}h^{\mu\nu}\partial^{\lambda}h_{\mu\nu}-\frac{1}{2}\partial_{\lambda}h\partial^{\lambda}h)+h^{\mu\nu}T_{\mu\nu}).$
(3)

Now, considering $h$ as a field, the first two terms refer to the free field
of the gravity and the third term indicates the interaction of the gravity
with the matter. The free gravitatioal field can be written as Zee (2010)

$h^{\mu\nu}=\frac{1}{(2\pi)^{3}}\int\frac{d^{3}k}{\sqrt{\omega}}\sum_{\sigma=\pm
2}(e^{\mu\nu}_{\sigma}(\vec{p})a_{\sigma}(\vec{p})e^{i\vec{p}.\vec{x}}+h.c.),$
(4)

where $\sigma$ denotes the spin of the graviton and $e^{\mu\nu}$ is the
polarization tensor with the properties
$e_{\sigma}^{\mu\nu}e_{\sigma^{\prime}}^{\mu\nu\star}=2\delta_{\sigma\sigma^{\prime}}$.

As a result, we can calculate the propagator and the scattering process of two
masses. To illustrate, in one loop level, after Fourier-transformation of
scattering amplitude in momentum space, and in the non-relativistic limit, the
potential can be obtained as Radkowski (1970); Kirilin and Khriplovich (2002);
Bjerrum-Bohr et al. (2003)

$V(r)=-\frac{Gm_{1}m_{2}}{r}(1+3\frac{G(m_{1}+m_{2})}{rc^{2}}+\frac{41G\hbar}{10\pi
r^{2}c^{3}}).$ (5)

The second term is the General Relativistic correction, and the third term is
related to the quantum gravity correction to the classical potential. These
two terms are extremely small compared to the first term. More specifically
the last terms is quite negligible, compared to the first two terms;
therefore, the detection of quantum nature of the gravity does not seem to be
feasible if one tries to detect it by using the last term of this equation.
This, in fact, shows why observation of the quantum mechanical nature of
gravity is such a difficult task.

Considering the extremely small contribution of quantum part in the potential
in Eq. (5), the substantially important quantum is whether it is possible to
observe the quantum nature of the gravity through the fist term (the Newtonian
approximation) of the potential? The answer to this question is surprisingly,
yes. This could be achieved by a purely information-theoretic approach to the
gravity. In particular, as recently was discovered, an interesting route to
this goal is the gravity induced entanglement, which circumvents the
challenges of the weak strength of the quantum contribution to the
gravitational interactions. The subtleness of the idea lies in the fact that
the induced entanglement can emerge even at the low energy limits, where we
can approximate the potential with its first term, i.e., the Newtonian
approximation Marshman et al. (2020); Andersen (2019). Interestingly, this
entanglement-assisted analyses of the quantum nature of gravity offers an
advantage for the accessibility of the induced quantumness in an empirical set
up, compared to the effect that are directly proportional to Planck constant
as appears in last term of the potential in Eq. (5). In the next sections we
discuss about this remarkable approach toward the investigating of quantum
gravity.

The important role of entanglement is inherited by the fact that if the
gravitational interaction can generate entanglement among two masses, the
graviational field itself should be of quantum nature. In other words, we
assume that two masses interact via gravity, in which the interaction
generates entanglement. In this setting, gravity acts as a mediator of the
entanglement between two masses. The interaction is induced by the exchange of
gravitons as the mediator of the entanglement. If the entanglement is
generated then graviational field needs to be coupled quantum mechanically to
each test mass in order for the generation of the entanglement. This is due to
the fact that no classical mediator can generate entanglement, as was shown in
Krisnanda et al. (2017). A similar conclusion can be drawn from LOCC theorem
which states that no local operation and classical communication (LOCC) can
increase the entanglement Nielsen and Chuang (2002). Therefore, starting with
two disentangled masses, the entire system remains disentangled unless some
quantum mechanical interactions are applied. As an important conclusion of
this fact, detection of entanglement in such a setting is suffix to conclude
that the gravity is of quantum nature.

