# Simulating spacetime with indefinite causal order via Rindler observers

Aleksandra Dimić  Faculty of Physics, University of Belgrade, Studentski Trg
12-16, 11000 Belgrade, Serbia<EMAIL_ADDRESS>Marko Milivojević
Faculty of Physics, University of Belgrade, Studentski Trg 12-16, 11000
Belgrade, Serbia<EMAIL_ADDRESS>Dragoljub Gočanin  Faculty of
Physics, University of Belgrade, Studentski Trg 12-16, 11000 Belgrade, Serbia
<EMAIL_ADDRESS>Časlav Brukner  Vienna Center for Quantum Science and
Technology (VCQ), University of Vienna, Faculty of Physics, Boltzmanngasse 5,
A-1090 Vienna, Austria  Institute for Quantum Optics and Quantum Information
(IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna,
Austria<EMAIL_ADDRESS>
###### Abstract

Realization of indefinite causal order, a theoretical possibility that even
causal relations between physical events can be subjected to quantum
superposition, apart from its general significance for the fundamental physics
research, would also enable quantum information processing that outperforms
protocols in which the underlying causal structure is definite. In this paper,
we propose a way to simulate specific spacetime with indefinite metric
structure by exploiting the equivalence between stationary observers sitting
in the vicinity of the event horizon of a Schwarzschild black hole and Rindler
observers in Minkowski space. Namely, by putting a Rindler observer, who
resides in causally definite Minkowski background, in a state of quantum
superposition of having two different values of proper acceleration, we can
simulate the experience of a stationary observer in gravitational field with
indefinite metric generated by a Schwarzschild black hole in a state of
quantum superposition of being at two different spatial locations with respect
to the observer. In this manner, a pair of entangled Rindler observers can be
used to simulate quantum communication protocols such as gravitational quantum
switch or the violation of Bell’s inequality for temporal order. We also
discuss the possibility of experimental realization by means of optomechanical
resonators.

##  1 Introduction

The principle of causality lays in the core of every physical theory and,
depending on the context, it has various interpretations. From an operational
point of view, causality can be understood as signaling/communication
relations between physical observers (systems, in general), an information
flow whose properties are intimately related to the nature of space and time,
the notions of which have evolved through several stages. In the old Newtonian
picture, space and time are two generically different entities, universal for
all observers. There is a single flat Euclidean space and a single global time
that enables us to universally distinguish between past, present and future.
Together, they constitute an absolute, independent background structure
relative to which every physical event takes place. Signals can propagate in
space with unlimited speed (action at a distance) and, consequently, each
event can be caused by any other in its present or past. Einstein’s theory of
special relativity (SR) changed this paradigm: space and time became united
into a four dimensional spacetime continuum - the Minkowski space - in which
signals cannot travel faster than the speed of light, enforcing them to stay
inside, or on the local light cone. Nevertheless, the structure of Minkowski
space adhered the character of an independent, fixed background on which
dynamical matter fields propagate. The radical change came with Einstein’s
theory of general relativity (GR). The dynamics of gravitational field, that
is, spacetime itself according to GR, is not given a priori, since it is
coupled to the dynamics of matter fields. There is no prescribed, independent
metric structure, no absolute background stage relative to which locations of
physical events are to be defined, there are just dynamical fields, spacetime
being one of them, and physical events are only located relative to one
another. Every event has its own past and future, a class of events from which
it can receive information and a class of events to which it can send
information and the possibility of communication between different observers
is completely determined by dynamical configuration of light cones. While in
flat Minkowski space all light cones have the same slope, in curved spacetimes
of GR, they can be tilted relative to each other, according to the
distribution of matter. For a given observer in a spacetime with definite
causal order, that is, an observer in a gravitational field with a definite
metric structure, the causal relations between physical events (as they appear
to the observer) are uniquely determined. Next logical question would be: can
spacetime have indefinite causal order, that is, can it be in a state in which
a particular observer experiences quantumly superposed metric structure?

It is generally expected that the unification of quantum mechanics (QM) and
gravitational physics will provide us with some deeper insights concerning the
nature of space and time at the microscopic (Planck) scale. That is the main
subject of quantum gravity. However, the standard methods of quantization of
matter fields employed in Quantum Field Theory (QFT) do not seem to work for
Einstein gravity; it holds a status of a non-renormalizabile theory with
undetermined high-energy degrees of freedom. In order to transcend the
traditional concepts established within GR and QFT, various radical approaches
of “quantizing” gravity are proposed so far, stemming from String Theory,
Quantum Loop Gravity, Noncommutative Field Theory, etc. Up to date, there has
been no empirical evidence that would support or disprove any of the proposed
“high-energy theories”. This state of affairs motivates us to take an
_operational point of view_ and reconsider in which sense and to what extent
can the fundamental principles of QM be applied to gravity while adhering the
tenets of GR [1, 2]. From a broader perspective, our goal is to find ways to
“lure out”, in laboratory conditions, effects that would distinctly
characterize gravity to have quantum features. An important question that
arises out of these considerations is whether it is possible to impose the
principle of quantum superposition upon the causal structure of spacetime,
that is, can we have a quantum superposition of two (or more) _macroscopically
distinct_ metric structures?

In the work of Oreshkov, Costa and Brukner [3], it was found that it is
possible to formulate quantum mechanics without any reference to a global
causal structure. The resulting framework - the process matrix formalism \-
allows for processes incompatible with any definite order between operations
performed on quantum systems. These indefinite causal structures are shown to
be advantageous for quantum computing [4, 5] and quantum communication [6, 7,
8]. One particular example that has experimental demonstration is “quantum
switch” [4, 9, 10, 11, 12, 13], where the main idea is to use an auxiliary
quantum system which can coherently control the order in which certain
operations are applied. In the case of the so called gravitational quantum
switch (GQS) [14] the role of the control system is played by a gravitating
object prepared in a state of quantum superposition of being at two different
spatial locations. Due to entanglement with the gravitating object, the
spacetime itself is expected to be in a state of quantum superposition of
having two macroscopically distinct metrics generated by the gravitating
object.

