# Relativity and decoherence of spacetime superpositions

Joshua Foo<EMAIL_ADDRESS>Centre for Quantum Computation &
Communication Technology, School of Mathematics & Physics, The University of
Queensland, St. Lucia, Queensland, 4072, Australia  Robert B. Mann  Perimeter
Institute, 31 Caroline St., Waterloo, Ontario, N2L 2Y5, Canada  Department of
and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
Magdalena Zych  Centre for Engineered Quantum Systems, School of Mathematics
and Physics, The University of Queensland, St. Lucia, Queensland, 4072,
Australia

###### Abstract

It is univocally anticipated that in a theory of quantum gravity, there exist
quantum superpositions of semiclassical states of spacetime geometry. Such
states could arise for example, from a source mass in a superposition of
spatial configurations. In this paper we introduce a framework for describing
such “quantum superpositions of spacetime states.” We introduce the notion of
the relativity of spacetime superpositions, demonstrating that for states in
which the superposed amplitudes differ by a coordinate transformation, it is
always possible to re-express the scenario in terms of dynamics on a single,
fixed background. Our result unveils an inherent ambiguity in labelling such
superpositions as genuinely quantum-gravitational, which has been done
extensively in the literature, most notably with reference to recent proposals
to test gravitationally-induced entanglement. We apply our framework to the
the above mentioned scenarios looking at gravitationally-induced entanglement,
the problem of decoherence of gravitational sources, and clarify commonly
overlooked assumptions. In the context of decoherence of gravitational
sources, our result implies that the resulting decoherence is not fundamental,
but depends on the existence of external systems that define a relative set of
coordinates through which the notion of spatial superposition obtains physical
meaning.

##  I Introduction

The twin discoveries of quantum theory and Einstein’s theory of general
relativity revolutionized our understanding of physical reality. Perhaps the
most distinctive features of these respective theories are the notions of
quantum superposition and spacetime. The former postulates that a quantum
system with a distinct set of possible configurations may also exist in a
general state featuring many of these configurations “at once.” The latter was
Einstein’s geometric interpretation of gravity as the curvature of a pseudo-
Riemannian manifold, generated by the presence of mass and energy [1].

The most significant challenge for modern physicists is finding a consistent
unification of quantum theory with general relativity. It is univocally
anticipated that any theory combining relativistic gravitation with quantum
mechanics must be able to describe spacetime as possessing quantum-mechanical
degrees of freedom, whose states reside in a complex Hilbert space and may be
placed in quantum superpositions of different configurations. Various formal
approaches such as the Wheeler-deWitt equation [2, 3] and loop quantum gravity
[4, 5, 6] have been developed, which (in principle) allow for such
environments to be constructed [7, 8, 9, 10, 11, 12].

In this article, we focus on the fundamental issue underlying many recent
studies, namely the physical meaning of “superpositions of spacetime metrics”
arising from the spatial superposition of a source mass. Such superposition
states are considered in the proposals for possible observations of
gravitationally-mediated entanglement [13, 14], in the context of decoherence
that they may induce on quantum systems [15, 16, 17, 18, 19, 20], and the
indefinite causal structures they give rise to [21, 22, 23, 24, 25]. Here we
demonstrate that any effects emerging due to a spatial superposition of a
massive object can be formally reproduced using a single spacetime metric, due
to two basic tenets of quantum mechanics and relativity: linearity of quantum
theory and invariance of dynamics under general coordinate transformations in
general relativity [26]. We show how these basic principles combined lead to
the notion of “relativity” of quantum superpositions and to invariance of
probabilities in quantum mechanics under quantum transformations between
coordinates, i.e. between sets of coordinates associated with the system
originally described as in a superposition. This is related to but is
conceptually simpler than and distinct from the approach of quantum reference
frames [27, 28, 29, 30, 31, 32, 33, 34].

Although many claims have been made that such scenarios represent genuine
examples of a quantum-gravitational metric [35, 36, 37, 38, 39, 40] (quantum-
gravitational insofar as such solutions cannot be described using the
frameworks provided by quantum theory and general relativity, minimally
requiring a weak-field formulation of quantum gravity), our result shows that
this is not unambiguously true. Just as for classical configurations, where
only relative distances between objects are of physical significance, the same
is also true for gravitational sources in a superposition of different
configurations, where the configuration states are related by a coordinate
transformation [41].

To understand the impact of the above result, we apply it in three distinct
contexts of recent interest: gravitationally-induced entanglement [35],
decoherence of spatial superpositions of black holes [16, 17], and of dark
matter [18]. We discuss the implications of our findings for the conclusions
that can be drawn from such scenarios and possibilities for lifting the
inherent ambiguities by considering superpositions of non-diffeomorphic
metrics.

##  II Results

###  II.1 Relativity of Quantum Superpositions

Before discussing superpositions of mass configurations sourcing gravity, it
is important to clarify the key concepts relevant to that discussion in the
general context of superpositions of quantum states, without including any
interactions.

General covariance demands that absolute position has no physical meaning
[42]. For example, we may consider a scenario with two particles, one
localized at some position $x_{1}$ and another at some translated position
labeled as $x_{1}+\delta x$. In any translationally invariant theory, this
scenario is equivalent to the scenario in which the first particle is assigned
a position $x_{1}-\delta x$ and the second one is assigned position $x_{1}$.
Any difference between the two scenarios is merely apparent, attributed to a
choice of a coordinate system, which emphasises that only relative distances
are physically meaningful, with coordinate systems playing a subordinate,
rather than fundamental role.

We now outline why this idea, despite intuition, does extend to quantum
theory, which we then present rigorously in the next section in the context of
superpositions of semiclassical states of spacetime.

Consider a scenario with two particles, where the first is assigned a state
$|x\rangle$ corresponding to a position $x$, while the second is in a
superposition of positions $(|x+\delta x_{1}\rangle+|x+\delta
x_{2}\rangle)/\sqrt{2}$. If indeed only relative distances are physically
meaningful in describing the configuration of the particles, then the
composite state above, $|x\rangle_{1}(|x+\delta x_{1}\rangle_{2}+|x+\delta
x_{2}\rangle_{2})/\sqrt{2}$, would be equivalent to a seemingly different
state where the second particle is localized at a position $x$ while the first
is in a superposition of translations, so that the two-particle state is
$(|x-\delta x_{1}\rangle_{1}+|x-\delta x_{2}\rangle_{1})|x\rangle_{2}$. For
each individual amplitude, the two states can be related by a passive
translation, represented as a unitary operator $\hat{U}(\delta x_{i})$ that
enacts a change of classical coordinates labelling the positions, and thus
relabelling the basis states of the particles: $|x\rangle_{1}|x+\delta
x_{i}\rangle_{2}=\hat{U}(\delta x_{i})|x-\delta
x_{i}\rangle_{1}|x\rangle_{2}$. Crucially, a mapping between the superposition
states, denoted simply by $\hat{U}$, also exists:

$\displaystyle\frac{1}{\sqrt{2}}|x_{1}\rangle_{1}(|x_{1}+\delta
x_{1}\rangle_{2}+|x_{1}+\delta x_{2}\rangle_{2})$
$\displaystyle\qquad=\frac{\hat{U}}{\sqrt{2}}(|x_{2}-\delta
x_{1}\rangle_{1}+|x_{2}-\delta x_{2}\rangle_{1})|x_{2}\rangle_{2},$ (1)

where $\hat{U}$ is now an operator acting jointly on both particles. While it
is uncontroversial such an operator $\hat{U}$ exists and can be seen as change
of basis in the two-particle Hilbert space, we argue in the next section that
it can also be seen as a quantum change of coordinates–to coordinates that are
“in a superposition” of different translations relative to the original
coordinate system.