##  III  Two-qubit Complementarity-entanglement relation

The space of a two-qubit state is the product of Hilbert space of each qubit
denoted by $H^{\mathbb{C}}_{1}\otimes H^{\mathbb{C}}_{2}$, offering a
4-dimensional Hilbert space for the system. Considering the fact that each
Hilbert space, in this setting, can be spanned by two orthogonal basis
$\\{|0\rangle,|1\rangle\\}$, the basis of two-qubit Hilbert space can be
written as $\\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\\}$, where
$|ij\rangle$ denotes the composite basis as $|i\rangle\otimes|j\rangle$, with
$i,j=0,1$. Therefore, the general form of a pure two-qubit state can be
expressed as

$\displaystyle|\psi\rangle=\alpha_{0}|00\rangle+\alpha_{1}|01\rangle+\alpha_{2}|10\rangle+\alpha_{3}|11\rangle.$
(6)

Interestingly, we can encode any two-state system, such as the paths in Young
double-slit experiments or Mach Zehnder interferometer as a correspondence to
a two-qubit system.

To characterise the set up for quantum analyses of the gravity in this work,
we consider two massive particles, each subjected to a Mach Zehnder
interferometer as depicted in Fig. 1. We can assign to the upper path and the
lower path states in the interferometer two orthogonal states $|u\rangle$ and
$|d\rangle$, respectively. This encoding is equivalent to $|1\rangle$ and
$|0\rangle$ in Eq. (6). Hence, assigning the basis $|u_{i}\rangle$ and
$|d_{i}\rangle$ for the $i$th interferometer ($i=1,2$), the system of the two
interferometers is described by the state of the form

$\displaystyle|\psi\rangle=\alpha_{0}|u_{1}u_{2}\rangle+\alpha_{1}|u_{1}d_{2}\rangle+\alpha_{2}|d_{1}u_{2}\rangle+\alpha_{3}|d_{1}d_{2}\rangle.$
(7)

Figure 1: Experimental set-up of interferometer for gravitationally induced
entanglement. The masses $m_{1}$ and $m_{2}$ undergo independent Mach Zehnder
type interference, and interact with each other by gravity. $BS_{i}$ ($i=1,2$)
indicates a beam splitter which is characterized by $(r_{i},t_{i})$. The
second beam splitter in each interferometer is assumed to be a 50:50 beam
splitter.

With this description, we can study the duality-entanglement concept in a two-
qubit system as shown the setup in Fig. 1. In our analyses of the wave-like
and particle-like features, without loss of generality, we consider the first
interferometer (first mass). Naturally, a similar discussion is also valid for
the second interferometer.

Now, wave-like feature in this system (first mass) is quantified by the
visibility, which is determined by Qian et al. (2018); de Melo et al. (2007);
Jakob and Bergou (2007); Maleki (2019)

$\displaystyle\mathcal{V}=\frac{p_{D}^{max}-p_{D}^{min}}{p_{D}^{max}+p_{D}^{min}},$
(8)

where $p_{D}$ is the probability of detecting the system after the second
beam-splitter of the first interferometer, and the upper indexes indicate the
maximum and minimum of the probability. Applying this relation to the quantum
state in Eq. (7), the visibility reduces to Maleki (2019)

$\displaystyle\mathcal{V}=2\mid{\alpha_{2}^{*}}\alpha_{0}+{\alpha_{3}^{*}}\alpha_{1}\mid.$
(9)

In a similar vein, the predictability is determined as Qian et al. (2018); de
Melo et al. (2007); Jakob and Bergou (2007); Maleki (2019)

$\displaystyle\mathcal{P}=\frac{\mid p_{u}-p_{d}\mid}{\mid p_{u}+p_{d}\mid},$
(10)

where the parameters $p_{u}$ and $p_{d}$ are the probabilities of finding
first particle in each of the chosen paths (i.e., the probability of finding
the first mass in the upper or the lower arm of its corresponding
interferometer). Hence, from this equation the particleness of system can be
attained as Maleki (2019)