Here we propose a method for simulating some specific indefinite causal
structures, potentially in laboratory conditions, by utilizing the equivalence
between stationary observers in the vicinity of the event horizon of a
Schwarzschild black hole and Rindler observers in Minkowski space [15]. It is
based on the generalization of Einstein’s equivalence principle to spacetimes
with indefinite metric structure by which we claim that _quantum superposition
of two macroscopically distinct metric structures of spacetime is locally
equivalent to a “quantum reference frame”_ [16] _in flat spacetime with two
superposed proper accelerations_. This allows us to understand the experience
of an observer sitting in indefinite gravitational field, in particular, in
two distinct “superposed Schwarzschild metrics”, in terms of a Rindler
observer in flat Minkowski space in the state of superposition of having two
different values of its proper accelerations. We present a Rindler-version of
GQS and the protocol for violation of Bell’s inequality for temporal order
that involves three Rindler observers. Finally, we discuss the possibility of
experimental realization of these Rindler-protocols by the means of
optomechanical oscillators [17, 18, 19, 20].

##  2 Rindler observers

Consider an arbitrary inertial observer in $(1+1)$ \- dimensional Minkowski
space, and a light cone of a single event of its worldline. This observer
defines global time $t$ running along its worldline. With respect to it, we
introduce an observer that has constant proper acceleration of magnitude
$\alpha$ in the $x$-direction, called the _Rindler observer_. The metric of
the $(1+1)$ \- dimensional Minkowski space covered by globally inertial
coordinates $x^{\mu}=(t,x)$ is given by $ds^{2}=-dt^{2}+dx^{2}$, where we set
$c=1$, implying the same dimensions of space and time. The worldline of a
Rindler observer, parameterized by its proper time $\tau$, is given by the
parametric equations:

$t(\tau)=\frac{1}{\alpha}\sinh(\alpha\tau),\;\;x(\tau)=\pm\frac{1}{\alpha}\cosh(\alpha\tau).$
(1)

Thus, the shape of the Rindler observer’s worldline is a hyperbola,
$t^{2}(\tau)-x^{2}(\tau)=-1/\alpha^{2}$, with branches embedded in the space-
like separated wedges of the above mentioned light cone, called the left (L)
and the right (R) Rindler wedge (see Fig. 1 (left panel)). Rindler observer
with larger proper acceleration has more curved worldline. The structure of
light cones in Minkowski space is such that Rindler observers in R-wedge can
only witness the events from regions R and P, and so, the null surface $t=x$
acts as an event horizon for these observers. Regions L and R are causally
disconnected from each other, meaning that Rindler observers in L-wedge cannot
communicate with Rindler observers in R-wedge.

Consider now two Rindler observers in the R-wedge, with different proper
accelerations $\alpha_{1}$ and $\alpha_{2}$. Let the second one be more curved
than the first one, that is, let $\alpha_{2}>\alpha_{1}$. A photon sent to the
left from the source $S$, with spacetime coordinates $t_{s}=0$ and
$x_{s}=x_{0}>0$, intersects worldlines of these Rindler observers at proper
times $\tau_{1}$ and $\tau_{2}$, respectively (see Fig. 1 (right panel)). At
$t=0$ both observers are closer to the origin than S. This implies that
$\alpha_{2}x_{0}>\alpha_{1}x_{0}>1$. This configuration has an interesting
feature that will turn out to be important. Namely, given the values of
$x_{0}$ and $\alpha_{1}$, there exists a unique value for $\alpha_{2}$,
defined as the non trivial solution ($\alpha_{2}\neq\alpha_{1}$) of the
equation

$\alpha_{2}x_{0}=(\alpha_{1}x_{0})^{\frac{\alpha_{2}}{\alpha_{1}}},$ (2)

for which $\tau_{1}=\tau_{2}$ (for details, see Appendix A).

Figure 1: Rindler observers. (Left panel): _Hyperbolic worldlines of left and
right Rindler observers in Minkowski space._ Patches L and R, called the left
and the right Rindler wedge, respectively, are causally disconnected. This
feature disables left and right Rindler observer to communicate with each
other. (Right panel): _Photon’s worldline intersects the Rindlers._ Two
Rindler observers in R-wedge with different proper accelerations $\alpha_{1}$
and $\alpha_{2}$, $\alpha_{1}<\alpha_{2}$. A photon sent from the point-like
source $S$, with spacetime coordinates $t_{s}=0$ and $x_{s}=x_{0}>0$,
intersects worldlines of these Rindler observers at proper times $\tau_{1}$
and $\tau_{2}$, respectively.

##  3 Indefinite causal order via Rindler observers

Imagine that we have a simple system involving a Schwarzschild black
hole111Having in mind that we are going to make a connection to Rindler
observers in Minkowski space, it is more suitable to talk about a
Schwarzschild black hole rather than some ordinary spherically symmetric
gravitating object, e.g. a planet, because of the importance of an event
horizon., and a stationary observer sitting in his/hers isolated laboratory
that is well enough localized and has negligible effect on the gravitational
field. We do not assume any fixed background spacetime to which we could refer
to and define the locations of objects. Black hole and the observer are
located relative to each other. Now suppose that the black hole and the
observer are in a state of quantum superposition of being at two different
relative distances from each other, where by relative distance we mean the
physical proper distance between the observer and the black hole’s horizon,
the length of a stationary observer’s meter stick if he/she would try to touch
the horizon with it.