Such an interpretation is consistent with the formal framework provided by
quantum reference frames (QRFs), which establishes transformations between
reference frames defined by physical degrees of freedom (DoFs); however in the
above case we do not need to invoke such additional DoFs. We stress that what
Eq. (II.1) represents is that the choice and labelling of basis states for a
quantum system is conventional, and we recognise that this freedom to relabel
the basis is true for any unitary mapping between different bases, even if no
classical coordinate transformation exists that such a unitary represents. We
sketch the above discussed interpretation of the unitary implied by Eq. (II.1)
as a transformation to coordinates in a “superposition” of translations in
Fig. 1.

Figure 1: Schematic diagram of an example of a relative state from two
different perspectives. In the top subplot, $A$ and $B$ are in a maximally
entangled state of their relative distance, with $C$ some “fixed” reference
coordinate system through which such distances acquire physical meaning. The
bottom subplot is physically equivalent to the top one, where the reference
coordinate system $C$ is now in quantum superposition.

###  II.2 Relativity of Spacetime Superpositions

Here we extend the ideas of the previous section to the case where the masses
in question are general relativistic sources of gravity. The results presented
here depend only on the linearity of quantum theory and the invariance of
physical laws under ordinary coordinate transformations. We first consider a
mass in some classical configuration that gives rise to a classical manifold
$g$ with coordinates parameterised by $x$. We can assign a semiclassical state
to this configuration and the manifold, which can be labelled by
$|g(x)\rangle$. Let us now generalise this to a mass in a superposition of two
classically distinct configurations $|g_{1}(x_{1})\rangle$,
$|g_{2}(x_{2})\rangle$, each of which sources a manifold:

$\displaystyle|\psi\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|g_{1}(x_{1})\rangle+|g_{2}(x_{2})\rangle\right),$
(2)

where $x_{1}$, $x_{2}$ are coordinates associated with the configurations, for
example coordinates centered at the position of the mass (we generalise this
to arbitrary superposition states in Methods). It is helpful here to
conceptualise the mass configuration as being quantum-controlled by an
ancillary system that can be prepared and measured in appropriate states. That
is, for each state of the control there is an associated mass configuration
with a classical manifold and gravitational field, relative to the other
quantum DoFs. This is a key assumption that underpins numerous recent
investigations in the area of spacetime superpositions, including analyses of
gravitationally-induced entanglement proposals [13, 14], spacetime quantum
reference frames [28, 29, 30], and decoherence [16, 17, 18].

As discussed, of increasing recent interest are scenarios in which source
masses are in a simple superposition of two configurations that differ by a
translation and are thus diffeomorphic (related by a coordinate
transformation). Formally, a (global) diffeomorphism between the manifolds
$M$, $M^{\prime}$ is a smooth map $\varphi:M\to M^{\prime}$ with smooth
inverse map $\varphi^{-1}$. Under this definition, the manifolds $M$,
$M^{\prime}$ represent the same geometry and are therefore physically
indistinguishable. In this section we predominantly focus on superpositions of
spacetimes related by a diffeomorphism, returning to general scenarios in the
Discussion.

Returning to Eq. (2), it follows that for the manifolds $g_{1}(x_{1})$,
$g_{2}(x_{2})$ related by a diffeomorphism, the semiclassical states
$|g_{1}(x_{1})\rangle$, $|g_{2}(x_{2})\rangle$ are related by some passive
unitary $|g_{1}(x_{1})\rangle=\hat{U}(\delta x)|g_{2}(x_{2})\rangle$ that
enacts this diffeomorphism, where $\delta x$ could be some potentially
complicated function of the spacetime parameters that characterises the
distance between every point between the two coordinate systems $x$,
$x^{\prime}$. It is possible to rewrite the state as the unitary
$\hat{U}(\delta x)$ that connects the two amplitudes in Eq. (2):

$\displaystyle|\tilde{\psi}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}\hat{I}+\hat{U}(\delta
x)\big{)}|g_{1}(x_{1})\rangle,$ (3)

where we have utilised the notation $|\tilde{\psi}\rangle\equiv|\psi\rangle$
to distinguish the representation of the state of the spacetime from that of
Eq. (2), though as emphasised, the two states are equivalent. This
relationship follows from classical general relativity and the basic tenets of
quantum theory, namely that symmetries can be represented with unitary
operators [43]. We emphasise that the spacetimes are related by a simple
relabelling of, for example, the origin of coordinates, and not an active
translation of the mass through some pre-existing curved spacetime. Such
relabelling of quantum coordinates is consistent with existing frameworks for
quantum reference frames and their more recent extensions to curved spacetime
[31, 32].

Let us now consider some additional quantum degree of freedom (DoF), $\phi$ in
some state $|\phi\rangle$, and a joint initial state of the form,

$\displaystyle|\psi\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|g_{1}(x_{1})\rangle+|g_{2}(x_{2})\rangle\right)|\phi\rangle.$
(4)

The physical system represented by the state $|\phi\rangle$ could be, for
example, a quantum field or a particle in first-quantization, or both these
interacting with each other–we will consider all of these in the applications
of our approach in the next sections and the Methods. The only assumption
behind Eq. (4) is that the state of the matter DoFs is uncorrelated with that
of the source mass (and thus of the spacetime), but apart from this, is
arbitrary. Again, the assumption of an uncorrelated initial state is not
necessary, and indeed we will also consider correlated initial states in the
Sec. II.4.