$\displaystyle\mathcal{P}=|{(|\alpha_{0}|}^{2}+{|\alpha_{1}|}^{2})-({|\alpha_{2}|}^{2}+{|\alpha_{3}|}^{2})|.$
(11)

On the other hand, the amount of entanglement in the above system can be
obtained using concurrence, which is defined through $R$ matrix such that
$R=\sqrt{\sqrt{\rho}\bar{\rho}\sqrt{\rho}}$, with
$\bar{\rho}=(\sigma_{x}\otimes\sigma_{x})\rho^{\star}(\sigma_{x}\otimes\sigma_{x})$.
By organizing the eigenvalues of the $R$ matrix in decreasing order, the
concurrence is defined as Wootters (1998)

$\displaystyle\mathcal{C}=\text{max}\\{0,\lambda_{0}-\lambda_{1}-\lambda_{2}-\lambda_{3}\\},$
(12)

In which the eigenvalues of matrix $R$ is denoted by $\lambda_{i}$ in the
deceasing order. In the case of the state in Eq. (7), the concurrence is given
by Maleki (2019)

$\displaystyle\mathcal{C}=2\mid\alpha_{0}\alpha_{3}-\alpha_{1}\alpha_{2}\mid.$
(13)

Recently, it was realized that concurrence plays a significant role in a
quantum complementarity setting, such that Qian et al. (2018); de Melo et al.
(2007)

$\displaystyle\mathcal{P}^{2}+\mathcal{V}^{2}+\mathcal{C}^{2}=1.$ (14)

This relation shows that entanglement can indeed control the duality in
quantum system, hence, the full description of the system entails wave-
particle-entanglement triality relation as above, rather than the duality
description alone. This relation also can be proven geometrically as outlined
in Ref. Maleki (2019), where it was shown that complementarity principle can
indeed be analysed from a completely geometric perspective.

##  IV gravitaionally induced complementarity and entanglement

As we discussed earlier, the ultimate theory of quantum gravity should enable
us to express gravity as a superposition of different states Marshman et al.
(2020); Andersen (2019). This is a main feature of quantum gravity which makes
it different from classical case and could be used in finding an experimental
approach to test the theory in a table-top experiment. Following this line of
thoughts, recently, it was found that quantum nature of gravity can be tested
in the light of entanglement generated by the gravity Bose et al. (2017);
Marletto and Vedral (2017). In these interesting works, it was shown that
interferometric setup enables testing the existence of the gravitationally-
induced entanglement between two masses.

Here, we lay out a rather general framework to consider quantum foundation of
gravity. To this end, we assume two massive particles, $m_{1}$ and $m_{2}$,
each subjected to an interferometer as depicted in Fig. 1. We also assume
that, in the first interferometer the first beam splitter, which particle
encounters, is characterized by $(r_{1},t_{1})$, where $r_{i}$ and $t_{i}$
represent the refectivity and transmissivity parameters of the $i$th beam
splitter. Similarly, in the second interferometer its first beam splitter is
characterized by $(r_{2},t_{2})$. For simplicity, we could assume that the
arms which is perpendicular to the initial direction of motion of the
particles in interferometer are negligible in comparison to the parallel arms
Bose et al. (2017); Marletto and Vedral (2017). Therefore, the initial state
of the two masses after passing through the first beam splitters can be
expressed as the product of states of the first and second particle paths,
which is given by

$\displaystyle|\psi(t=0)\rangle=(t_{1}|u_{1}\rangle+r_{1}|d_{1}\rangle)\otimes(t_{2}|u_{2}\rangle+r_{2}|d_{2}\rangle).$
(15)

Note that the setting considered here is much more general compared to the
previous studies Bose et al. (2017); Marletto and Vedral (2017) where only
50:50 beam splitters were taken into the consideration.