The observer would “feel” that he/she resides in a gravitational field with
indefinite metric structure, a kind of a “quantum spacetime” the nature of
which we want to comprehend. From an operational viewpoint, one could
equivalently say that the observer is in a state of superposition of being at
two different locations in spacetime with definite Schwarzschild metric. In
the latter case, having a definite spacetime structure, we can introduce the
standard Schwarzschild coordinates $(t,r,\theta,\phi)$ outside the horizon,
The metric is given by:

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}_{2},$ (3)

with $f(r)=1-\frac{R_{S}}{r}$ and
$\Omega_{2}^{2}=d\theta^{2}+\sin^{2}{\theta}d\phi^{2}$, the metric of a
$2$-sphere $S^{2}$. The radial distance between the stationary observer
sitting at $r_{lab}$ and the event horizon at $R_{S}$ is

$\rho=\int_{R_{S}}^{r_{lab}}\frac{dr}{f(r)}.$ (4)

To be in “two superposed Schwarzschild gravitational fields” is effectively
the same as being in a state of superposition of having two different radial
distances from the horizon of a Schwarzschild black hole. In general, for
every stationary $(r=\rm{const}.)$ observer in a gravitational field of a
Schwarzschild black hole there is an equivalent Rindler observer in Minkowski
space and vice versa (according to the Einstein’s principle of equivalence
gravitational field is _locally equivalent_ to an accelerating reference frame
in flat spacetime). Even more natural correspondence holds if a stationary
Schwarzschild observer is close to the horizon, because spacetime metric in
the vicinity of the horizon reduces to the metric of Minkowski space in
Rindler coordinates. In that case, if the Schwarzschild observer’s proper
distance from the horizon is $\rho$, then its corresponding Rindler observer’s
distance from the origin will also be $\rho$. These two observers have the
same proper acceleration, inversely proportional to $\rho$. For Rindler
observer this relation between its proper acceleration and its distance from
the origin always holds, but for a stationary Schwarzschild observer it holds
only in the vicinity of black hole’s horizon, and so, this condition becomes
important for obtaining genuine equivalence (see Appendix B for details).
Hence, we can effectively transcribe the original system (an observer and a
black hole in a state of quantum superposition of being at two different
relative distances from each other) in terms of the corresponding Rindler
observer in Minkowski space having two superposed values of its proper
acceleration. This entails the claim that Einstein’s equivalence principle
holds even in spacetime with indefinite metric structure, in other words, that
quantum superposition of two macroscopically distinct metric structures of
spacetime is _locally equivalent_ to a “quantum reference frame” in flat
spacetime with two superposed proper accelerations. It is an extension of
Einstein’s equivalence principle that assumes its compatibility with the
linearity of quantum mechanics applied to spacetime.

Figure 2: Possible configurations involving two stationary observers, $A_{S}$
and $B_{S}$, and a Schwarzschild black hole that is in a state of
superposition of being at two different locations relative to them, and the
corresponding configurations that involve entangled Rindler observers in
Minkowski space.

Let us now take two stationary observers, Schwarzschild-Amber $(A_{S})$ and
Scwarzchild-Blue $(B_{S})$222Note that amber and blue are, conveniently, the
actual names of the colors designating the two observers., sitting in their
isolated laboratories, and a Schwarzschild black hole that is in a state of
superposition of being at two different locations with respect to them.
$A_{S}$ and $B_{S}$ then reside in a spacetime with indefinite metric
structure. These two observers correspond to the pair of Rindler observers in
Minkowski space, Rindler-Amber $(A_{R})$ and Rindler-Blue $(B_{R})$, with
entangled proper accelerations. By examining all possible configurations of
this system, we conclude that there are four nonequivalent cases (see Fig. 2).
For example, the first configuration corresponds to the superposition of a
state in which communication between observers $A_{S}$ ($A_{R}$) and $B_{S}$
($B_{R}$) is impossible due to the presence of the horizon, and the other
state in which they can communicate. The correspondence can be extended to the
case of many observers residing in spacetime with indefinite metric structure,
and here we will use it to simulate two simple quantum information protocols
that can be naturally established using indefinite causal structures -
gravitational quantum switch (for this the second configuration will be
relevant) and the protocol for the violation of Bell’s inequality for temporal
order (for which we need three observers).

##  4 Simulating gravitational quantum switch via entangled Rindler observers

The idea that a gravitating object in spatial superposition can induce a
superposition of two gravitational fields dates back to Feynman [21] and it
was promoted, for example, in [14, 22, 23, 24]. Most importantly for this
work, it was employed in [14] as a way of obtaining gravitational quantum
switch. Basically, a gravitating object is prepared in a state of quantum
superposition of being at two different spatial locations, thus producing, due
to its entanglement with the gravitational field, a spacetime with indefinite
metric structure. This opens a possibility of defining a communication
protocol in which one can obtain a superposition of temporal order for two
operationally defined physical events. The state of the gravitating object
plays the role of a quantum control for the order of these events (see [14]
for the complete account). We will now present a somewhat simpler version of
gravitational quantum switch that can be more easily transcribed in terms of
Rindler observers. The setup is illustrated in Fig. $3$. It involves two
observers, $A_{S}$ and $B_{S}$, sitting in their isolated laboratories, a
photon source $S$, and a Schwarzschild black hole that is in the state of
superposition of having two different locations relative to the observers. In
the first case the black hole is closer to $A_{S}$, state $|L\rangle$, and in
the other it is closer to $B_{S}$, state $|R\rangle$. In both superposed
states, $A_{S}$ and $B_{S}$ lay on the same radial ray, they are sitting at
some fixed (but different) distance from the horizon, and have the same
distance between each other. We can think of this as having two observers,
with fixed relative distance, in a gravitational field with two superposed
macroscopically distinct Schwarzschild metrics. A photon source is connected
to the observers and it can send photons to them. The source is gravity-
sensitive and it is adjusted so that it emits a photon in the polarization
state $|{\Psi}\rangle$ towards the black hole. The position of the black hole
thus plays the role of a quantum control for the whole process. Due to the
fact that time runs slower for the observer closer to the horizon, we can
arrange things so that the photon passes through both laboratories at the same
moment of their local proper time (see Appendix C for details). This is
analogues to the case of two Rindler observers from Section $2$. When the
photon gets inside the laboratory, instantaneously, a unitary transformation,
$U_{A}$ or $U_{B}$, depending on the laboratory, is applied on its
polarization state. The meeting of the photon and the laboratory $A_{S}$ and
instantaneous application of unitary $U_{A}$ is the event a, and likewise, the
meeting of the photon and the laboratory $B_{S}$ and instantaneous application
of unitary $U_{B}$ is the event b.