Now, in quantum theory, the physically meaningful quantities are probability
amplitudes between the states of the system. Let us denote some arbitrary
final state $|\Omega\rangle$ and consider their evolution–including free
dynamics as well as interactions between the systems–denoted by $\hat{U}$. In
general, the time-evolution of the matter DoFs can depend on the state of the
source mass which formally can be represented as
$\hat{U}_{\mathrm{int.}}=\hat{U}_{1}\otimes|g_{1}\rangle\langle
g_{1}|+\hat{U}_{2}\otimes|g_{2}\rangle\langle g_{2}|$, where $\hat{U}_{1}$ and
$\hat{U}_{2}$ individually govern the time-evolution of all the DoFs on the
respective spacetime and we suppress the symbols $x_{1},x_{2}$ denoting the
coordinates associated with a given mass configuration and manifold henceforth
as it does not lead to any ambiguity. This gives the general form of the
probability amplitude,

$\displaystyle\langle\Omega|\hat{U}|\psi\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}\langle\Omega|\hat{U}_{1}|g_{1}\rangle+\langle\Omega|\hat{U}_{2}|g_{2}\rangle\big{)}|\phi\rangle.$
(5)

Recall that we can re-express $|g_{2}\rangle$ in terms of the coordinate
transformation applied to $|g_{1}\rangle$ enacted by the passive unitary
$\hat{U}(\delta x)$. This unitary necessarily acts on the coordinates of all
DoFs, and Eq. (5) can be written as

$\displaystyle\langle\Omega|\hat{U}|\psi\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}\langle\Omega|\hat{U}_{1}|g_{1}\rangle|\phi\rangle$
$\displaystyle+\langle\Omega|\hat{U}_{2}\hat{U}(\delta x)|g_{1}\rangle|\delta
x^{-1}\phi\rangle\big{)},$ (6)

where the notation $|\delta x^{-1}\phi\rangle$ is understood as the inverse
transformation of the coordinates applied to $|\phi\rangle$. Inserting the
identity operator expressed as $\hat{I}=\hat{U}(\delta
x)\hat{U}^{\dagger}(\delta x)$ in the second term gives,

$\displaystyle\langle\Omega|\hat{U}|\psi\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}\langle\Omega|\hat{U}_{1}|g_{1}\rangle|\phi\rangle$
$\displaystyle+\langle\delta x^{-1}\Omega|\hat{U}^{\dagger}(\delta
x)\hat{U}_{2}\hat{U}(\delta x)|g_{1}\rangle|\delta x^{-1}\phi\rangle\big{)}.$
(7)

Importantly, as long as $\hat{U}(\delta x)$ encodes a diffeomorphism,
$\hat{U}^{\dagger}(\delta x)\hat{U}_{2}\hat{U}(\delta x)\equiv\hat{U}_{1}$
(enacting a coordinate relabelling of the unitary) and thus

$\displaystyle\langle\Omega|\hat{U}|\psi\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}\langle\Omega|\hat{U}_{1}|g_{1}\rangle|\phi\rangle$
$\displaystyle+\langle\delta x^{-1}\Omega|\hat{U}_{1}|g_{1}\rangle|\delta
x^{-1}\phi\rangle\big{)}.$ (8)

Equations (5) and (II.2) describe the same scenario–they demonstrate that in
this case, the same dynamics can be interpreted as taking place in space-time
that is in superposition, Eq. (5), or in a “single” spacetime where the matter
DoFs represented by $\phi$ and the measurements encoded by $|\Omega\rangle$
are correlated accordingly, Eq. (II.2).

This is a key point of this article, and one that is commonly overlooked. So-
called “quantum superpositions” of spacetimes can be ambiguous–as in the
example given here–and can be re-expressed in terms of modified initial states
and measurements of the remaining DoFs, but where the state associated with
the source mass, and thus with the spacetime, can be treated as classical.
This equivalence between representations is especially important given the
recent interest in identifying purportedly quantum-gravitational effects
produced by spatial superpositions of a source mass.

As an illustration of this idea, take $\hat{U}(\delta x)$ to be a a simple
translation of the coordinates by some distance $\delta x$. Equation (II.2)
can thus be written as

$\displaystyle\langle\Omega|\hat{U}|\psi\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}\langle\Omega|\hat{U}_{1}|g_{1}\rangle|\phi\rangle$
$\displaystyle\qquad+\langle\Omega-\delta
x|\hat{U}_{1}|g_{1}\rangle|\phi-\delta x\rangle\big{)}.$ (9)

Again, we have here a situation in which all the dynamics and measurements
between the considered DoFs occur on a single “fixed” spacetime background
described by $|g_{1}\rangle$, compared with Eq. (5) which for this case
describes the source mass in a spatial superposition with the distance $\delta
x$ in some coordinates; we have sketched this schematically in Fig. 2.

Figure 2: Illustration of how the same scenario viewed from different
coordinates acquires the interpretation of (a) superposition of spacetimes,
(b) superposition of locations of matter DoFs (e.g. a test particle) with
respect to one mass configuration.

The above discussed notion of the relativity of spacetime superpositions
should be understood as a statement about the invariance of physical
predictions on the choice of coordinates and the related fact that only
relative quantities have physical meaning and relevance (e.g. for the strength
of an interaction)–in the present context this will mostly refer to relative
distances. Recall that a superposition of diffeomorphically related spacetimes
can equivalently be represented via suitably adapted states and measurements
via dynamics occurring on a single spacetime background.

One can think, however, that once a complete (e.g. interferometric) scenario
is considered such a change of representation can no longer be achieved. By
the above arguments, this can be done. If in addition the matter DoFs only
evolve by a phase like in many recently considered scenarios, even further
identifications between apparently different scenarios can be made as follows.

Consider the representation given by Eq. (2), with some additional quantum
DoFs coupled to the spacetime i.e. Eq. (4). Time-evolving the system and
measuring the source mass in the superposition basis
$|\eta\rangle=(|g_{1}\rangle+|g_{2}\rangle)/\sqrt{2}$ gives the final
conditional state of the remaining DoFs:

$\displaystyle\langle\eta|\hat{U}|\psi\rangle$
$\displaystyle=\frac{1}{2}\big{(}\hat{U}_{1}+\hat{U}_{2}\big{)}|\phi\rangle.$
(10)

Note that the state on which we condition is chosen to be the same as the
initial state just for simplicity. However this can be arbitrary, for example
imprinted with some relative phase $\varphi$ as in typical interferometric
setups. The DoFs described by the state $|\phi\rangle$ evolve in a coherent
superposition of “paths” parametrised by the coordinates $x_{1},x_{2}$,
associated with the different amplitudes of the spacetime superposition.
Probability amplitudes for the remaining DoFs can be obtained by calculating
the overlap with some final state $|\Omega\rangle$, giving the probability
amplitude $\langle\Omega|\langle\eta|\hat{U}|\psi\rangle$, which explicitly
reads

$\displaystyle\frac{1}{2}\langle\Omega|\big{(}\hat{U}_{1}+\hat{U}_{2}\big{)}|\phi\rangle.$
(11)

On the other hand, if we adopt the representation given by Eq. (II.2) (for the
specific example of a superposition of translations), the amplitude
equivalently reads

$\frac{1}{{2}}\big{(}\langle\Omega|\hat{U}_{1}|\phi\rangle+\langle\Omega-\delta
x|\hat{U}_{1}|\phi-\delta x\rangle\big{)},$ (12)

where we used the fact that the two mass configurations and metrics are
macroscopically distinct, meaning $\langle g_{1}(x)|\hat{U}^{\dagger}(\delta
x)|g_{1}(x)\rangle=\langle g_{1}(x)|\hat{U}(\delta x)|g_{1}(x)\rangle=0$, and
that $\hat{U}^{\dagger}(\delta x)\hat{U}_{1}\hat{U}(\delta x)=\hat{U}_{2}$ by
virtue of the fact that $\hat{U}(\delta x)$ encodes a diffeomorphism.