Figure 2: The concurrence of the system as a function of different parameters
in the setup: (a) shows the dependence of concurrence on the gravitaionally
induced phase and reflectivity of the first beam splitter in the first
interferometer. (b) depicts the concurrence as a function of reflectivity of
the first beam splitters in each interferometer, where the gravitaionally
induced phase is set to $\phi=\pi$. Figure 3: The visibility of the system as
a function of different parameters in the setup: (a) the dependence of
visibility on the gravitaionally induced phase and reflectivity of the first
beam splitter in the first interferometer. (b) visibility as a function of
reflectivity of first beam splitters in each interferometer, assuming that the
gravitaionally induced phase is $\phi=\pi$.  Figure 4: The predictability of
the system as a function of different parameters in the setup: (a) dependence
of predictability on the gravitaionally induced phase and reflectivity of the
first beam splitter in the first interferometer. (b) predictability as a
function of reflectivity of first beam splitters in each interferometer, where
the gravitaionally induced phase is fixed to $\phi=\pi$.

The interaction of a quantum particle with gravitational field results in an
induced phase, which its first experimental demonstration was attained in a
famous experiment by Colella, Overhauser and Werner (which is known as the COW
experiment) in 1975 Colella et al. (1975). Now, similar to this experiment, we
consider the setting where the gravitational field can indeed induce phase
shift on the quantum systems Bose et al. (2017); Marletto and Vedral (2017).
In this setting gravity induces a phase shift and decouples from the system
once the phase shift is attained Bose et al. (2017); Marletto and Vedral
(2017). Therefore, the initial state of the systems given by
$|\psi(t=0)\rangle$ will evolve into

$\displaystyle|\psi(t=T)\rangle$
$\displaystyle=r_{1}r_{2}e^{i\phi_{1}}|d_{1}\rangle|d_{2}\rangle+r_{1}t_{2}e^{i\phi_{2}}|d_{1}\rangle|u_{2}\rangle$
$\displaystyle+t_{1}r_{2}e^{i\phi_{3}}|u_{1}\rangle|d_{2}\rangle+t_{1}t_{2}e^{i\phi_{4}}|u_{1}\rangle|u_{2}\rangle,$
(16)

before entering the second beam splitter in each interferometer. Here, the
phases are given by

$\displaystyle\phi_{1}$ $\displaystyle=G\frac{m_{1}m_{2}}{\hbar
d}T,\qquad\phi_{2}=G\frac{m_{1}m_{2}}{\hbar(d+x)}T,$ $\displaystyle\phi_{3}$
$\displaystyle=G\frac{m_{1}m_{2}}{\hbar(d-x)}T,\qquad\phi_{4}=\phi_{1}.$ (17)

In which $T$ is the time duration of gravitational interaction in each arm and
$x$ is the width of interferometer.

Hence, this state at hand, we can calculate the entanglement using Eq. (13),
from which we obtain

$\displaystyle\mathcal{C}=2r_{1}r_{2}t_{1}t_{2}\mid
1-e^{i(2\phi_{1}-\phi_{2}-\phi_{3})}\mid.$ (18)

This immediately results in

$\displaystyle\mathcal{C}^{2}=8(r_{1}r_{2}t_{1}t_{2})^{2}(1-\cos(\phi)),$ (19)

in which $\phi=2\phi_{1}-\phi_{2}-\phi_{3}$. It is easy to check that when
$\phi=2n\pi$ (for $n$ being an integer number) the entanglement is zero, while
it becomes maximum when $\phi=n\pi$ ($n$ being an odd number).

The behavior of this equation is presented in Fig. 2. In Fig. 2(a), the
dependence of the concurrence on the gravitaionally induced phase and
reflectivity of the first beam splitter in the first interferometer is
illustrated. In Fig. 2(b) also, the concurrence as a function of reflectivity
of first beam splitters in each interferometer is plotted, in which the
gravitaionally induced phase is set to be $\phi=\pi$. In both cases when we
choose the reflectivity of beam splitters as $r_{1}=r_{2}=1/\sqrt{2}$, the
entanglement becomes maximum.

We note that the entanglement generated in this setting is indeed induced by
gravitational field. Here, gravity acts as a mediator of the entanglement, and
if the entanglement could be observed we can conclude that the gravity is of
quantum nature. This is due to the fact that no classical mediator can
generate entanglement Krisnanda et al. (2017).