Figure 3: Gravitational quantum switch. The system involves a photon source
$S$, two stationary observers, $A_{S}$ and $B_{S}$, and a black hole in the
state of superposition of having two different locations relative to them.
Photon source shoots the photon towards the black hole, and so, the position
of the black hole plays the role of a quantum control for the whole process.
The photon is in a superposition of traveling in two opposite directions.

If $A_{S}$ and $B_{S}$ are, in both superposed states, close to the horizon,
than this whole system corresponds to the system involving two Rindler
observers in Minkowski space with _entangled proper accelerations_ (Fig. $4$).
In the Rindler-scenario, worldline of a photon emitted by the source $S$
intersects worldlines of the two Rindler observers $A_{R}$ and $B_{R}$ with
entangled proper accelerations. On the left panel, the proper acceleration of
$A_{R}$ ($\alpha_{1}$) is smaller than the proper acceleration of $B_{R}$
($\alpha_{2}$) and on the right panel it is other way around, $A_{R}$ has
larger proper acceleration ($\alpha_{2}$) and $B_{R}$ has smaller proper
acceleration ($\alpha_{1}$). By choosing a suitable values for the proper
accelerations $\alpha_{1}$ and $\alpha_{2}$, with $\alpha_{1}<\alpha_{2}$,
these meetings occur at the same proper time $\tau^{*}$ (see discussion in
Section $2$). The photon is in the polarization state $|{\Psi}\rangle$ and it
is acted upon by the Rindler observers333By “observer” we mean the internal
dynamical degrees of freedom inside the laboratory, whatever they may be.
according to their “color” degree of freedom, “amber” ($A$) or “blue” ($B$),
that identifies and distinguishes them. Observer $A_{R}$ performs a unitary
transformation $U_{A}$ at $\tau^{*}$ without disturbing the trajectory of the
photon, and observer $B_{R}$ performs a unitary transformation $U_{B}$ at
$\tau^{*}$ also without disturbing the trajectory of the photon. In general,
degrees of freedom of laboratory (its kinematic mode and internal degrees of
freedom) can get entangled with the state of the photon. Internal state of the
laboratories evolves according to the observer’s proper time, since it
accounts for the physical rate of change. Thus, we will take a state
$|{\tau_{\alpha}}\rangle$ of observer’s “clock” and its “ticking” (which
depends on $\alpha$) to be abstractions of its entire actual state and its
evolution, respectively, without getting into details of what are observer’s
actual degrees of freedom and its hamiltonian. We now define event $a$ to be
the meeting of the photon with $A_{R}$ and instantaneous application of the
unitary $U_{A}(\tau^{*})$, and likewise, we define event $b$ to be the meeting
of the photon with $B_{R}$ and instantaneous application of the unitary
$U_{B}(\tau^{*})$.

Figure 4: Quantum switch via Rindler observers. Worldline of a photon emitted
by the source $S$ intersects worldlines of the two Rindler observers $A_{R}$
and $B_{R}$ that have entangled proper accelerations. On the left panel,
$A_{R}$ has smaller proper acceleration ($\alpha_{1}$) than $B_{R}$
($\alpha_{2}$), and on the right panel, $A_{R}$ has larger proper acceleration
($\alpha_{2}$) than $B_{R}$ ($\alpha_{1}$). By choosing a suitable values for
the proper accelerations, these meetings occur at the same proper time
$\tau^{*}$. Rindler observers act upon the photon according to their ”color”
degree of freedom, ”amber” ($A$) or ”blue” ($B$), that identifies them.
Observer $A_{R}$ performs a unitary transformation $U_{A}$ at $\tau^{*}$
without disturbing the trajectory of the photon, and observer $B_{R}$ performs
a unitary transformation $U_{B}$ at $\tau^{*}$ also without disturbing the
trajectory of the photon. When laboratories meet the photon, they
instantaneously come to rest and remain that way until some particular moment
$t_{m}$ of inertial observer $C$’s proper time at which a projective
measurement is performed in order to disentangle the state of the photon from
that of the laboratories. Observer $C$ eventually receives the photon.

It would also be convenient to neutralize the accelerations of Rindler
laboratories at the moment they meet the photon because kinematic state of the
laboratory can also get entangled with the state of the photon444This is
analogues to letting the corresponding Schwarzschild observers $A_{S}$ and
$B_{S}$ to become free falling at the time they meet the radially falling
photon. In this way we can avoid the difficulty of disentangling these
kinematic degrees of freedom from the state of the photon afterwards.
Neutralization can be achieved by instantaneously putting to rest each of the
laboratories when they meet the photon, making them inertial from that point
on.