Equation (12) is a formal statement of the fact that a scenario involving
quantum systems in a spacetime sourced by a spatial superposition of mass
configurations is fully equivalent to a scenario where the particle follows
superposed trajectories in one spacetime. The equivalence between the two
scenarios can be interpreted as describing the same physical situation simply
using two sets of coordinates related by a “superposition” of classical
transformations [41], similar to scenarios arising in the context of quantum
reference frames [27] operationally established using quantum systems. This
general equivalence further implies that probability amplitudes of matter DoFs
will be identical in both scenarios (e.g. the transition probability of
quantum matter coupled to a quantum field).

The crux of the argument is that only relative configurations between the
interacting systems are physically relevant, such as a superposition of two
distances between a pair of particles; a global (joint) coordinate
transformation enacted on all degrees of freedom is irrelevant. Scenarios
where quantum probes are situated on a classical background while the time-
evolution occurs in a superposition of trajectories were recently considered
in Refs. [44, 45, 46, 47, 48].

Let us take an approximation that two amplitudes of the matter remain
orthogonal (e.g. only evolve by a phase as in Refs [13, 14]) which here means
that $\langle\Omega-\delta
x|\hat{U}_{1}|\phi\rangle=\langle\Omega|\hat{U}_{1}|\phi-\delta x\rangle=0$,
so that the above can be equivalently written as

$\frac{1}{{2}}\big{(}\langle\Omega|+\langle\Omega-\delta
x|\big{)}\hat{U}_{1}\big{(}|\phi\rangle+|\phi-\delta x\rangle\big{)}.$ (13)

This reflects that the scenario (a) in which the source mass is prepared and
measured in superposition while the other DoFs evolve in the resulting
superposition of metrics is equivalent to the scenario (b) in which the other
DoFs are prepared and measured in superposition while the source mass remains
in some fixed state sourcing a classical metric.

We conclude this section by mentioning that the formalism presented here has a
well-defined non-relativistic limit, in which it can describe matter that
sources a Newtonian potential and through this potential interacts with other
massive particles. This is of course also true for general relativistic
descriptions of a classical metric that reduce to the non-relativistic
Newtonian description in such a limit. Such “superpositions of Newtonian
potentials,” which have garnered just as much interest as their general
relativistic counterparts (see for example Refs [49, 50, 51, 52, 53] and the
following section on gravitationally-induced entanglement proposals), are
fully characterised by the present framework. We now turn to specific
scenarios of practical interest as applications of our results.

###  II.3 Superposition of Geometries in the Gravitationally-Induced
Entanglement Proposals

The previous analysis demonstrated how in general, spacetime superpositions in
which the respective amplitudes are related by a diffeomorphism can be re-
expressed in terms of a single background spacetime with the states and
measurements of the remaining DoFs suitably adapted. This motivates us to
revisit the conclusions one can draw concerning quantum gravity from such
scenarios.

In this section, we apply our approach and construction to recent proposals by
Bose et. al. [13] and Marletto and Vedral [14], which have attracted
significant recent interest as presenting possibilities for witnessing the
quantization of the gravitational field. These “gravitationally-induced
entanglement” (GIE) setups suggest that observing entanglement between two
spatially superposed source masses, interacting through the Newtonian
potential, is not equivalent to matter interferometry on a fixed background
and would provide evidence of the quantum nature of gravity (for example, that
entanglement is mediated via gravitons). Here, we focus on a recent
interpretation of this argument by Christodolou and Rovelli [35] who argue
that such a setup represents a superposition of genuinely distinct spacetime
metrics [35].

Figure 3: Schematic diagram of the GIE proposal. Each of the states
$|x_{i}^{L,R}\rangle$ is assumed to be in a highly localised Gaussian
wavepacket such that their overlap is zero.

The GIE proposal is depicted in Fig. 3. Two masses, $m_{1}$ and $m_{2}$, are
each prepared and measured in spatial superpostion of freely falling
trajectories in the uniform gravitational field of Earth (for example via
Stern-Gerlach interferometry techniques). Thus initially, each particle is in
a state of the form

$\displaystyle|\psi_{i}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\big{(}|x_{i}^{L}\rangle+|x_{i}^{R}\rangle\big{)},$
(14)

where $i=1,2$ labels the particles, while the labels $L$, $R$ denote mutually
orthogonal eigenstates of the $x$-component of the spin-1/2 particle,
controlling the path that each particle takes. For simplicity, we have chosen
the separation between the paths, $\delta x$, to be equal for each particle,
though the analysis applies for arbitrary separations between any of the
branches of the state. Evolution under mutual gravitational interaction for a
time $\tau$ entangles the test masses by imparting distance-dependent phases
to the components of the superposition.

Christodolou and Rovelli note that immediately after the superpositions are
created, we have a product state of the particles and some external metric
$|g\rangle$ that has not had time to change appreciably,

$\displaystyle|\psi\rangle$
$\displaystyle=\frac{1}{2}\big{(}|x_{1}^{L}\rangle+|x_{1}^{R}\rangle\big{)}\big{(}|x_{2}^{L}\rangle+|x_{2}^{R}\rangle\big{)}\otimes|g\rangle,$
$\displaystyle=\frac{1}{2}\big{(}|x_{1}^{L},x_{2}^{L}\rangle+|x_{1}^{L},x_{2}^{R}\rangle$
$\displaystyle\qquad+|x_{1}^{R},x_{2}^{L}\rangle+|x_{1}^{R},x_{2}^{R}\rangle\big{)}\otimes|g\rangle.$
(15)

However, once the gravitational disturbance has had time to propagate, each
branch of the two-particle state will source, in general, a different metric
depending on the relative distance between the particles in that branch:

$\displaystyle|\psi\rangle$
$\displaystyle\to\frac{1}{2}\big{(}|x_{1}^{L},x_{2}^{L},g_{LL}\rangle+|x_{1}^{L},x_{2}^{R},g_{LR}\rangle$
$\displaystyle\qquad+|x_{1}^{R},x_{2}^{L},g_{RL}\rangle+|x_{1}^{R},x_{2}^{R},g_{RR}\rangle\big{)}$
(16)

where $g_{ij}$, $i,j=R,L$ denote the respective metrics. They further argue
that such a scenario no longer represents a semiclassical spacetime but a
genuine superposition of metrics. This is true when referring to the metric
generated by both particles, each in superposition.