Next, we consider the visibility of the interference in the first
interferometer. Using Eq. (9), we can obtain for the visibility of the first
particle

$\mathcal{V}=2\mid(r_{1}r_{2}e^{i\phi_{1}})(t_{1}r_{2}e^{-i\phi_{3}})+(t_{1}t_{2}e^{-i\phi_{4}})(r_{1}t_{2}e^{i\phi_{2}})\mid.$
(20)

Therefore, the square of the visibility provides

$\displaystyle\mathcal{V}^{2}$ $\displaystyle=$ $\displaystyle
4(r_{1}t_{1})^{2}\mid r_{2}^{2}+t_{2}^{2}e^{-i\phi}\mid^{2}$ (21)
$\displaystyle=$ $\displaystyle
4(r_{1}t_{1})^{2}[r_{2}^{4}+t_{2}^{4}+2(r_{2}t_{2})^{2}\cos{(\phi)}].$

As can readily be seen from this result, if the refectivity in the second
interferometer becomes zero ($r_{2}=0$), the the gravitational interaction
would have no effect on the visibility of the first interferometer.

In Fig. 3 the results for visibility is plotted. In Fig. 3(a), the visibility
of the system as a function of reflectivity of first beam splitters in each
interferometer is depicted, where the gravitaionally induced phase is set to
$\phi=\pi$. Fig. 3(b) illustrates the dependence of visibility on the
gravitaionally induced phase and reflectivity of the first beam splitter in
the first interferometer.

Finally we attain the predictability of the first mass from Eq. (11) as

$\displaystyle\mathcal{P}^{2}=(r_{1}^{2}-t_{1}^{2})^{2}.$ (22)

Considering these relations the equality in Eq. (14) is fulfilled. As a
result, the gravity induces a tradeoff between entanglement and
complementarity.

In Fig. 4 the results for the predictability in terms of various involving
parameters is illustrated. In Fig. 4(a) predictability as a function of
reflectivity of beam splitters in each interferometer is depicted. Here the
gravitaionally induced phase is fixed to $\phi=\pi$. In Fig. 4(b) the
dependence of predictability on the gravitaionally induced phase and
reflectivity of the first beam splitter in first interferometer is
illustrated. As we expect from Eq. 22 it only dependence on the reflectivity
of the first beam splitter and for $r_{1}=1/\sqrt{2}$, the path information
entirely vanishes.

Taking $r_{1}=r_{2}=1/\sqrt{2}$, the entanglement becomes maximum, while
visibility reduces. Practically, choosing 50:50 beam splitter observing the
quantum entanglement features of the gravity becomes more feasible. In this
case, the relations for the first particle simplifies to

$\displaystyle\mathcal{C}^{2}$ $\displaystyle=$ $\displaystyle
2(r_{1}t_{1})^{2}(1-\cos(\phi)),$ $\displaystyle\mathcal{V}^{2}$
$\displaystyle=$ $\displaystyle 2(r_{1}t_{1})^{2}(1+\cos(\phi)),$
$\displaystyle\mathcal{P}^{2}$ $\displaystyle=$
$\displaystyle(r_{1}^{2}-t_{1}^{2})^{2}.$ (23)

This readily provides $\mathcal{P}^{2}+\mathcal{V}^{2}+\mathcal{C}^{2}=1$.
Therefore, quantum nature of the gravity provides complementarity and
entanglement, enabling a feasible experimental analysis of the quantum
gravity, and providing important features of the quantum nature of the
gravity.

