# Gravitationally modulated quantum correlations: Discriminating classical and
quantum models of ultra-compact objects with Bell nonlocality

Luciano<EMAIL_ADDRESS>0.85358pt1,2,3}$ and
Fabrizio<EMAIL_ADDRESS>0.85358pt1,2}$
1Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno,
Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy.
2INFN, Sezione di Napoli, Gruppo collegato di Salerno, Fisciano (SA), Italy.
3Institut für Theoretische Physik, Albert-Einstein-Allee 11, Universität Ulm,
89069 Ulm, Germany.

(April 21, 2023)

###### Abstract

We investigate the relation between quantum nonlocality and gravity at the
astrophysical scale, both in the classical and quantum regimes. Considering a
gedanken experiment involving particle pairs orbiting in the strong
gravitational field of ultra-compact objects, we find that the violation of
Bell inequality acquires an angular modulation factor that strongly depends on
the nature of the gravitational source. We show how such gravitationally-
induced modulation of quantum nonlocality readily discriminates between black
holes (both classical and inclusive of quantum corrections) and string
fuzzballs, i.e., the true quantum description of ultra-compact objects
according to string theory. These findings promote Bell nonlocality as a
potentially key tool in putting quantum gravity to the test.

##  I Introduction

The development of a consistent and predictive theory of quantum gravity is
one of the main unresolved conundrums in contemporary physics loll .
Relentless efforts in the attempt to reconcile quantum mechanics and general
relativity have produced a number of promising candidate models, including
asymptotic safety ass , causal dynamical triangulations causal , non-
commutative geometry ncg , loop quantum gravity lqg , doubly special
relativity dsr  string theory st  and the more recent proposal of
“gravitizing” quantum mechanics gravquant . All the aforementioned theoretical
schemes have their own characteristics and predictions which make them
profoundly different among each other. Despite that, it is still possible to
recognize similar aspects which are thus likely to be part of a general
treatment of quantum gravity (see for instance Refs. rec1 ; rec2  and therein
for a review on this topic). Prominent examples of features foreseen by many
of the above models are the emergence of an intrinsic non-local behavior in
the theoretical description of quantum gravity (_i.e._ , see Refs. nl1 ; nl2 )
and the existence of a minimal length at the Planck scale with the ensuing
modifications of the canonical commutation relations of quantum mechanics and
the associated Heisenberg uncertainty principle gup1 ; gup2 .

Concerning the notion of a minimal spatial resolution, this can be deduced
also from gedanken experiments involving large large  and micro micro  black
holes, in proximity of which quantum gravitational effects are expected to
become dominant. As a matter of fact, the strong gravity regime near a black
hole333For the purpose of the present analysis, henceforth we assume to work
with the strong gravity regime comprising the region of spacetime in proximity
of the horizon. prevents the use of any known approximation in the study of
quantum systems. To quote a relevant example along this direction, it is worth
observing that, although the extension of quantum field theory to curved
backgrounds has provided successful fundamental predictions (such as the
Hawking radiation), these findings are still plagued by unphysical divergences
when considered beyond their limits of applicability. However, this fact does
not undermine the validity of the aforementioned results, but it rather points
towards the quest for a unified description of quantum and gravitational
phenomena. Achievements in addressing these difficulties would yield major
progress towards a viable theory of quantum gravity able to settle open issues
such as the information paradox and the singularity problem that arise in the
context of classical and semi-classical approaches to gravitational phenomena.

An interesting resolution for both of the above issues in the framework of
superstring theory is represented by the fuzzball proposal fuzz1 ; fuzz2 ,
according to which the supposed black hole is in fact conceived as a massive
object made of a very large number of microscopic strings which, by
definition, feature a minimal length extension qualitatively of the order of
the Planck scale. Even though the original arguments leading to the fuzzball
solution were purely theoretical, it has been recently pointed out that
concrete realizations of fuzzballs lead to a phenomenology that might be
accessible, for instance via the observational investigation of gravitational
waves bianchi .

Nevertheless, the fuzzball proposal is not the only self-consistent and robust
alternative concerning the generalization of black hole physics that
incorporates quantum gravitational effects. Indeed, also loop quantum gravity
predicts the settlement of the problems discussed above by relying on the
underlying spacetime discretization lqgbh , whilst for the asymptotic safety
paradigm the solution is to be found in the quantum scale invariance asbh . On
a final note, it is worth stressing that another important class of black
holes can be derived in the context of higher-derivative theories and non-
local gravity hdgbh . These models cure the issues stemming from merging
quantum field theory and gravitation (_i.e._ , non-renormalizability, non-
unitarity, etc.) by adding higher-derivative terms in the Lagrangian of the
gravitational interaction. Interestingly, these contributions are able to
remove the unwanted features that affect the canonical quantization of
Einstein’s general relativity and may give rise to potentially detectable
effects in a significant number of physical phenomena.

In a parallel development, the community active in quantum information
science, atomic physics and quantum optics has picked up in recent years on
the original ideas by Bronstein and Feynman Bronstein1 ; Bronstein2 ; Feynman
, suggesting to test the hypothetical quantum nature of gravity in the
laboratory by measuring witnesses of the bipartite entanglement between two
test masses induced by a quantized gravitational field, i.e. a quantum
gravitational mediator Bose ; Marletto .

Motivated by the above considerations, in the present work we address the
broader question of gravitationally-induced modifications of nonlocal quantum
correlations. Given that a classical gravitational mediator cannot induce any
form of quantum nonlocality, be it, in ascending hierarchical order,
entanglement, steering, or Bell nonlocality, we investigate whether classical
and quantum gravity can have different effects on already existing quantum
correlations, previously established by other physical interactions on pairs
of test masses. To this end, we study a gedanken experiment which revolves
around the dynamics of the Bell nonlocality of particle pairs in the
gravitational field generated by ultra-compact objects of diverse nature, such
as black holes and fuzzballs, in order to assess whether and how different
gravitational sources affect the dynamical evolution of quantum nonlocality.