However the GIE proposal does not depend on a geometry that is sourced by both
particles. The result of the proposed interferometric measurements depend on
the gravitational interaction between the two particles. Thus, the metric
involved can be interpreted as that sourced by, say, particle $i=1$ (this
choice being arbitrary), and despite the point made by Christodolou and
Rovelli, the proposed setup of Fig. 3 can be reinterpreted as an interference
experiment on classical space-time, where a ‘source’ particle follows a fixed
trajectory while the other undergoes a suitably adapted interference
experiment, using our framework developed in the earlier section.

To see this explicitly, note that the states $|\psi_{i}^{L}\rangle$,
$|\psi_{i}^{R}\rangle$ can be related by a unitary operator $\hat{U}(\delta
x)$ implementing a translation by the distance $\delta x$, so that we can
express Eq. (II.3) as

$\displaystyle|\psi\rangle$
$\displaystyle=\frac{1}{2}\big{(}|x_{1}^{L},x_{2}^{L}\rangle+|x_{1}^{L},x_{2}^{R}\rangle+\hat{U}(\delta
x)|x_{1}^{L},x_{2}^{L}-\delta x\rangle\vphantom{\frac{1}{2}}$
$\displaystyle\qquad+\hat{U}(\delta x)|x_{1}^{L},x_{2}^{R}-\delta
x\rangle\big{)}.\vphantom{\frac{1}{2}}$ (17)

In the GIE proposal, after some interval of time-evolution each mass is
measured in a superposition, and so the relevant amplitude is
$\langle\eta|\hat{U}|\psi\rangle$ where
$|\eta\rangle=|\eta_{1}\rangle\otimes|\eta_{2}\rangle$.

Using our general result, leading to Eq. (12), we obtain

$\displaystyle\langle\eta|\hat{U}|\psi\rangle$
$\displaystyle=\frac{1}{2}\big{(}\langle\eta_{1}|\langle\eta_{2}|\hat{U}|x_{1}^{L},x_{2}^{L}\rangle\vphantom{\frac{1}{2}}$
$\displaystyle+\langle\eta_{1}|\langle\eta_{2}|\hat{U}|x_{1}^{L},x_{2}^{R}\rangle\vphantom{\frac{1}{2}}$
$\displaystyle+\langle\eta_{1}-\delta x|\langle\eta_{2}-\delta
x|\hat{U}|x_{1}^{L},x_{2}^{L}-\delta x\rangle\vphantom{\frac{1}{2}}$
$\displaystyle+\langle\eta_{1}-\delta x|\langle\eta_{2}-\delta
x|\hat{U}|x_{1}^{L},x_{2}^{R}-\delta x\rangle\big{)}\vphantom{\frac{1}{2}}$
(18)

where we have used the fact that GIE assumes an interaction given by
$\hat{U}=\int\mathrm{d}x_{1}\mathrm{d}x_{2}\exp(-it\frac{Gm_{1}m_{2}}{|\hat{x}_{1}-\hat{x}_{2}|})|x_{1}\rangle\langle
x_{1}|\otimes|x_{2}\rangle\langle x_{2}|$, i.e. the Newtonian potential
depending on the distance $x_{1}-x_{2}$ between each of the respective
branches of the superposition, which is explicitly invariant under passive
translations (i.e. translations of both coordinates).

To finalise our argument, note that this interaction allows us to consider
either of the particles as the source mass of the potential at the location of
the other particle (if there were a third system considered, e.g. a small test
mass, then the metric in the vicinity of that mass would be sourced by both
particles $i=1,2$. However in the present scenario, only two particles are
involved). Considering the particle $i=1$ as the source, the associated metric
in the above is fixed (defined by the position of that particle) and thus
allows us to interpret the scenario as the evolution of the second particle,
which is in a superposition of four amplitudes, taking place on a single
spacetime. This becomes particularly clear when including the assumption made
in this scenario, that the states only evolve by a phase while the distance
between the particles remains unchanged, and that the measurements are
performed in a superposition of the paths, i.e.
$|\eta_{i}\rangle=(|x_{i}^{L}\rangle+|x_{i}^{R}\rangle)/\sqrt{2}$. These two
together mean that in the original formulation, the relevant amplitude is

$\langle\eta|\hat{U}|\psi\rangle=\frac{1}{4}\sum_{n,m}\langle
x_{1}^{n},x_{2}^{m}|\hat{U}|x_{1}^{n},x_{2}^{m}\rangle,$ (19)

where $n,m=L,R$ implies a sum over the four amplitudes, while using our result
in Eq. (II.3), the same amplitude reads

$\langle\eta|\hat{U}|\psi\rangle=\langle\tilde{x}_{2}|\langle
x_{1}^{L}|\hat{U}|x_{1}^{L}\rangle|\tilde{x}_{2}\rangle,$ (20)

where $|\tilde{x}_{2}\rangle:=(|x_{2}^{L}\rangle+|x_{2}^{L}-\delta
x\rangle+|x_{2}^{R}\rangle+|x_{2}^{R}-\delta x\rangle)/\sqrt{4}$. Clearly we
can write this as $\langle\tilde{x}_{2}|\hat{U}_{L}|\tilde{x}_{2}\rangle$
where
$\hat{U}_{L}=\int\mathrm{d}x_{2}\exp(-it\frac{Gm_{1}m_{1}}{|x_{1}^{L}-\hat{x}_{2}|})|x_{2}\rangle\langle
x_{2}|$ is the gravitational potential sourced by particle $1$ positioned at
$x_{1}^{L}$ and experienced by particle $2$.

The calculation above shows that GIE, originally explained as due to a quantum
superposition of gravitating source masses, can be explained in terms of a
test mass in a classical potential sourced by the other mass. Thus,
independently of any other considerations, there is an inherent ambiguity in
the interpretation of such experiments as testing quantum features of
gravitational degrees of freedom. Indeed, using linearity of quantum theory
the same amplitude is obtained from the expression Eq. (20) in which the
gravitational field only features as an external (classical) potential. We
stress that this ambiguity is present because in the considered scenario only
relative distances between the particles play a role in determining the final
probability, and not their absolute locations relative to, say, the laboratory
reference frame.