For the sake of reference, we will update the state of the whole system
(Rindlers $\otimes$ photon) by using the proper time $t$ of inertial observer
$C$ sitting at $x=0$. If the system is prepared at $t=0$ in the composite
state
$|{\tau_{\alpha_{1}}(0),A}\rangle|{\tau_{\alpha_{2}}(0),B}\rangle|{\Psi}\rangle$,
the photon is first sent to the $A_{R}$ (left panel), and transmitted in the
same direction to $B_{R}$. In the other case, when the state of the system is
$|{\tau_{\alpha_{1}}(0),B}\rangle|{\tau_{\alpha_{2}}(0),A}\rangle|{\Psi}\rangle$
(right panel), the signal first gets to $B_{R}$ and then to $A_{R}$. If the
system is prepared in a state of superposition of these two states, at $t=0$
we have

$\displaystyle\frac{1}{\sqrt{2}}\Big{(}$
$\displaystyle|{\tau_{\alpha_{1}}(0),A}\rangle|{\tau_{\alpha_{2}}(0),B}\rangle+|{\tau_{\alpha_{1}}(0),B}\rangle$
$\displaystyle|{\tau_{\alpha_{2}}(0),A}\rangle\Big{)}|{\Psi}\rangle.$ (5)

At $t<t_{1}$ (where $t_{1}$ is the $C$’s time coordinate of the intersection
of the photon’s worldline with the less curved Rindler worldline) the state is

$\displaystyle\frac{1}{\sqrt{2}}\Big{(}$
$\displaystyle|{\tau_{\alpha_{1}}(t),A}\rangle|{\tau_{\alpha_{2}}(t),B}\rangle+|{\tau_{\alpha_{1}}(t),B}\rangle$
$\displaystyle|{\tau_{\alpha_{2}}(t),A}\rangle\Big{)}|{\Psi}\rangle.$ (6)

When the photon passes through the laboratories, unitary transformation
$U_{A}(\tau^{*})$ or $U_{B}(\tau^{*})$ is applied on it, depending on the
laboratory. After the passage of the photon through both laboratories, at some
instant $t$ such that $t>t_{2}$ (where $t_{2}$ is the $C$’s time coordinate of
the intersection of the photon’s worldline with the more curved Rindler
worldline) the state of the whole system is given by

$\displaystyle\frac{1}{\sqrt{2}}\Big{(}$
$\displaystyle|{\tau^{*}+t-t_{1},{A}}\rangle$
$\displaystyle|{\tau^{*}+t-t_{2},{B}}\rangle
U_{B}(\tau^{*})U_{A}(\tau^{*})|{\Psi}\rangle+$ (7)
$\displaystyle|{\tau^{*}+t-t_{1},{B}}\rangle$
$\displaystyle|{\tau^{*}+t-t_{2},{A}}\rangle
U_{A}(\tau^{*})U_{B}(\tau^{*})|{\Psi}\rangle\Big{)},$

where the differences $t-t_{1}$ and $t-t_{2}$ are the time intervals during
which respective Rindler laboratories are at rest relative to $C$. Finally, we
need to disentangle the state of the photon from the internal state of Rindler
laboratories. To this end, at some fixed moment $t_{m}$ of the $C$’s global
time, a projective measurement (postselection of the internal state of the
Rindler laboratories) is performed in the superposition basis
$\\{|{m_{i}}\rangle,|{m_{i}^{\perp}}\rangle|i=1,2\\}$, separately for each
laboratory. The basis states are given by

$\displaystyle|{m_{i}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\Big{(}|{\tau^{*}+t_{m}-t_{i},A}\rangle+|{\tau^{*}+t_{m}-t_{i},B}\rangle\Big{)},$
$\displaystyle|{{m_{i}}^{\perp}}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\Big{(}|{\tau^{*}+t_{m}-t_{i},A}\rangle-|{\tau^{*}+t_{m}-t_{i},B}\rangle\Big{)}.$
(8)

Postselection on any pair of possible measurement results leads to the
following state of the photon

$\frac{1}{\sqrt{2}}(U_{B}(\tau^{*})U_{A}(\tau^{*})\pm
U_{A}(\tau^{*})U_{B}(\tau^{*}))|{\Psi}\rangle.$ (9)

which corresponds to indefinite temporal order of events $a$ and $b$.

##  5 Violation of Bell’s inequality for temporal order

In Ref. [14] it was shown that by using a massive object in a spatial
superposition (and by extension, superposition of spacetime with two different
geometries) as a control system, one can realize events with “entangled
temporal order”. This allows violation of Bell’s inequalities [25, 26, 27, 28,
29, 30, 31] for temporal order. Here we present an alternative realization of
this protocol using Rindler observers. To this end, we consider the following
situation: laboratory $A_{R}$ is in the left Rindler wedge, laboratory $B_{R}$
is in the right Rindler wedge and laboratory $C_{R}$ is in the superposition
of being in right and left Rindler wedge.

Figure 5: Protocol for the violation of Bell’s inequality. In the
“gravitational” scenario, black hole is in a superposition of being between
$A_{S}$ and $C_{S}$ and between $C_{S}$ and $B_{S}$. This corresponds to
Rindler scenario where $C_{R}$ is in a superposition of being in both wedges,
left and right, while $A_{R}$ and $B_{R}$ have swapped magnitudes
accelerations, while staying in the same respective wedge.