Historically, establishing a relation between cosmological objects and quantum
entanglement was the central result of a celebrated paper by Maldacena and
Susskind epepr , where it was conjectured that the entanglement shared by two
particles can be interpreted as a non-traversable wormhole; such a
correspondence may be viewed as a precondition for the unification of quantum
and gravitational effects. In addition to the aforesaid achievement, the
interplay between gravity and entanglement can be identified in a significant
number of relevant frameworks. For instance, it is worth recalling that, by
means of entanglement entropy, it is possible to deduce another theoretical
evidence of black hole thermodynamics related to the area law arealaw . Along
the same direction, by means of thermodynamical arguments one can also show
that Einstein’s field equations of general relativity must necessarily be
fulfilled if entanglement equilibrium is established thermogr .

Here, instead, by relying on Einstein-Podolsky-Rosen (EPR) nonlocal
correlations epr  shared by particle pairs orbiting around ultra-compact
objects, we investigate what insights Bell nonlocality, rather than
entanglement, can provide on the nature and properties of gravitational
structures. In this respect, it is important to recall once more that
nonlocality and entanglement are distinct concepts that stand in a
hierarchical relation: whilst a violation of Bell inequality always implies
entanglement, the opposite implication does not necessarily hold, a notorious
counterexample being that of the Werner mixed two-qubit states werner , which
can be entangled without violating Bell inequality.

Proceeding to evaluate explicitly the amount of Bell nonlocality in an extreme
astrophysical scenario, we resort to the physically transparent Clauser-Horne-
Shimony-Holt (CHSH) form of Bell inequality chsh ; bell ; bellreview  for
massive spin-$1/2$ particle pairs, and we find that gravity in the strong-
field regime affects significantly the quantum nonlocality shared by the test
particles. Indeed, the overall degree of violation of the CHSH inequality is
modulated by an angular factor that strictly depends on the nature of the
ultra-compact object under consideration.

This result, which is completely general and may be adapted also to different
frameworks, is elucidated by focusing on three relevant cases: the classical
Schwarzschild black hole, the Schwarzschild black hole within a quantum-
corrected treatment at leading (perturbative) order and the string fuzzball
solution. We find that the gravitational modulation of bipartite Bell
nonlocality discriminates unambiguously between all of them. In order to
proceed in our investigation of the thought experiment, we make use of some
recently introduced techniques that allow to evaluate EPR correlations in
different gravitational scenarios ueda ; adami ; our ; as a side result of the
main analysis, we generalize such techniques and extend their range of
validity to include any static and spherically symmetric spacetime whose
metric tensor is expressed in isotropic coordinates.

The paper is organized as follows: in Sec. II we introduce the necessary
mathematical tools based on the concept of Wigner rotation in curved
spacetime, as it is needed in the analysis of the EPR correlations shared by
two spin-$1/2$ particles in the gravitational field of an ultra-compact
object. Section III is devoted to the explicit computation of the Wigner
rotation in various regimes; this result is then applied to the evaluation of
the EPR correlations in Sec. IV. In Sec. V we discuss and compare three
relevant instances of static and spherically symmetric ultra-compact
gravitational objects, i.e. the string fuzzball, the classical Schwarzschild
black hole and the quantum-corrected Schwarzschild black hole, and we show how
for each of them the orbiting particle pairs feature a different degree of
Bell nonlocality. Finally, in Sec. VI we comment on our results and
perspectives on future research.

Throughout the manuscript, we adopt Planck units ($c=\hbar=G=1$) and the
mostly positive signature for the metric, i.e.,
$\eta_{ab}=\mathrm{diag}(-,+,+,+)$.

##  II Wigner rotation in curved spacetime

For a consistent treatment of spin-$1/2$ particles in curved spacetime it is
necessary to make use of the tetrad (or vierbein) formalism; for a
comprehensive introduction on this subject, the interested reader can consult
Ref. gravitation . A tetrad field $e^{\mu}_{a}$ evaluated at a spacetime point
$x$ is completely characterized by the relation

$g_{\mu\nu}(x)e^{\mu}_{a}(x)e^{\nu}_{b}(x)=\eta_{ab}\,,$ (1)

where the summation over repeated indexes is understood, $g_{\mu\nu}(x)$ is
the metric tensor defined on the Riemannian manifold and $\eta_{ab}$ is the
Minkowski metric acting on the flat plane tangent to the manifold in the point
$x$. Henceforth, to discriminate between the indexes of the manifold and of
the tangent bundle, we employ Greek letters for the former and Latin letters
for the latter.

The expression (1) is essential to analyze spin-$1/2$ particle states in
curved backgrounds, since they are defined as the states that belong to the
spin-$1/2$ representation of the local Lorentz transformation (LLT) group,
while general relativity is based upon invariance under diffeomorphisms.
Precisely in order to build a bridge between these two notions, one can
introduce tetrads, as they allow to “project” diffeomorphism-covariant tensors
of the differentiable manifold onto local Lorentz-covariant quantities defined
on a flat tangent plane.

By virtue of this procedure, a generic spin state with four-momentum
$k^{\mu}=mu^{\mu}$ (where $u^{\mu}u_{\mu}=-1$) at the spacetime point $x$ can
be unambiguously labeled with $|k^{a},\sigma;x\rangle$, where
$k^{a}=e^{a}_{\mu}k^{\mu}$ and $\sigma=\uparrow,\downarrow$ is the third
component of the spin. Naturally, the field $e^{a}_{\mu}$ is the inverse of
the one appearing in Eq. (1); consequently, the following identities hold:

$e^{\mu}_{a}e^{a}_{\nu}=\delta^{\mu}_{\nu}\,,\qquad
e^{\mu}_{a}e^{b}_{\mu}=\delta^{b}_{a}\,.$ (2)

If we now want to describe the dynamical evolution of a spin-$1/2$ particle
moving in curved spacetime, we have to account for different flat tangent
spaces, each of which is associated to a given point of the particle’s
trajectory. As a first step, we consider what happens after an infinitesimal
interval of proper time $d\tau$, after which the particle is located at the
new point $x^{\prime\mu}=x^{\mu}+u^{\mu}d\tau$. Accordingly, the shift in
momentum is given by

$k^{a}(x^{\prime})=k^{a}(x)+\delta k^{a}(x)\,,$ (3)

where the variation is made of two distinct contributions, namely444When there
is no need for disambiguation, the dependence on the spacetime position will
be omitted.:

$\delta k^{a}=\delta k^{\mu}e^{a}_{\mu}+k^{\mu}\delta e^{a}_{\mu}\,.$ (4)

By defining the four-acceleration $a^{\mu}=u^{\nu}\nabla_{\nu}u^{\mu}$
originated by an external force and recalling that $k^{\mu}k_{\mu}=-m^{2}$ as
well as $k^{\mu}a_{\mu}=0$, it is straightforward to observe that the first
variation of Eq. (4) becomes

$\delta
k^{\mu}=ma^{\mu}d\tau=-\frac{1}{m}\left(a^{\mu}k_{\nu}-k^{\mu}a_{\nu}\right)k^{\nu}d\tau\,.$
(5)

On the other hand, the second factor of Eq. (4) can be rewritten by
introducing the expression for the connection one-form gravitation , that is,
$\omega^{a}_{\mu b}=e^{a}_{\nu}\nabla_{\mu}e^{\nu}_{b}$, and hence

$\delta e^{a}_{\mu}=-u^{\nu}\omega^{a}_{\nu
b}e^{b}_{\mu}d\tau=\xi^{a}_{b}e^{b}_{\mu}d\tau\,.$ (6)

In so doing, exploiting Eqs. (5) and (6) to rewrite Eq. (4), one can identify
an infinitesimal local Lorentz transformation occurring for the quantity
$k^{a}$. As a matter of fact

$\delta k^{a}=\lambda^{a}_{b}k^{b}d\tau\,,$ (7)

where

$\lambda^{a}_{b}=-\frac{1}{m}\left(a^{a}k_{b}-k^{a}a_{b}\right)+\xi^{a}_{b}$
(8)

is an infinitesimal LLT. This means that the momentum of the particle as
viewed by a local reference frame (i.e., the one belonging to the tangent
space) undergoes the transformation

$k^{a}(x^{\prime})=\Lambda^{a}_{b}(x)k^{b}(x)\,,\qquad\Lambda^{a}_{b}=\delta^{a}_{b}+\lambda^{a}_{b}\,,$
(9)

which is precisely a LLT. Consequently, the evolution of a spin-$1/2$ state
must be described in terms of a representation of the spin-$1/2$ local Lorentz
group. Bearing this in mind, we recall that, in the context of flat spacetime,
under the action of a given Lorentz transformation $\Lambda^{a}_{b}$, the
spin-$1/2$ one-particle state $|k^{a},\sigma\rangle$ transforms as follows
tung ; weinberg :

$U(\Lambda)|k^{a},\sigma\rangle=\sum_{\sigma^{\prime}}D_{\sigma^{\prime}\sigma}^{(1/2)}\left(W\left(\Lambda,k\right)\right)|\Lambda
k^{a},\sigma^{\prime}\rangle\,,$ (10)

with $D_{\sigma^{\prime}\sigma}^{(1/2)}\left(W\left(\Lambda,k\right)\right)$
being a $2\times 2$ unitary matrix that allows for the the Wigner rotation
$W^{a}_{b}(\Lambda,k)$ of the spin. The Wigner rotation wigner  can be written
as

$W^{a}_{b}(\Lambda,k)=\left[L^{-1}(\Lambda k)\Lambda L(k)\right]^{a}_{b}\,,$
(11)

where $L^{a}_{b}$ is the Lorentz boost

$L^{0}_{0}=\Xi\,,\quad L^{i}_{0}=L^{0}_{i}=\frac{k^{i}}{m}\,,\quad
L^{i}_{j}=\delta_{ij}+\left(\Xi-1\right)\frac{k^{i}k^{j}}{|\vec{k}|^{2}}\,,$
(12)

with $\Xi=\sqrt{|\vec{k}|^{2}+m^{2}}$ and the indexes $i,j=1,2,3$.

When generalizing to include the case of a curved spacetime, we have to resort
to local Wigner rotations stemming from the LLTs described in Eq. (9).
Accordingly, Eq. (11) becomes

$U(\Lambda(x))|k^{a},\sigma;x\rangle=\sum_{\sigma^{\prime}}D_{\sigma^{\prime}\sigma}^{(1/2)}\left(W(x)\right)|\Lambda
k^{a},\sigma^{\prime};x\rangle\,.$ (13)

Notice that a similar scenario holds true not only for spinors, but for Dirac
bispinors as well; for recent applications of the latter, see Refs. bispinors
and references therein.

The form of the infinitesimal local Wigner rotation can be extracted from Eq.
(9); indeed, one can verify that

$W^{a}_{b}=\delta^{a}_{b}+\vartheta^{a}_{b}d\tau\,,\qquad\vartheta^{i}_{j}=\lambda^{i}_{j}+\frac{\lambda^{i}_{0}k_{j}-\lambda_{j0}k^{i}}{k^{0}+m}\,,$
(14)

where $i,j=1,2,3$, as they are the only non-vanishing terms of
$\vartheta^{a}_{b}$.

##  III Wigner rotation for a generic metric in isotropic coordinates

In the following, we compute the Wigner rotation angle for a general class of
static and spherically symmetric spacetime solutions, thus going beyond the
standard Schwarzschild case treated in Ref. ueda  and the weak-field limit
considered in Ref. our . To this aim, we make use of a generic line element
that can be cast in isotropic spherical coordinates as follows:

$ds^{2}=-f(r)dt^{2}+g(r)\left[dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}\right]\,.$ (15)

We note in passing that, by setting $f(r)=(1-M/2r)^{2}/(1+M/2r)^{2}$ and
$g(r)=(1+M/2r)^{4}$, one recovers the results of Ref. ueda  within a different
coordinate system, while if $f(r)=1+2\phi(r)$ and $g(r)=1-2\psi(r)$, with
$\phi(r)$ and $\psi(r)$ being weak gravitational potentials arising in
extended theories of gravity, one recovers the findings of Ref. our .