##  V Bell inequality and testing quantum gravity

The non-local correlation exhibited by quantum entanglement was first
addressed in 1935 by Einstein, Podolsky and Rosen in a famous paper (usually
called the EPR paper) Einstein et al. (1935), which demonstrated the
apparently paradoxical context of quantum mechanics. Almost three decades
later, John Stewart Bell found an experimentally manageable and quantitative
platform to investigate this controversial feature of composite quantum
systems and how to discriminate it from classical descriptions Bell (1987).
Here, we discuss Bell inequality in the context of quantum gravity, which
provides an experimentally testable approach for quantum nature of the
gravity. The most commonly used Bell inequality is the so-called Bell-CHSH
inequality Clauser et al. (1969); Horodecki et al. (1995). The CHSH operator
is given by Clauser et al. (1969); Horodecki et al. (1995)

$\hat{B}=\vec{a}\cdot\vec{\sigma}\otimes\left(\vec{b}+\vec{b^{\prime}}\right)\cdot\vec{\sigma}+\vec{a^{\prime}}\cdot\vec{\sigma}\otimes\left(\vec{b}-\vec{b^{\prime}}\right)\cdot\vec{\sigma},$

where $\vec{a},\vec{a^{\prime}},\vec{b},\vec{b^{\prime}}$ are unit vectors.
Given the above relation, the Bell-CHSH inequality reads Clauser et al.
(1969); Horodecki et al. (1995)

$|\langle\hat{B}\rangle|=|\operatorname{tr}(\rho\hat{B})|\leq 2.$

Violation of this inequality, i.e., $|\langle\hat{B}\rangle|>2$, indicates
existence of quantum correlations in a quantum state, and hence, it provides
an experimentally appealing method for the test of quanta of the gravity.

As an interesting connection, there is a relation between entanglement and the
maximum violation of Bell-CHSH inequality given as Ghosh et al. (2001);
Verstraete and Wolf (2002)

$\displaystyle\mathcal{B}=2\left(\sqrt{1+\mathcal{C}^{2}}\right).$ (24)

This in turn shows that Bell inequality can be violated by all entangled
states ($\mathcal{C}\neq 0$). Since the quantum mechanical non-locality
appears when this parameter exceeds two, we plot $\mathcal{I}=\mathcal{B}-2$
in Fig. 5. Accordingly, Fig. 5(a) indicates the violation of Bell inequality
as a function of reflectivity of first beam splitter in first interferometer
and the gravitationally induced phase. In Fig. 5(b) we demonstrate Bell
parameter $\mathcal{I}$ as a function of the reflectivity of the first beam
splitters in each interferometer, where we have assumed that the
gravitaionally induced phase is $\phi=\pi$. Therefore, the maximum violation
of the Bell-CHSH inequality occurs when both of the beam splitters are 50:50.

To establish an interesting relation between complementarity and Bell
parameter, using Eq. (24), we can attain a new relation as

$\displaystyle 4(\mathcal{P}^{2}+\mathcal{V}^{2})+\mathcal{B}^{2}=8.$ (25)

This relation demonstrates that in order to measure quantum non-locality,
which is sufficient condition for the quantum nature of the gravity, we can
simply measure the path-information and the visibility. This observation is
insightful for practical detection of non-classical correlations through
gravity.

Figure 5: The figure(a) indicates the divergence of Bell parameters as
function of reflectivity of the first beam splitter in the first
interferometer, and the gravitationally induced phase. In the figure(a) , the
divergence form Bell parameter as a function of reflectivity of first beam
splitters of first interferometers is plotted, in which the gravitaionally
induced phase is fixed to $\phi=\pi$.

##  VI Uncertainty relations in interferometeric quantum gravity

Now that we have considered entanglement and complementarity as the two
fundamental aspects of the quantum mechanics, we now address the uncertainty
principle in the context of quantum gravity, as the other fundamental concept
of the quantum mechanics Bagchi and Pati (2016). The concept of quantum
uncertainty was first proposed by Werner Heisenberg for the position and
momentum operators Heisenberg (1985). Later Robertson generalized it for any
pair of non-commutative operators, such that considering a pair of non-
commutative operators A and B the uncertainty relation reads Robertson (1929)

$\displaystyle\Delta{A}\Delta{B}\geq\frac{1}{2}\mid\langle[A,B]\rangle\mid,$
(26)

where $\Delta{A}$ and $\Delta{B}$ are the standard deviations of these
operators and $\langle[A,B]\rangle$ is the expectation value of the commutator
of the operators. We could define operators corresponding to the
predictability and visibility as follows Bosyk et al. (2013)