This corresponds to the three stationary laboratories $A_{S}$, $B_{S}$ and
$C_{S}$ residing in spacetime generated by a Schwarzschild black hole that is
in a state of superposition of being between $A_{S}$ and $C_{S}$ and between
$C_{S}$ and $B_{S}$ (see Fig. 5). The protocol for violation of Bell’s
inequality goes as follows: two sources $S_{1}$ and $S_{2}$ send two photons
(one photon each), which are initially in the uncorrelated state
$|{\Psi_{L}}\rangle\otimes|{\Psi_{R}}\rangle$, into right and left Rindler
wedge, respectively. When photon meets a Rindler laboratory $X_{R}$ ($A_{R}$,
$B_{R}$ or $C_{R}$), local unitary transformation $U_{X}(\tau^{*})$ is
performed on it and the photon proceeds in the same direction. As in the
previous protocol, after the passage of the photons, the states of the
laboratories are decoupled from the states of the photons. Initial state of
the laboratories and the photons is given by

$\displaystyle\frac{1}{\sqrt{2}}\Big{(}|{\tau_{-\alpha_{2}}^{A}}\rangle|{\tau_{\alpha_{2}}^{C}}\rangle|{\tau_{\alpha_{1}}^{B}}\rangle+|{\tau_{-\alpha_{1}}^{A}}\rangle|{\tau_{-\alpha_{2}}^{C}}\rangle|{\tau_{\alpha_{2}}^{B}}\rangle\Big{)}|{\Psi_{L}}\rangle|{\Psi_{R}}\rangle,$
(10)

where $|{\tau_{\beta}^{\rm lab}}\rangle=|{\tau_{\beta}(0),{\rm lab}}\rangle$,
$\beta\in\\{\alpha_{1},\alpha_{2},-\alpha_{1},-\alpha_{2}\\}$, ${\rm
lab}\in\\{A,B,C\\}$. One can readily check that the joint state of the two
photons, after performing appropriate measurements on the internal degrees of
freedom of the Rindler laboratories555Trace out of the degrees of freedom of
laboratories can be done, in principle, in many ways; one suggestion is given
in the previous section. is

$\frac{1}{\sqrt{2}}(U_{A}(\tau^{*})|{\Psi_{L}}\rangle
U_{C}(\tau^{*})U_{B}(\tau^{*})|{\Psi_{R}}\rangle\pm
U_{C}(\tau^{*})U_{A}(\tau^{*})|{\Psi_{L}}\rangle
U_{B}(\tau^{*})|{\Psi_{R}}\rangle).$ (11)

If we perform unitary transformations such that states
$U_{A}|{\Psi_{L}}\rangle$, $U_{C}U_{B}|{\Psi_{R}}\rangle$ are orthogonal to
$U_{C}U_{A}|{\Psi_{L}}\rangle$, $U_{B}|{\Psi_{R}}\rangle$, respectively, then
the state (11) is maximally entangled. One possible choice of states and
operations is
$|{\Psi_{L}}\rangle=|{+}\rangle=1/\sqrt{2}(|{0}\rangle+|{1}\rangle)$,
$|{\Psi_{R}}\rangle=|{0}\rangle$,
$U_{A}=H=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\ 1&-1\\\
\end{array}\right)$, $U_{B}=\sigma_{z}=\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\
\end{array}\right)$ and $U_{C}=\sigma_{x}=\left(\begin{array}[]{cc}0&1\\\
1&0\\\ \end{array}\right)$. Finally, we can imagine two inertial observers
sitting at $x=0$, for example, that can choose suitable measurements on the
corresponding photons and perform Bell’s test, thus confirming the violation
of Bell’s inequality for temporal order.

##  6 Conclusion

To conclude, we applied an equivalence between stationary observers near the
event horizon of a Schwarzschild black hole and Rindler observers in Minkowski
space to simulate quantum information protocols in gravitational field with
indefinite metric structure. We claim that such gravitational field is locally
equivalent to a quantum non-inertial reference frame in Minkowski space that
has superposed proper acceleration. An important example is gravitational
quantum switch, where one uses a gravitating object in a state of
superposition of being at two different spatial positions as a quantum
control, for which we need two Rindler observers in Minkowski space with
entangled proper accelerations. Likewise, the violation of Bell’s inequality
for temporal order can be simulated by using three Rindler observers. Thus, we
are able to “mimic” the experience of a stationary observer in spacetime with
two superposed Schwarzschild metrics by preparing the corresponding Rindler
observer in a state of superposition of having two different proper
accelerations.

There is a growing effort in demonstrating quantum features of nano-to-
mesoscale optomechanical systems. This may provide a challenging, yet feasible
experimental realizations for the proposed Rindler protocols [17]. Recently,
mesoscopic mechanical resonators were considered as quantum non-inertial
reference frames [18, 19] and entanglement of two massive mechanical
oscillators is achieved [20]. It has been proposed to utilize quantum optical
fields in order to prepare and measure the quantum states of mechanical
resonators, conceivably opening the possibility to quantumly control the
acceleration of such quantum non-inertial reference frames [17].

A potential drawback of the proposed protocols might arise due to the Unruh
effect, that is, the fact that Rindler observer experiences ordinary Minkowski
vacuum as a thermal state. In this context, Rindler observer should be viewed
as Unruh-DeWitt detector [32], which interacts with, for example, a scalar
quantum field in Minkowski space. The temperature detected by the Rindler
observer is related to its proper acceleration $\alpha$ by the relation
$T=\hbar\alpha/2\pi k_{B}c$. The increase of the thermal noise may affect the
final state, such that it can no more be considered as a coherent
superposition, but rather a (convex) classical mixture. However, since in our
scheme we can choose $\alpha$ to be arbitrarily small by tuning the other
parameters, the Unruh effect can always be made negligible.

In future work, it would be interesting to explore further the possibility of
transcribing quantum information protocols in general spacetimes with
indefinite causal order in terms of equivalent quantum non-inertial reference
frames in Minkowski space, with a goal of establishing the full
correspondence. This could be a step towards a better understanding of quantum
nature of spacetime.