As the metric tensor is diagonal, we can compute the tetrads rather
straightforwardly

$e^{t}_{0}=\frac{1}{\sqrt{f}}\,,\quad e^{r}_{1}=\frac{1}{\sqrt{g}}\,,\quad
e^{\theta}_{2}=\frac{1}{r\sqrt{g}}\,,\quad
e^{\varphi}_{3}=\frac{1}{r\sin\theta\sqrt{g}}\,.$ (16)

From the above equation, it is possible to deduce the non-vanishing components
of the connection one-form

$\displaystyle\omega^{0}_{t1}=\omega^{1}_{t0}=\frac{f^{\prime}}{2\sqrt{gf}}\,,\qquad\omega^{2}_{\theta
1}=-\omega^{1}_{\theta 2}=1+\frac{rg^{\prime}}{2g}\,,\qquad\omega^{3}_{\theta
3}=\cot\theta\,,$ $\displaystyle\omega^{3}_{\varphi 1}=-\omega^{1}_{\varphi
3}=\left(1+\frac{rg^{\prime}}{2g}\right)\sin\theta\,,\qquad\omega^{3}_{\varphi
2}=-\omega^{2}_{\varphi 3}=\cos\theta\,,\qquad$ (17)

where the prime denotes derivation with respect to the coordinate $r$.

Without loss of generality, we can investigate the circular motion555In
principle, one could analyze geodesic orbits and/or other kind of closed
trajectories in proximity of the ultra-compact object. However, the circular
motion has been selected to streamline computations, but such a peculiar
choice does not spoil the qualitative behavior of the final result. of the
entangled particles around the ultra-compact object by assuming that the
dynamics takes place on the equatorial plane $\theta=\pi/2$. Additionally, we
let the EPR source be located at $\varphi=0$ and the two observers performing
the local spin measurements at $\pm\varphi$. A sketch of the physical setup is
shown in Fig. 1.

Figure 1: A pair of spin-$1/2$ particles initially sharing a perfect EPR
correlation is produced at $\varphi=0$. Particles travel along a circular
orbit around the ultra-compact object in opposite directions. At the end of
each propagation, the spins are rotated due to the presence of a non-trivial
background spacetime.

Due to the above requirements, the expression of the four-velocity is
simplified, as

$u^{\mu}(x)=\left(\frac{\cosh\zeta}{\sqrt{f}},0,0,\frac{\sinh\zeta}{r\sqrt{g}}\right)\,,$
(18)

where $\zeta$ denotes the rapidity in the local reference frame.

We recall that the motion under investigation is not a geodesic one;
therefore, there must be an external force acting on the system that perfectly
compensates the presence of gravity and prevents the emergence of geodesic
instabilities in proximity of the horizon. Such a force produces the non-
vanishing acceleration

$a^{\mu}=\left(0,\frac{1}{g}\left[\frac{f^{\prime}}{2f}\cosh^{2}\zeta-\left(\frac{1}{r}+\frac{g^{\prime}}{2g}\right)\sinh^{2}\zeta\right],0,0\right)\,.$
(19)

We can now compute the Wigner angle introduced in Eq. (14). First, by means of
simple algebraic manipulations one determines the quantities $\xi^{a}_{b}$ in
Eq. (6), finding the following non-vanishing components:

$\xi^{1}_{0}=\xi^{0}_{1}=-\frac{f^{\prime}}{2f\sqrt{g}}\cosh\zeta\,,\qquad\xi^{1}_{3}=-\xi^{3}_{1}=\left(\frac{2g+rg^{\prime}}{2rg\sqrt{g}}\right)\sinh\zeta\,.$
(20)

By virtue of the above expressions, we can write the infinitesimal LLT (8)
explicitly. Indeed, recalling that $k^{a}=mu^{a}$ and
$u^{a}=e^{a}_{\mu}u^{\mu}$, then
$u^{a}=\left(\cosh\zeta,0,0,\sinh\zeta\right)$. In a similar fashion,
$a^{a}=e^{a}_{\mu}a^{\mu}$, and thus the only non-vanishing component of
$a^{a}$ is $a^{1}=e^{1}_{r}a^{r}=\sqrt{g}a^{r}$. Therefore,

$\lambda^{1}_{0}=\lambda^{0}_{1}=-\frac{1}{r\sqrt{g}}\left[1+\frac{rg^{\prime}}{2g}-\frac{rf^{\prime}}{2f}\right]\cosh\zeta\sinh^{2}\zeta\,,\qquad\lambda^{1}_{3}=-\lambda^{3}_{1}=\frac{1}{r\sqrt{g}}\left[1+\frac{rg^{\prime}}{2g}-\frac{rf^{\prime}}{2f}\right]\cosh^{2}\zeta\sinh\zeta\,.$
(21)

After the evaluation of the infinitesimal LLT, the only quantity left to
compute is the infinitesimal Wigner rotation (14). Because of the choice of
the physical setup summarized in Fig. 1, the only terms of $\vartheta^{a}_{b}$
different from zero are

$\vartheta^{3}_{1}=-\vartheta^{1}_{3}=-\frac{1}{r\sqrt{g}}\left[1+\frac{rg^{\prime}}{2g}-\frac{rf^{\prime}}{2f}\right]\cosh\zeta\sinh\zeta\,.$
(22)

Next, we consider the finite transformation as a Dyson series of infinitesimal
ones ueda , whose formal sum reads

$W^{3}_{1}=T\exp\left[\int^{\tau_{f}}_{\tau_{i}}\vartheta^{3}_{1}\,d\tau^{\prime}\right]=\exp\left[\vartheta^{3}_{1}\left(\tau_{f}-\tau_{i}\right)\right]\,,$
(23)

where $T$ denotes the time ordering operator.

##  IV Gravitationally-induced modulation of Bell nonlocality

According to the setting of our gedanken experiment, the EPR source emits a
pair of particles, $A$ and $B$, moving away from the source in opposite
directions with constant four-momenta $k^{a}_{\pm}=(m\cosh\zeta,0,0,\pm
m\sinh\zeta)$ after having been prepared in the maximally entangled spin
singlet

$|\psi\rangle=\frac{1}{\sqrt{2}}\left(|k^{a}_{+},\uparrow;\varphi=0\rangle_{A}|k^{a}_{-},\downarrow;\varphi=0\rangle_{B}-|k^{a}_{+},\downarrow;\varphi=0\rangle_{A}|k^{a}_{-},\uparrow;\varphi=0\rangle_{B}\right)\,.$
(24)

The CHSH inequality chsh  and the associated CHSH measurements are a powerful
toolbox to access and test the degree of quantum nonlocality in the
correlations between two dichotomous variables; for the problem at hand, such
variables are the spins of the entangled particles.