Consequently, the interpretation that the initial state Eq. (II.3) evolves
into Eq. (II.3) describing an entangled state of semiclassical spacetime
metrics, while correct, is not required to predict the amplitude of interest
in this proposal. Instead, the result of the experiment can be adequately
described in terms of a superposition of distances between masses where the
one designated as the source is fixed, and consequently the spacetime is
fixed. This highlights again the inherent ambiguity in the notion of
“spacetime superposition” if the superposed amplitudes differ by a
diffeomorphism (or any symmetry of the dynamics).

Removing the above ambiguity requires superpositions of non-diffeomorphic
metrics. By definition such metrics are not related by any change of
coordinates, since they effectively give rise to unique solutions to the
Einstein field equations. It then follows that the prior analysis, in which
various amplitudes in superposition could be related by unitaries representing
a general coordinate transformation, is no longer applicable. In such cases,
the superposition of geometries that arises is unambiguously quantum-
gravitational, insofar as the resulting metrics associated with different
amplitdues cannot be re-expressed as a single classical metric with suitably
transformed dynamics and measurement bases for the remaining DoFs. Applied to
the GIE proposals, this could involve atoms that are prepared in a
superposition of energy eigenstates [54], with each eigenstate generating a
different curvature. Related examples that have been studied in the literature
include a black hole [55, 56] or dark matter clump [18] in a superposition of
masses, and an expanding universe in a superposition of expansion rates [47].

###  II.4 Apparent Decoherence of Spacetime Superpositions

In the previous section, we showed how the linearity of quantum theory and the
fact that the notion of position is a relative one (only relative distances
between systems are physically meaningful) leads to the relativity of
spacetime superpositions, and we demonstrated that this has significant
implications for the interpretation of scenarios that appear to give rise to
quantum superpositions of geometries. In this section, we apply our approach
to scenarios exploring the phenomenon of decoherence. We show how following a
common approach may lead to different conclusions about the decoherence of a
mass configuration, depending on how one defines the coordinate system to
describe the configuration.

Let us consider for example, the case of a black hole at some position and its
Hawking radiation, described by the state
$|\chi_{i}\rangle\equiv|\chi_{i}(x_{i})\rangle$ with $x_{i}$ being the
position of the black hole. A black hole in superposition of different
positions $x_{1}$ and $x_{2}$ would evolve into an entangled state

$\displaystyle|\psi\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|x_{1},\chi_{1}\rangle+|x_{2},\chi_{2}\rangle\right),$
(21)

since a black hole cannot be ultimately isolated from its Hawking radiation,
and we have assumed the shorthand
$|x_{i},\chi_{i}\rangle\equiv|x_{i}\rangle\otimes|\chi_{i}\rangle$. Upon
tracing out the radiation degrees of freedom from the state in Eq. (21), one
finds

$\displaystyle\mathrm{Tr}_{\chi}|\psi\rangle\langle\psi|$
$\displaystyle=\frac{1}{2}\begin{pmatrix}1&|\nu|^{2}\\\
|\nu|^{2}&1\end{pmatrix}\to\frac{1}{2}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},$
(22)

where $\nu=\langle\chi_{1}|\chi_{2}\rangle$ is the overlap between the
radiation states associated with the different positions of the black hole.
For distinguishable states of the radiation, over time the off-diagonal
elements of the reduced density matrix will become suppressed, leaving a
classical mixture of the two locations of the black hole. This has been argued
to lead to fundamental decoherence [16, 17, 20] , due to the above mentioned
fact that a black hole cannot be isolated from its Hawking radiation.

However based on the framework introduced thus far, the relevant question is,
with respect to what physical system is the black hole superposed? As we have
discussed, the use of physical systems that can interact with the source mass
and measurements of final probability amplitudes is crucial for understanding
the physical implications of scenarios involving superpositions of spacetime
metrics. For example, choosing coordinates whose origin is defined as the
location of the black hole, Eq. (21) by construction becomes a product state
of the black hole and its radiation,

$\displaystyle|\psi\rangle$ $\displaystyle=|x_{1},\chi_{1}\rangle,$ (23)

from which one would not deduce decoherence. Based on our framework for the
relativity of superpositions, the transformation between Eq. (21) and Eq. (23)
involves or acts on all DoFs. The tacit assumption behind Eq. (22) is that in
computing the overlap $\nu$, one necessarily has access to an uncorrelated
measuring device (for example a first-quantised particle) that can probe
different spacetime points of the radiation field $\chi(x)$ [57, 58].

To see this, it is instructive to consider probability amplitudes that are
obtained by coupling a measuring device (or devices) that are perfectly
correlated with the position of the black hole. The measuring device may
function as a proxy for arbitrary matter DoFs that are correlated with the
black hole-radiation system. We can take the state of this device to be

$\displaystyle|\Omega\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|x_{1},\Omega_{1}\rangle+|x_{2},\Omega_{2}\rangle\right).$
(24)

In this scenario the relative distance between the measuring device and the
black hole is identical in each branch of the superposition. For ease of
comparison, let us consider the approach of Ref. [16], which models the black
hole as a dielectric sphere that interacts with (isotropically) scattered and
emitted (Hawking) photons. Time-evolving the state with the quantum-controlled
unitary $\hat{U}=\sum_{i}\hat{U}_{i}|x_{i},\chi_{i}\rangle\langle
x_{i},\chi_{i}|$ gives

$\displaystyle\hat{U}\hat{\rho}\hat{U}^{\dagger}$
$\displaystyle=\frac{1}{2}\sum_{i,j}\hat{U}_{i}|x_{i},\chi_{i}\rangle\langle\chi_{j},x_{j}|\hat{U}_{j}^{\dagger},$
(25)

where the sum runs over $i,j=1,2$. Crucially, the radiation states are here
correlated with the black hole position, and hence the relative distance
between these subsystems is fixed in either branch of the superposition.
Measuring the system in the correlated state given by Eq. (24) yields