Acknowledgments
The authors thank Natália Móller, Dejan Simić, Marko Vojinović, Nikola
Paunković and Ämin Baumeler for helpful comments. A.D. and M.M. acknowledge
support from the project No. ON171035 and D.G. from the project No. ON171031
of Serbian Ministry of Education and Science. Additionally, A.D. acknowledges
support from scholarship awarded from The Austrian Agency for International
Cooperation in Education and Research (OeAD-GmbH). A.D. and D.G. acknowledge
grant FQXi-MGA-1806 that supported their stay in Vienna. A.D. and D.G. would
also like to thank University of Vienna and IQOQI for hospitality during their
stay. Č.B. acknowledges the support of the Austrian Academy of Sciences
through Innovationsfonds Forschung, Wissenschaft und Gesellschaft, Austrian
Science Fund (FWF) through the projects I-2526-N27 and I-2906 and the
University of Vienna through the research platform TURIS. This publication was
made possible through the support of a grant from the John Templeton
Foundation. The opinions expressed in this publication are those of the
authors and do not necessarily reflect the views of the John Templeton
Foundation.
Author Contribution. A.D., M.M., D.G. and Č.B. equally contributed to all
aspects of the research.

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## APPENDIX

##  Appendix A Equal proper time condition for two Rindler observers

Here we give a simple derivation of the relation between proper accelerations
of two Rindler observers that must be satisfied so that a photon’s worldline
intersects worldlines of the Rindler observers at the same moment of their
individual proper time.
From (Fig. A1) we can see that

$x_{0}-x_{i}=t_{i},$ (12)

where $i=\\{1,2\\}$. Given the parametric equations of Rindler observer’s
worldline,

$t_{i}=\frac{1}{\alpha_{i}}\sinh(\alpha_{i}\tau_{i}),\;\;x_{i}=\frac{1}{\alpha_{i}}\cosh({\alpha_{i}\tau_{i}}),$
(13)

we can deduce the instant of the observer’s proper time $\tau_{i}$ in which
he/she receives the signal from the point-source $S$ at $x_{0}>0$. From (12)
and (13) it follows that

$\frac{{\rm e}^{\alpha_{i}\tau_{i}}+{\rm
e}^{-\alpha_{i}\tau_{i}}}{2\alpha_{i}}+\frac{{\rm e}^{\alpha_{i}\tau_{i}}-{\rm
e}^{-\alpha_{i}\tau_{i}}}{2\alpha_{i}}=x_{0},$ (14)

Figure A1: A photon intersecting two Rindler worldlines. Worldline of a photon
sent from a point-like source $S$ intersects worldlines of two Rindler
observers at instances $\tau_{1}$ and $\tau_{2}$ of their respective proper
time. Inertial reference frame coordinates of the intersection points are
denoted by $t_{i}$, $x_{i}$, $i=\\{1,2\\}$. Figure A2: Numerical analysis.
Ratio of accelerations for Rindler observers. If we take that
$X=\alpha_{1}x_{0}$ and $Y=\alpha_{2}x_{0}$ then (16) becomes
$Y=\Phi_{X}(Y)=X^{\frac{Y}{X}}$. We have found three classes of solutions, but
we can regard only one class as relevant if we take an assumption $Y>X$. One
trivial solution in all three cases is $Y=X$. In a), case $X<1$ is
illustrated, with $X=1/2$, where we have only trivial solution. When
$X\in(1,{\rm e})$, we have non-trivial solution, $Y>X$. That case is
represented in b), for $X={\rm e}/2$. Finally, in the case $X>{\rm e}$, we get
two solutions - trivial one and the other for which $Y<X$. That case is
illustrated in c), where $X$ is chosen to be $2{\rm e}$.

and so,

$\tau_{i}=\frac{1}{\alpha_{i}}\ln(\alpha_{i}x_{0}).$ (15)

Note that both the prefactor and the argument of the logarithm are positive.
The condition of equality of the proper times $\tau_{1}$ and $\tau_{2}$ gives
us the following relation between $\alpha_{1}$ and $\alpha_{2}$:

$\alpha_{2}x_{0}=(\alpha_{1}x_{0})^{\frac{\alpha_{2}}{\alpha_{1}}}.$ (16)

By introducing new variables, $X:=\alpha_{1}x_{0}$ and $Y:=\alpha_{2}x_{0}$,
previous equation can be formulated as

$Y=X^{\frac{Y}{X}}.$ (17)

Numerical analysis of (17) shows that solution $Y=X$, which exists for each
value of $X$, is unique for $X<1$ (Fig. A2 (a)). This solution is trivial,
since it corresponds to a single Rindler observer (or two overlapping Rindler
observers with $\alpha_{1}=\alpha_{2}$). If $X>1$, there are two possible
solutions for $Y$, one which is trivial and the other that can be greater or
less than $X$ (Fig. A2 (b,c)). Note that for every value of $x_{0}$ (position
of the source) we have a continuous infinity of pairs of Rindler trajectories
that satisfy (16).

##  Appendix B Schwarzschild metric in the vicinity of a horizon

Worldlines of stationary observers in curved spacetime do not conform to the
geodesics defined by the spacetime metric. To maintain a fixed position they
must oppose their inertia with some proper acceleration. Generally,
$4$-acceleration of an observer in curved spacetime is given by
$a^{\mu}=U^{\nu}\nabla_{\nu}U^{\mu}$, where $U^{\mu}$ is its $4$-velocity, and
its proper acceleration by $a=\sqrt{g_{\mu\nu}a^{\mu}a^{\nu}}$. In particular,
for a stationary observer in Schwarzschild metric

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}_{2},$ (18)

with $f(r)=1-\frac{R_{S}}{r}$ and
$\Omega_{2}^{2}=d\theta^{2}+\sin^{2}{\theta}d\phi^{2}$, the metric of a
$2$-sphere $S^{2}$, we have

$a=\frac{R_{S}}{2r^{2}\sqrt{f(r)}}.$ (19)

Analogously to the proper time $d\tau=\sqrt{f(r)}dt$ of an observer sitting at
$r=const$, we can introduce proper radial distance,

$d\rho=\frac{dr}{\sqrt{f(r)}}.$ (20)

Integrating from $R_{S}$ to $r$ we get the proper radial distance of the
stationary observer at some fixed $r$ from the event horizon,