As a key ingredient, we need two sets of measurements
$\\{\hat{A}_{1},\hat{A}_{2}\\}$ and $\\{\hat{B}_{1},\hat{B}_{2}\\}$ performed
on parties $A$ and $B$, respectively, with the aim of detecting the
orientation of the third component of the spin. If correlations of the spins
in a given shared state are local in the sense of Bell theorem bell ;
bellreview , then the inequality chsh ; bellreview

$\mathcal{S}[|\Psi\rangle]=|\langle\hat{A}_{1}\hat{B}_{1}\rangle+\langle\hat{A}_{1}\hat{B}_{2}\rangle+\langle\hat{A}_{2}\hat{B}_{1}\rangle-\langle\hat{A}_{2}\hat{B}_{2}\rangle|\leq
2\,,$ (25)

holds, where
$\langle\hat{A}_{i}\hat{B}_{j}\rangle=\langle\Psi|\hat{A}_{i}\hat{B}_{j}|\Psi\rangle$.
If Eq. (25) is violated, spin correlations are nonlocal and local hidden-
variable theories are falsified.

Together with the state described in Eq. (24), the employment of the
observables

$\hat{A}_{1}=\hat{\Sigma}_{x}^{(A)}\,,\quad\hat{A}_{2}=\hat{\Sigma}_{y}^{(A)}\,,\quad\hat{B}_{1}=-\frac{\hat{\Sigma}_{x}^{(B)}+\hat{\Sigma}_{y}^{(B)}}{\sqrt{2}}\,,\quad\hat{B}_{2}=-\frac{\hat{\Sigma}_{x}^{(B)}-\hat{\Sigma}_{y}^{(B)}}{\sqrt{2}}\,,$
(26)

allows to reach the maximum violation of the inequality allowed by quantum
mechanics, namely $\mathcal{S}[|\psi\rangle]=2\sqrt{2}$, also known as the
Tsirelson bound tsir .

Now, the maximally entangled initial state evolves in a curved spacetime, and
because of the Wigner rotation the spins of the entangled particles undergo a
precession motion that prevents the perfect EPR correlation of the initial
state from being preserved. Clearly, we expect that whether and how much the
propagation of the particles along a closed path will change the orientation
of the spins and, in turn, the violation of the CHSH inequality, should depend
on the nature of the gravitational object around which the particles are
orbiting.

Now, assume that, after a finite proper time
$\tau_{f}-\tau_{i}={r\sqrt{g}\,\varphi}/\sinh\zeta$, particles $A$ and $B$
have reached their respective detection points; in this proper time interval,
the Wigner transformation can be viewed as a rotation about the 2-axis ueda ;
adami ; our

$\mathbb{W}\left(\pm\varphi\right)=\begin{pmatrix}1&0&0&0\\\
0&\cos\Theta&0&\pm\sin\Theta\\\ 0&0&1&0\\\
0&\mp\sin\Theta&0&\cos\Theta\end{pmatrix}\,.$ (27)

where $\Theta$ can be derived from Eq. (23)

$\Theta=\frac{r\sqrt{g}\,\varphi}{\sinh\zeta}\vartheta^{1}_{3}=\varphi\cosh\zeta\left[1+\frac{rg^{\prime}}{2g}-\frac{rf^{\prime}}{2f}\right]\,.$
(28)

The physical meaning of the rotation angle $\Theta$ and the spin precession
can be readily visualized from Fig. 1.

Having the explicit expression for the Wigner rotation, the transformation
acting on the spin states can be computed as shown in Eq. (13). Specifically,
one can verify that ueda ; adami ; our

$D_{\sigma^{\prime}\sigma}^{(1/2)}=e^{\mp i\frac{\sigma_{y}}{2}\Theta}\,,$
(29)

with $\sigma_{y}$ being the Pauli matrix with imaginary entries.

Crucially, we see that, as the particles progress travelling along the orbit
that circles the ultra-compact object, the initial spin-singlet state gets
embroiled in a linear superposition with the spin-triplet states, which
implies that measurements of the spin along the same direction are no longer
perfectly correlated in the local reference frame for $\pm\varphi$ ueda ;
adami ; our .

In order to preserve perfect correlation in the local reference frame, it is
sufficient to rotate the bases by $\mp\varphi$ while keeping the 2-axis fixed
in the point that is denoted by $\pm\varphi$. In so doing, we obtain

$|k^{a}_{\pm},\uparrow;\pm\varphi\rangle^{\prime}=\cos\frac{\varphi}{2}|k^{a}_{\pm},\uparrow;\pm\varphi\rangle\pm\sin\frac{\varphi}{2}|k^{a}_{\pm},\downarrow;\pm\varphi\rangle\,,$
(30)

and

$|k^{a}_{\pm},\downarrow;\pm\varphi\rangle^{\prime}=\mp\sin\frac{\varphi}{2}|k^{a}_{\pm},\uparrow;\pm\varphi\rangle+\cos\frac{\varphi}{2}|k^{a}_{\pm},\downarrow;\pm\varphi\rangle\,,$
(31)

so that the evolved state reads

$|\psi^{\prime}\rangle=\frac{1}{\sqrt{2}}\Bigl{[}\cos\Delta\Bigl{(}|k^{a}_{+},\uparrow;\varphi\rangle^{\prime}|k^{a}_{-},\downarrow;-\varphi\rangle^{\prime}-|k^{a}_{+},\downarrow;\varphi\rangle^{\prime}|k^{a}_{-},\uparrow;-\varphi\rangle^{\prime}\Bigr{)}+\sin\Delta\Bigl{(}|k^{a}_{+},\uparrow;\varphi\rangle^{\prime}|k^{a}_{-},\uparrow;-\varphi\rangle^{\prime}+|k^{a}_{+},\downarrow;\varphi\rangle^{\prime}|k^{a}_{-},\downarrow;-\varphi\rangle^{\prime}\Bigr{)}\Bigr{]}\,,$
(32)

where

$\Delta=\Theta-\varphi=\varphi\,\Bigl{[}\cosh\zeta\left(1+\frac{rg^{\prime}}{2g}-\frac{rf^{\prime}}{2f}\right)-1\Bigr{]}\,.$
(33)