$\displaystyle\langle\Omega|\hat{U}\hat{\rho}\hat{U}^{\dagger}|\Omega\rangle\vphantom{\frac{1}{2}}$
$\displaystyle\>\>\>=\frac{1}{4}\big{(}\langle\Omega_{1},x_{1}|\hat{U}_{1}|x_{1},\chi_{1}\rangle\langle\chi_{1},x_{1}|\hat{U}_{1}^{\dagger}|x_{1},\Omega_{1}\rangle\vphantom{\frac{1}{2}}$
$\displaystyle\>\>\>\qquad+\langle\Omega_{1},x_{1}|\hat{U}_{1}|x_{1},\chi_{1}\rangle\langle\chi_{2},x_{2}|\hat{U}_{2}^{\dagger}|x_{2},\Omega_{2}\rangle\vphantom{\frac{1}{2}}$
$\displaystyle\>\>\>\qquad+\langle\Omega_{2},x_{2}|\hat{U}_{2}|x_{2},\chi_{2}\rangle\langle\chi_{1},x_{1}|\hat{U}_{1}^{\dagger}|x_{1},\Omega_{1}\rangle\vphantom{\frac{1}{2}}$
$\displaystyle\>\>\>\qquad+\langle\Omega_{2},x_{2}|\hat{U}_{2}|x_{2},\chi_{2}\rangle\langle\chi_{2},x_{2}|\hat{U}_{2}^{\dagger}|x_{2},\Omega_{2}\rangle\vphantom{\frac{1}{2}}\big{)}$
$\displaystyle\>\>\>\qquad+\mathrm{orthogonal\>terms}.\vphantom{\frac{1}{2}}$
(26)

where in the momentum basis, the density matrix for the radiation (correlated
with the black hole position) is given by
$|\chi_{i}\rangle\langle\chi_{j}|=\int\mathrm{d}^{3}k\frac{p(k)}{4\pi
k^{2}}\exp(-i\hat{k}\cdot\hat{x}_{i})|k\rangle\langle
k|\exp(i\hat{k}\cdot\hat{x}_{j})$, and $p(k)$ is the probability of emitting a
particle with momentum $k$ [16]. The “orthogonal terms” are those that vanish
under the assumption that the spacetimes are macroscopically distinct.
Utilising the fact that the black hole states (including all additional
coupled DoFs, $\phi=\chi,\Omega$) are related via the unitary translation
$|x_{2},\phi(x_{2})\rangle=\hat{U}(\delta x)|x_{1},\phi(x_{1})\rangle$, and
that $\hat{U}^{\dagger}(\delta x)\hat{U}_{2}\hat{U}(\delta
x)\equiv\hat{U}_{1}$, we find that

$\displaystyle\langle\Omega|\hat{U}\hat{\rho}\hat{U}^{\dagger}|\Omega\rangle$
$\displaystyle=\left|\langle\Omega_{1},x_{1}|\hat{U}_{1}|x_{1},\chi_{1}\rangle\right|^{2}.$
(27)

This of course, is the standard probability amplitude between the initial and
final states $|\chi_{1}\rangle$, $|\Omega_{1}\rangle$, computed on the
classical spacetime metric $|x\rangle$ produced by the black hole. The black
hole is in a “classical position state” with respect to the correlated
measuring device; in principle, this could be extended to all other matter
DoFs in the universe. In this scenario, there does not exist an absolute set
of relative coordinates with which decoherence can be meaningfully computed,
and thus one concludes that there is no decoherence at all.

The alternative scenario is to consider a measuring device that is not
initially correlated with the position of the black hole, for example
described by the state
$|\Omega\rangle=|\Omega_{1}\rangle(|x_{1}\rangle+|x_{2}\rangle)/\sqrt{2}$. In
this case,

$\displaystyle\langle\Omega|\hat{U}\hat{\rho}\hat{U}^{\dagger}|\Omega\rangle\vphantom{\frac{1}{2}}$
$\displaystyle\>\>\>=\frac{1}{2}\big{(}\langle\Omega_{1},x_{1}|\hat{U}_{1}|x_{1},\chi_{1}\rangle\langle\chi_{1},x_{1}|\hat{U}_{1}^{\dagger}|x_{1},\Omega_{1}\rangle\vphantom{\frac{1}{2}}$
$\displaystyle\qquad+\langle\Omega_{1},x_{2}|\hat{U}_{2}|x_{2},\chi_{2}\rangle\langle\chi_{2},x_{2}|\hat{U}_{2}^{\dagger}|x_{2},\Omega_{1}\rangle\big{)}.\vphantom{\frac{1}{2}}$
(28)

The conditional state Eq. (II.4) will in general lead to off-diagonal terms
given by the overlap between the state of the measurement device and the
radiation state $\chi_{2}$, $\langle\Omega_{1}|\chi_{2}\rangle$. It follows
that the black hole will decohere over time as information about its position
is exchanged through its radiation to the measurement device.

In order to obtain decoherence then, one must necessarily account for the fact
that the black hole is not perfectly isolated from external DoFs through which
its state is defined. In this sense, decoherence of black holes due to Hawking
radiation is not fundamental. Unruh pointed out such examples as demonstrating
a “false loss of coherence” [59]. The basic idea is that decoherence is often
inferred by virtue of the coupling between a system and its environment, which
in this case is represented by the position of the black hole $|x\rangle$ with
its radiation field $|\chi(x)\rangle$. However, so long as any changes to the
system are made adiabatically, the radiation field is effectively enslaved to
the position of the black hole, and hence correlated with the black hole
position. If the black hole states are brought back together adiabatically,
interference will be observed and coherence is recovered.

###  II.5 Coherence of Spacetime Superpositions

To conclude, we now demonstrate using our framework how spacetime
superpositions do not fully decohere even in circumstances where the presence
of additional matter DoFs reveals which-way information about the quantum
spacetime. Let us consider for concreteness the case where a probe particle
resides outside a black hole in a superposition of relative distances from the
horizon. For simplicity we can model the particle as a qubit with internal
states $\\{|0\rangle,|1\rangle\\}$, which interacts with the spacetime
superposition through the quantum field $\hat{\phi}$, whose initial state is
denoted by $|\phi\rangle$. The evolution of the black hole, quantum field, and
matter system takes the form
$\hat{U}|\psi\rangle=(\hat{U}_{1}|g_{1}\rangle+\hat{U}_{2}|g_{2}\rangle)|\phi\rangle|0\rangle/\sqrt{2}$.
As usual, the interaction takes place in a quantum-controlled superposition of
coordinate parametrisations, wherein the individual evolutions depend on the
field operator $\hat{\phi}\equiv\hat{\phi}(x_{1}),\hat{\phi}(x_{2})$ on the
different manifolds in superposition. We have written the state of the field
$|\phi\rangle$ as separable from the control degree of freedom. This is a
particular choice following a recent investigation that showed this to be
convenient in the case of the (2+1)-dimensional Banados-Zanelli-Teitelboim
black hole, where the ground state of the field can be taken to be the anti-de
Sitter vacuum state [55]. However we are not constrained by this choice; one
could in principle assume the field state to be correlated with the black hole
(or the control system correlated with the black hole) as well.