$\displaystyle\hat{\mathcal{P}}$ $\displaystyle=$ $\displaystyle\sigma_{z},$
(27) $\displaystyle\hat{\mathcal{V}}$ $\displaystyle=$
$\displaystyle\cos{\theta}\ \sigma_{x}+\sin{\theta}\ \sigma_{y},$

where

$\displaystyle\sigma_{z}$ $\displaystyle=$
$\displaystyle\ket{u_{1}}\bra{u_{1}}-\ket{d_{1}}\bra{d_{1}},$
$\displaystyle\sigma_{x}$ $\displaystyle=$
$\displaystyle\ket{u_{1}}\bra{d_{1}}+\ket{d_{1}}\bra{u_{1}},$
$\displaystyle\sigma_{y}$ $\displaystyle=$
$\displaystyle-i\ket{u_{1}}\bra{d_{1}}+i\ket{d_{1}}\bra{u_{1}}.$ (28)

Since the expectation values of an arbitrary operator $O$, with density matrix
$\rho$, is defined as $\langle O\rangle=Tr(\rho O)$, one can write the
uncertainty relation for the predictability and the visibility operators as

$\displaystyle\Delta\mathcal{P}\Delta\mathcal{V}$ $\displaystyle\geq
2(r_{1}t_{1})$ $\displaystyle\times\mid
r_{2}^{2}\sin({\phi_{1}-\phi_{3}-\theta})+t_{2}^{2}\sin({\phi_{2}-\phi_{4}-\theta})\mid.$
(29)

Even though this uncertainty relation is fundamentally significant in the
context of quantum mechanics, some tighter, hence better, uncertainty
relations are considered as the alternative uncertainty relations Bagchi and
Pati (2016); Białynicki-Birula and Mycielski (1975); Maccone and Pati (2014);
Song et al. (2017). One useful alternative uncertainty is defined as the sum
of the uncertainties of each operator. As a result, for two Hermitian unitary
operators $A=\vec{a}.\vec{\sigma}$ and $B=\vec{b}.\vec{\sigma}$, where
$\vec{a}$ and $\vec{b}$ are the unit vectors, simultaneously acting on the
density matrix $\rho=1/2(I+\vec{r}.\vec{\sigma})$, which $\vec{r}$ is a unit
vector, the uncertainty is defined as Bagchi and Pati (2016)

$\displaystyle\Delta{A}^{2}+\Delta{B}^{2}$ $\displaystyle\geq
1+\mid\vec{a}.\vec{b}\mid^{2}-2\mid\vec{a}.\vec{b}\mid$
$\displaystyle\times\sqrt{1-\Delta{A}^{2}}\sqrt{1-\Delta{B}^{2}}\mid(\vec{a}\times\vec{b}).\vec{r}\mid^{2}.$
(30)

Now, considering such an uncertainty in the context of the problem in hand, we
attain

$\displaystyle\Delta\mathcal{P}^{2}+\Delta\mathcal{V}^{2}$ $\displaystyle\geq
1+4(r_{1}t_{1})^{2}$ $\displaystyle\times\mid
r_{2}^{2}\sin({\phi_{1}-\phi_{3}-\theta})+t_{2}^{2}\sin({\phi_{2}-\phi_{4}-\theta})\mid^{2}$
(31)

The minimum of the sum uncertainty is one, which can be obtained when one of
the operators has zero uncertainty. The second term, in the right-hand-side
can never exceed one, and interestingly is quadrant of the uncertainty
relation in Eq. (29). Therefore, the sum uncertainty bound is more tighter
than the uncertainty relation in Eq. (29).

In an experimental setting, one can control the quantum uncertainty effects of
the gravity by controlling the gravitationally induced phases. This, in fact,
provides a deep connection between the quantum mechanics and gravity from the
uncertainty perspective.