Before we can evaluate the CHSH inequality (25) for the observables introduced
in Eq. (26), the measurement operators must be rewritten in the new reference
frame obtained as a result of the rotation, that is

$\displaystyle\hat{A}_{1}^{\prime}=\cos{\Theta}\hat{\Sigma}_{x}^{(A)}-\sin{\Theta}\hat{\Sigma}_{z}^{(A)}$
$\displaystyle\,,\quad\hat{A}_{2}^{\prime}=\hat{\Sigma}_{y}^{(A)}\,,$ (34)
$\displaystyle\hat{B}_{1}^{\prime}=-\frac{\cos{\Theta}\left(\hat{\Sigma}_{x}^{(B)}+\hat{\Sigma}_{y}^{(B)}\right)+\sin{\Theta}\hat{\Sigma}_{z}^{(B)}}{\sqrt{2}}$
$\displaystyle\,,\quad\hat{B}_{2}^{\prime}=\frac{\cos{\Theta}\left(\hat{\Sigma}_{x}^{(B)}-\hat{\Sigma}_{y}^{(B)}\right)+\sin{\Theta}\hat{\Sigma}_{z}^{(B)}}{\sqrt{2}}\,.$

Note that, in terms of experimental complexity, the preparation of the
detectors does not require difficult steps. Indeed, as the observers lie in a
locally inertial reference frame, the only required additional action with
respect to a standard CHSH test would consist in rotating the experimental
apparatus so as to match the angular distance spanned by the pair of particles
(that is, $\pm\varphi$). However, this distance is known _a priori_ once the
endpoints (and thus the location of the observers) have been determined on the
circular orbit.

Collecting all the above results, we finally obtain

$\mathcal{S}^{\prime}[|\psi^{\prime}\rangle]=2\sqrt{2}\cos^{2}\Delta\,.$ (35)

We can interpret Eq. (35) as follows: when a CHSH-like experiment is carried
out after the observables have been rotated, the maximal initial violation of
the CHSH inequality $2\sqrt{2}$ becomes modulated by a factor
$\cos^{2}\Delta$.

Inspecting Eq. (33), we see that, in the presence of the gravitational
interaction, the phase shift parameter $\Delta$ responsible for the overall
violation of the CHSH inequality acquires contributions that depend on the
details of the spacetime in which the entangled particles propagate.
Therefore, depending on the actual nature of the ultra-compact gravitational
source being considered, we expect to find distinct and possibly significantly
different modulations in the violation of the CHSH inequality. A comment is in
order here: since the action of the Wigner rotation on the initial state is
but a local unitary map, the total degree of nonlocality should not be
influenced by it. As a matter of fact, one can check that, with a suitable
selection of different directions for the observables ueda , the perfect EPR
correlation would be restored even after the propagation of the two spins
along the orbit. Hence, the effect of gravity and of the external acceleration
required to maintain a circular trajectory only amounts to changing the
orientation of the measurements; in other words, nonlocality remains
essentially unaffected.

On a final note, it is worth observing that, as long as the initial state is
prepared in close analogy to the one introduced in Eq. (24) and the ensuing
observables are chosen in such a way to reproduce the maximally allowed degree
of nonlocality, no qualitative deviation from the current analysis is expected
to arise when a different pair of particles with arbitrary spins is used to
study the problem at hand.

##  V Comparing models of ultra-compact objects with Bell nonlocality

The gravitational modulation of the Bell nonlocality derived in the previous
Section, i.e., Eqs. (33) and (35), can be exploited to compare relevant
alternative models of ultra-compact structures, such as the string fuzzball
and the black hole (classical or with perturbative quantum corrections). We
would like to stress again that these equations are completely general, and
can thus be applied to an arbitrary black hole-like solution whose metric can
be cast in the form (15).

###  V.1 String fuzzballs

Within string-theory inspired cosmology, fuzzballs fuzz1 ; fuzz2 ; rew  are
spheres of strings of definite, finite volume that simulate the behavior of
black holes, but having the two main problems plaguing the latter (the
singularity and the information paradox) removed by the finite length
extension of their microscopic components.

A concrete fuzzball solution amenable to quantitative investigation is
obtained from $\mathcal{N}=2$ four-dimensional supergravity, with a non-
minimal coupling between gravity, four $U(1)$ gauge fields and three complex
scalars. This particular case allows for some explicit phenomenological
predictions that might soon be tested via gravitational waves detection by
studying ringdown (gravitational wave peak in merging events), quasi-normal
modes, and spectroscopy bianchi .

In isotropic spherical coordinates, a four-dimensional fuzzball geometry can
be described by the line element bianchi

$ds^{2}=-\frac{dt^{2}}{\sqrt{H_{1}H_{2}H_{3}H_{4}}}+\sqrt{H_{1}H_{2}H_{3}H_{4}}\left[dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}\right]\,,$ (36)

where $H_{A}=1+Q_{A}/r$, $A=1,2,3,4$, with $Q_{A}$ being electric and magnetic
charges. The total mass of the fuzzball is given by
$M=\left(Q_{1}+Q_{2}+Q_{3}+Q_{4}\right)/4$, and one recovers the extremal
Reissner-Nordström black hole solution when all the charges are equal. A
straightforward comparison with Eq. (15) yields the identification

$f(r)=\frac{1}{g(r)}=\frac{1}{\sqrt{H_{1}H_{2}H_{3}H_{4}}}\,.$ (37)

Before concluding, we observe that, for the sake of comparing different
spherically-symmetric scenarios, we consider a metric that does not include
“tidal” effects due to the non-overlapping of the three centers with which
fuzzball multicenter microstate solutions are built in bianchi . Effectively,
this amounts to considering only the metric of the extremal black hole
solution. However, this approximation becomes more precise the more massive
the compact object is, since the impact of tidal forces on nearby test masses
scales as the inverse of the mass spag . Therefore, by focusing on
supermassive gravitational sources, we could actually regard such effects as
negligible from a physically sound perspective.