$\rho=r\left(1-\frac{R_{S}}{r}\right)^{1/2}+\frac{R_{S}}{2}\ln\left[\frac{2r}{R_{S}}-1+\frac{2r}{R_{S}}\left(1-\frac{R_{S}}{r}\right)^{1/2}\right].$
(21)

From (21) it follows that $\rho\sim r$ in the limit
$r/R_{S}\rightarrow+\infty$, and so from (19) we get $a\sim R_{S}/2\rho^{2}$,
which is just the Newtonian inverse square law. Now, let $r=R_{S}+\epsilon$
for some small $\epsilon$. In the limit $\epsilon/R_{S}\rightarrow 0$ we have
$\rho\sim 2\sqrt{R_{S}\epsilon}$, and the proper acceleration of a stationary
observer in the vicinity of event horizon is inversely proportional to its
proper distance from the horizon, $a\sim 1/\rho$. In the intermediate region,
proper acceleration is some very complicated function of observers proper
radial distance.
Schwarzschild metric near the horizon becomes

$ds^{2}=-\rho^{2}d\eta^{2}+d\rho^{2}+R_{S}^{2}d\Omega^{2}_{2}.$ (22)

where we introduced new time coordinate $\eta=\frac{t}{2R_{S}}$. The non-
angular part of the above metric is the metric of $(1+1)$-dimensional
Minkowski space, denoted by $\mathcal{M}_{2}$, in Rindler coordinates. This
becomes evident if we start with the metric of $\mathcal{M}_{2}$ in Minkowski
coordinates $(T,X)$

$ds^{2}_{\mathcal{M}_{2}}=-dT^{2}+dX^{2},$ (23)

and introduce Rindler coordinates $(\rho,\eta)$ by $T=\rho\sinh\eta$ and
$X=\rho\cosh\eta$ in which the above metric takes the form

$ds^{2}_{\mathcal{M}_{2}}=-\rho^{2}d\eta^{2}+d\rho^{2}.$ (24)

Coordinate $\rho$ is time-like and $\eta$ is space-like. Since
$X^{2}-T^{2}=\rho^{2}\geq 0$, coordinates $(\rho,\eta)$ cover only part of
$\mathcal{M}_{2}$ \- the right Rindler wedge. Rindler coordinates
$(\rho,\eta)$ become singular at $\rho=0$ but, using the Minkowski coordinates
$(T,X)$, one could analytically continue them from the right Rindler wedge to
the whole Minkowski space. Similarly, in the case of a Schwarzschild black
hole, we use Kruskal coordinates to make an analytic continuation of
Schwarzschild coordinates $(t,r)$ across the horizon thus obtaining their
maximal extension. The event horizon of a black hole, defined by $\rho=0$ as
seen from (22), corresponds to the light cone $T=\pm X$, and near-horizon
black hole geometry is $Rindler\times S^{2}$.
An observer at $r=const$ $(r\approx R_{S})$ in Schwarzschild metric
corresponds to a uniformly accelerating observer with $\rho=const$ in the
Rindler wedge, i.e. an observer in Minkowski space whose worldline is a
hyperbola $X^{2}-T^{2}=\rho^{2}=const$, whose constant proper acceleration is
given by

$\alpha=\frac{1}{\rho}=\frac{1}{2\sqrt{R_{S}}\sqrt{r-R_{S}}}.$ (25)

Figure C1: Schematic representation of black hole and laboratories $A_{1}$ and
$A_{2}$. Radially falling photon passes through stationary laboratories
$A_{1}$ and $A_{2}$ sitting at two different radial distances from the black
hole. The different ticking rates of local clocks make it possible to arranged
the relative distances so that meetings of the photon with the laboratories
occur at the same moment of their local proper times.

##  Appendix C Equal proper times for free falling photon

Start with the equation for radially free falling photon in Schwarzschild
coordinates:

$\frac{dt}{dr}=-\frac{1}{1-\frac{R_{S}}{r}}=-\frac{r}{r-R_{S}},$ (26)

where $R_{S}$ is the Schwarzschild radius. A radially falling photon, emitted
at $r=R_{0}$, passes through the laboratory $A_{1}$, sitting at $r=r_{1}$, at
the time $t_{1}$. A simple calculation gives us that for $t_{1}$ we have

$t_{1}=R_{0}-r_{1}+R_{S}\ln{\frac{R_{0}-R_{S}}{r_{1}-R_{S}}}.$ (27)

The same photon passes through the laboratory $A_{2}$ sitting at $r=r_{2}$, at
the time $t_{2}$ given by (see Fig. C1)

$t_{2}=R_{0}-r_{2}+R_{S}\ln{\frac{R_{0}-R_{S}}{r_{2}-R_{S}}}.$ (28)

We are looking for the condition for both events to happen at the same local
proper times measured in laboratories $A_{1}$ and $A_{2}$, i.e.

$\tau_{1}=\tau_{2}.$ (29)

Proper times of the laboratories are related to the global time $t$ (proper
time of a stationary observer at infinity) by

$\tau_{i}=\sqrt{1-\frac{R_{S}}{r_{i}}}t_{i}=\sqrt{1-\frac{R_{S}}{r_{i}}}\left(R_{0}-r_{i}+R_{S}\ln{\frac{R_{0}-R_{S}}{r_{i}-R_{S}}}\right).$
(30)

Numerical analysis of the previous equation shows that it is not possible to
obtain the relation (29) for arbitrary values of $R_{0}$ and $R_{S}$. However,
if the ratio $R_{S}/R_{0}<10^{-4}$, two possible solutions, pairs of $r_{1}$
and $r_{2}$ that satisfy the condition of equal proper times, are always
present. It is important to mention that one position, $r_{1}$, is very close
to the black hole, that is, $r_{1}<10R_{S}$, while the other position,
$r_{2}$, can be further away.