##  VII experimental challenges

The experimental investigations of quantum gravity via induced entanglement
addresses one of the most important problems of the physics. Therefore, it
would be important to consider the feasibility of the induced entanglement
generations via current technology. In this section, we briefly review the
important challenges that such an experiments is subjected to and discuss the
the feasibility of the study.

One important challenge for generating entanglement in this realm is that one
needs to create a large enough induced phase shift that can be measurable in
the experiments. Considering the relation for the induced phase,
$G{m_{1}m_{2}T}/{\hbar d}$, there are three parameters that we could adjust in
reaching to a measurable phase in the order of unit. The first parameter is
masses of the particles. The product of the masses must be big enough to
enhance the entanglement degree. This requires preparing a large system in a
quantum superposition. This problem has intensively considered in the context
of quantum mechanics, and many improvements have been made thus far Leggett
(1980); Julsgaard et al. (2001); Lee et al. (2011); Klimov et al. (2015);
Ockeloen-Korppi et al. (2018). Therefore, we can consider a system with mass
of about $10^{-14}$kg a reasonable mass size, for which the possible
candidates could be a massive molecule Arndt et al. (1999); Gerlich et al.
(2011), micro crystals Bose et al. (2017) or Bose-Einstein condensates
Helmerson and You (2001); Peise et al. (2015). The other factor in controlling
the induced phase is the distance between two interferometers. In a realistic
setting, if the two objects get more closer to each other than a certain
limit, the other interactions such as electrostatic interactions can dominate
the gravitational interaction Bose et al. (2017); Marletto and Vedral (2017).
To overcomes this challenge, it has been proposed that by using a conductor in
between the two masses, hence shielding other interactions, we can adjust the
distance of the interferometer as close as about $d=1\mu m$ van de Kamp et al.
(2020). The last parameter that we can control is the time that each particles
travels in the interferometer arms. To illustrate, if we adjust the travel
time of particles in the interferometer arms in the order of $T=0.1$sec, the
induced phase could be adjust in the order of unit and consequently the
maximum entanglement of two particles through gravitational interaction would
be achievable. As another practical setup, one can consider two coupled nano-
mechanical oscillators with the mass $10^{-12}$kg, and the interaction time
$T=10^{-6}$sec that would enable the phase shifts large enough to prepare
maximally entangled states with the distance $d$ in the order of a few
micrometers Marletto and Vedral (2017). Therefore, in such a timescales we
must protect the system from decoherence from the environment. To eliminate
the environments effect on the system we need a high vacuum and a low
temperature. The estimated pressure is in the order of $10^{-15}$pa and
temperature must be about $0.1$K Bose et al. (2017), which is an achievable
condition with the current technologies Ishimaru (1989).

##  VIII Conclusion

Quantization of the gravity remains as one of the most important, yet
extremely illusive, challenges at the heart of modern physics. It has been
argued that a direct empirical evidence for the quantum nature of the gravity
(i.e., detecting gravitons) can shed light on the charactristics of the
ultimate theory of the quantum gravity. However, such a task is far beyond the
current capabilities and it seems not to be achievable in terms of near future
technologies. To circumvent this impasse, it was recently shown that
gravitationally-induced entanglement, tailored in the interferometric
frameworks, can be used to detect the quantum nature of the gravity. However,
many fundamental and empirical aspects of these schemes are yet to be
discovered. Considering the fact that, beside quantum entanglement, quantum
uncertainty and complementarity principles are the two other foundational
aspects of quantum physics, the quantum nature of the gravity needs to
manifest all of these features. In this work, we considered an interferometric
setup for testing these three nonclassical aspects of quantum mechanics in
quantum gravity setting, which can shed light on the connections between
gravity and quantum physics in a broader and deeper discipline. As we show in
this work, all of these fundamental features of quantum gravity can be framed
and fully analyzed in an interferometric scheme. We showed the relation
between gravitationally-induced entanglement and complementarity principle and
investigated its features. We also developed a relation between the violation
of Bell inequality as a sufficient criteria for the quantumness of the
entanglement and complementarity principle.

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