###  V.2 Classical and quantum Schwarzschild black holes

Black holes can be investigated both in a general-relativistic classical
context as well as in a quantum-corrected one. The last instance occurs when
one considers gravity as an effective field theory, so that quantum
gravitational radiative corrections influence the energy-momentum tensor
appearing in Einstein equations. In turn, such a modification gives rise to
long-range corrections appearing in the expression of the metric tensor
$g_{\mu\nu}$. In general, the magnitude of such corrections is extremely small
and can be neglected, but in the proximity of a black hole they actively
affect the metric and the ensuing gravitational phenomenology. Therefore, we
can investigate the implications of the CHSH experiment in the strong-gravity
regime, both in the classical and in the quantum-corrected framework.

Figure 2: The gravitational potential of an ultra-compact object as a function
of the distance (radius) from the object center in units of the Schwarzschild
radius for different spacetimes. Sample values are fixed at $M=2.5$,
$Q_{1}=1$, $Q_{2}=2$, $Q_{3}=3$, and $Q_{4}=4$.

Specifically, we are interested in the extrapolation of the Schwarzschild-like
solution inclusive of quantum corrections in the isotropic coordinate system
don . The line element associated with the quantum-corrected spacetime reads

$ds^{2}=-\left[\frac{1-\frac{M}{2r}+\frac{31}{30\pi}\frac{M}{r^{3}}}{1+\frac{M}{2r}-\frac{31}{30\pi}\frac{M}{r^{3}}}\right]^{2}dt^{2}+\left(1+\frac{M}{2r}-\frac{7}{30\pi}\frac{M}{r^{3}}\right)^{4}\left[dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\varphi^{2}\right]\,,$ (38)

where $M$ is the mass of the black hole. The standard Schwarzschild solution
is recovered when the additive corrections that depend on $1/r^{3}$ in Eq.
(38) are removed. By comparing the metric of Eq. (38) with the one of Eq.
(15), the following identification holds:

$f(r)=\left[\frac{1-\frac{M}{2r}+\frac{31}{30\pi}\frac{M}{r^{3}}}{1+\frac{M}{2r}-\frac{31}{30\pi}\frac{M}{r^{3}}}\right]^{2}\,,\qquad
g(r)=\left(1+\frac{M}{2r}-\frac{7}{30\pi}\frac{M}{r^{3}}\right)^{4}\,.$ (39)

###  V.3 Comparison

We can now compare the different ultra-compact objects and establish if and
how the fuzzball and black hole solutions differ in their response to the CHSH
quantum nonlocality test.

Figure 3: The magnitude of the gravitational modulation parameter $\Delta$ as
a function of the radius of the circular orbit in units of the Schwarzschild
radius for different spacetimes. Blue solid line: $\Delta_{SF}$, Eq. (43),
corresponding to the string fuzzball solution. Red solid line: $\Delta_{QS}$,
Eq. (41), corresponding to the quantum-corrected Schwarzschild solution. Green
solid line: $\Delta_{SF}$, Eq. (40), corresponding to the classical
Schwarzschild solution. Sample values are fixed at $\varphi=\pi/4$, $M=2.5$,
$Q_{1}=1$, $Q_{2}=2$, $Q_{3}=3$, $Q_{4}=4$ and $\zeta=10$.

Firstly, we observe that the two classes of objects already differ at the
classical level in the behavior of the gravitational potential in regions
sufficiently close to the event horizon, as illustrated in Fig. 2. Next, in
order to estimate quantitatively the distinct predictions in the quantum
regime, we need to evaluate the gravitational modulation parameter of Bell
nonlocality $\Delta$, i.e., Eq. (33), in each case. Since we have specialized
the general line element (15) to the instances (36) and (38), we are left with
the task of taking advantage of the expressions of $f(r)$ and $g(r)$ appearing
in Eqs. (37) and (39) and derive the explicit form of $\Delta$.

Figure 4: CHSH correlation as a function of the orbit radius in units of the
Schwarzschild radius for different spacetimes. Blue solid line: string
fuzzball solution. Red solid line: quantum-corrected Schwarzschild solution.
Green solid line: classical Schwarzschild solution. Sample values are fixed at
$\varphi=\pi/4$, $M=2.5$, $Q_{1}=1$, $Q_{2}=2$, $Q_{3}=3$, $Q_{4}=4$ and
$\zeta=10$.

As a preliminary step, we determine the form $\Delta_{CS}$ of the parameter
$\Delta$ holding for the isotropic, classical standard Schwarzschild solution,
that is

$\Delta_{CS}=\varphi\left\\{\frac{\cosh\zeta}{2}\frac{M^{2}+10Mr-8r^{2}}{\left(M-2r\right)\left(M+2r\right)}-1\right\\}\,.$
(40)

Accounting for the quantum perturbative corrections appearing in Eq. (38), one
can derive the form $\Delta_{QS}$ of the parameter $\Delta$ holding for the
quantum-corrected Schwarzschild solution at leading order

$\Delta_{QS}=\varphi\left\\{\frac{\cosh\zeta}{2}\frac{\beta(r)}{\left[30\pi
r^{3}+M\left(31-15\pi r^{2}\right)\right]\left[30\pi r^{3}-M\left(31-15\pi
r^{2}\right)\right]\left[30\pi r^{3}-M\left(7-15\pi
r^{2}\right)\right]}-1\right\\}\,,$ (41)

where

$\beta(r)=54000\pi^{3}r^{9}-M^{3}\left(31-15\pi r^{2}\right)^{2}\left(79+15\pi
r^{2}\right)+900M\pi^{2}r^{6}\left(451-45\pi r^{2}\right)-60M^{2}\pi
r^{3}\left(2263-3930\pi r^{2}+675\pi^{2}r^{4}\right)\,.$ (42)

Finally, making use of the potentials $f(r)$ and $g(r)$ in Eq. (37), we obtain
the form $\Delta_{SF}$ of the parameter $\Delta$ holding for the string
fuzzball solution

$\Delta_{SF}=\varphi\left\\{\frac{\cosh\zeta}{2}\frac{r^{3}\left(Q_{3}+Q_{4}+2r\right)+Q_{2}\left(r^{2}-Q_{3}Q_{4}\right)r-Q_{1}\left[Q_{3}Q_{4}r-r^{3}+Q_{2}\left(2Q_{3}Q_{4}+Q_{3}r+Q_{4}r\right)\right]}{(Q_{1}+r)(Q_{2}+r)(Q_{3}+r)(Q_{4}+r)}-1\right\\}\,.$
(43)

The three expressions $\Delta_{CS}$, $\Delta_{QS}$ and $\Delta_{SF}$ and the
ensuing modulations differ significantly when evaluated along orbits
sufficiently close to the ultra-compact objects, as illustrated in Fig. 3 and
Fig. 4, respectively. We see that, in the strong-gravity regime, the specific
nature of the gravitational source affects dramatically the degree of
violation of the CHSH inequality; hence, the gravitational modulation of Bell
nonlocality allows to distinguish different models of ultra-compact objects
and to discriminate between the string theory and quantum field theory
approaches to quantum gravity.

In particular, in Fig. 4 we plot the oscillatory modulation of Bell
nonlocality as measured by the degree of violation of the CHSH inequality for
orbits close to the event horizon, that is, for an interval of the radial
coordinate that does not exceedingly deviate from the Schwarzschild radius.
For a wider range of values of the radial coordinate, the frequency of the
oscillations $\cos^{2}\Delta$ grows very rapidly, thereby blurring the
interpolation patterns. Moreover, as the quantities $\Delta_{S}$,
$\Delta_{QS}$ and $\Delta_{F}$ share the same behavior in the limit $r\gg 2M$,
the phase shifts of the CHSH correlations can no longer be resolved in this
regime.

In connection with the above reasoning, it is important to recall that, by
selecting a given circular trajectory, we are essentially fixing the value of
the radius. Therefore, the figures and the modulating parameters do not really
have to be interpreted as functions of $r$, because once its value is chosen
they cannot vary with the particles’ dynamical evolution. Hence, the
evaluation of the differences between the predictions of the distinct models
has to be intended for a fixed value of the radius. For instance, by looking
at Fig. 4, it is immediate to verify that, for certain values of $r$ (_i.e._ ,
for certain circular orbits), the magnitude of $\mathcal{S}^{\prime}$ is
exactly zero according to some models, thereby signaling a complete absence of
correlations between the two spins. For these exact values, instead, the other
models predict a non-vanishing degree of CHSH correlations, thus entailing
that the outcome of the gedanken experiment is unambiguously distinguishable.

##  VI Discussion

We have discussed a gedanken experiment that shows how quantum nonlocality in
strong gravitational fields leads to predictions that discriminate between
different models of quantum gravity phenomenology, both perturbative and non-
perturbative ones. CHSH nonlocality tests with entangled particle pairs on
circular orbits near ultra-compact objects show that the spin precession
occurring in curved spacetime is responsible for a modulation of the degree of
quantum nonlocality. In the presence of a non-trivial spacetime background,
the standard maximally allowed violation $2\sqrt{2}$ of the CHSH inequality
becomes $2\sqrt{2}\cos^{2}\Delta$, with the angular modulation factor $\Delta$
strictly dependent on the metric tensor components and heavily influenced by
the conjectured underlying nature of the ultra-compact object considered. A
simple measurement of quantum nonlocality can thus be used to validate or
falsify some phenomenological models of quantum gravitational effects in the
strong-field regime.

The thought experiment we have conceived provides further evidence supporting
the use of quantum information concepts like entanglement and Bell nonlocality
as a key tool in the investigation of yet hypothetical quantum gravitational
phenomena. In deriving the main result of our work, we have also generalized
the formalism of Refs. ueda ; our  on the study of quantum correlations in
gravitational fields, so as to make it applicable in general, beyond the
Schwarzschild solution and weak-field limit in the isotropic coordinate
system.

Concerning possible experimental tests probing the quantum nature of the
gravitational field, while high-energy scattering processes are still too far
from the experimental scales needed to detect coexisting quantum and
gravitational effects, it is a largely shared belief that tiny signatures of
such a coexistence might still be revealed with currently available means
through satellite experiments, gravitational wave analysis or tabletop
laboratory tests centered around foundational aspects of quantum mechanics. In
this respect, it is worth remarking that credited proposals which aim at
collecting a direct measurement of quantum gravity phenomenology are
essentially based either upon decoherence/gravity-based wave function collapse
models d1 ; d2 ; d3 ; d4 ; d5 ; d6 ; d7 ; d8 ; d9 ; d10  or, as already
mentioned, the detection of gravitationally-induced quantum correlations by
quantum gravitational mediators Bose ; Marletto ; Schmole ; Carney ;
Pedernales ; Weiss ; Cosco ; Christodoulou ; Bosso . The hard challenge facing
the experimental implementation of these table-top laboratory tests is that of
realizing correlated and delocalized superpositions of large enough masses. So
far, spatial superpositions have been observed with masses at most of the
order of $10^{-23}$ Kg (large molecules) Fein , while the faintest
gravitational field that can be currently measured is the one generated by
masses of the order of $10^{-4}$ Kg Westphal .

While one may hope to significantly improve these numbers by considering some
amplification mechanisms, table-top probing of gravitational effects on Bell
inequality might turn out to be significantly less challenging and might open
the way to laboratory simulations of extreme cosmological conditions suitable
for the verification of quantum-gravity-induced modulations of quantum
nonlocality as described in the present work. In this respect, a particularly
promising avenue might involve designing experimental tests of the CHSH
inequality near the horizon of sonic and optical analogues of black holes
Visser ; Unruh ; Philbin ; Kolobov .

On a final note, our findings strongly suggest that resorting to the entire
spectrum of quantum resources, from nonlocality, entanglement and steering to
discord, coherence and complementarity may provide very useful insights in the
investigation on the actual nature of gravity.

## Acknowledgements

The authors acknowledge support by MUR (Ministero dell’Università e della
Ricerca) via the project PRIN 2017 “Taming complexity via QUantum Strategies:
a Hybrid Integrated Photonic approach” (QUSHIP) Id. 2017SRNBRK. L. P.
acknowledges networking support by the COST Action CA18108 and financial
support from the “Angelo Della Riccia” Foundation. The authors would like to
thank the anonymous reviewers for their suggestions and comments which helped
to improve the quality of the manuscript.

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