To compute the decoherence of the black hole, we can consider an
interferometric setup in which the black hole (or the control system
correlated with the black hole) undergoes a Mach-Zehnder-type trajectory with
a controllable phase $\varphi$ on one of the arms [21]. Measuring the control
in the superposition basis
$|\eta\rangle=(|g_{1}\rangle+|g_{2}\rangle)/\sqrt{2}$ and tracing out the
field and matter DoFs leaves,

$\displaystyle\mathrm{Tr}_{\mathrm{FM}}\langle\eta|\hat{U}|\psi\rangle$
$\displaystyle=\frac{1}{2}\big{(}1+V\cos(\varphi)\big{)},$ (29)

where $V=\langle
0|\langle\phi|\hat{U}_{1}^{\dagger}\hat{U}_{2}|\phi\rangle|0\rangle$.
Decoherence here is quantified by the visibility $V$ of the interference
pattern produced as the relative phase $\varphi$ is varied. $V$ quantifies the
contrast in the maxima and minima of the interference fringes observed in such
an interferometric setup. If $V=0$, the black hole state has fully decohered,
while if $V=1$, coherence is maximally retained. Now in general, the
visibility will take the form (see Methods)
$V=1+L_{12}-\frac{1}{2}(P_{1}+P_{2})$. Here, $L_{12}$ is a contribution
originating as an “interference” term between the spacetime amplitudes in
superposition, while $P_{1}$ and $P_{2}$ are contributions from the individual
spacetime amplitudes themselves. In Methods, we give an explicit example of
the form of these terms assuming a particular interaction known as the Unruh-
deWitt model [60, 61, 62], which well approximates the light-matter
interaction [63]. In general, $P_{1,2}>L_{12}$ and thus $V<1$: some amount of
coherence is lost. However in the regime of weak coupling between the qubit,
field, and spacetime, the decrease in coherence will be very small,
$\sim\mathcal{O}(\lambda^{2})$ where $\lambda\ll 1$. This is consistent with
Unruh’s observation concerning rapid or strong interactions leading to
decoherence, while weak coupling or adiabatic evolution leads to false loss of
coherence [64, 65, 66, 67]. Moreover, it aligns with the intuition that
decoherence results from an irretrievable or irreversible loss of information
(coherence) from the system to the environment [59]. This may occur for
example when this information is carried off to infinity; however here, the
weak coupling to the probe particle allows for coherence to be retained.

As more and more particles are introduced into the system, the decrease in
visibility becomes more significant. The visibility for a system of $N$
particles coupled to the black hole is given by

$\displaystyle V$
$\displaystyle=1+\frac{1}{2}\sum_{i,j=1}^{N}L_{ij}-\frac{1}{2}\sum_{i=1}^{N}P_{i}.$
(30)

Here, $L_{ij}$ is the interference term between the $i$th and $j$th components
of the superposition, while $P_{i}$ are the “local” contributions from the
$i$th amplitude of the superposition. The factor of $(1/2)$ in front of the
first summation is to avoid double counting the $L_{ij}=L_{ji}$ terms. Since
$P_{i}>L_{ij}$ for all $(i,j)$ in general, (58) says that as one adds more and
more matter to the system, the visibility steadily decreases.

##  III Discussion

In this article, we have introduced the notion of “relativity of spacetime
superpositions” for quantum superpositions of mass configurations whose
amplitudes differ by a coordinate transformation. Such superposition states
have garnered significant interest for their direct relevance in low-energy
tests of quantum gravity, and for understanding emergent quantum-gravitational
phenomena from the “bottom-up” (i.e. investigations of quantum-gravitational
effects that do not rely upon a formal theory of quantum gravity). Our
framework only depends upon the basic tenets of linearity in quantum theory
and the invariance of dynamics under coordinate transformations in general
relativity. We have drawn attention to the fact that the physical
configuration of a particular system and the way one labels such a
configuration, even when that system exhibits quantum-mechanical properties
like superposition and the subsystems involved are sources of the
gravitational field, is not fundamental but merely conventional. So long as
relative distances between the considered amplitudes remain invariant, one can
perform arbitrary transformations of quantum coordinates that map
superposition states of the semiclassical metric to a scenario in which the
metric is fixed and classical, while the dynamics of remaining DoFs undergo a
modified time-evolution. Our approach is complementary to recent work in the
field of QRFs, though it relies upon a noticeably simpler and distinct
mathematical framework. Moreover, it is generally applicable to arbitrary
quantum systems coupled to the spacetime, such as quantum fields.

The relativity of spacetime superpositions has significant implications for
both conceptual and practical proposals for witnessing quantum-gravitational
effects arising from so-called quantum superpositions of geometries. The main
point emerging from our framework is that quantum superpositions of
gravitational sources whose states are related by a coordinate transformation
(which includes all spatial and even temporal superpositions) are not
unambiguously quantum-gravitational. This is particularly important in ongoing
discussions and proposals for deriving and observing phenomena not describable
within the current paradigms of quantum mechanics and general relativity.

A pertinent example of where our framework is particularly important is the
phenomenon of decoherence. It is often tacitly assumed but not explicitly
stated that measurements can be performed on environmental degrees of freedom.
It is not usually recognised that such measurements are not only required to
see decoherence, but only through such measurements can superpositions of mass
configurations acquire physical meaning. Without them, and assuming complete
isolation from all externally coupled systems, superpositions of spatial
states of spacetime are operationally indistinguishable from dynamics
occurring on a single classical background. This highlights the necessity of
including matter degrees of freedom coupling to the mass configuration that
define coordinates via relative distances to the respective spatial
amplitudes. However as we have demonstrated, the inclusion of such matter does
not make the superposition more fundamentally “quantum-gravitational,” for it
is again always possible to re-express the spacetime as a single, fixed
background and the matter in a superposition of configurations.

As a final point, and as alluded to above, the symmetry between
representations of scenarios involving superpositions of spatial
configurations is broken once one considers superpositions of metrics that
cannot be related by a simple diffeomorphism. Such scenarios, characterised
for example by a source in a superposition of masses do not admit a simple
unitary transformation between the superposed amplitudes; these amplitudes
represent unique solutions to Einstein’s field equations and thus are
physically distinguishable. This point motivates an important extension to the
current GIE proposals, namely the feasibility of experimental schemes
involving the interaction of test particles in superpositions of energy
eigenstates, and the possibility of witnessing genuinely quantum-gravitational
effects induced thereby.

##  IV Methods

###  IV.1 Coupling a Quantum Field and Matter DoF to a Spacetime Superposition

In this section, we apply the relativity of spacetime superpositions to the
specific interaction between a quantum field and a first-quantized particle
with the background sourced by a mass in superposition of configurations.

Consider as usual a classical mass configuration $|g\rangle$ with a
corresponding classical manifold. Consider also the quantum field $\hat{\phi}$
on this manifold in the state $|\phi\rangle$ and a first-quantised particle
(this need not be an elementary particle), modelled as a qubit with internal
states $\\{|0\rangle,|1\rangle\\}$, which is a proxy for some additional
matter degrees of freedom that can interact with the field and background
superposition. We take the initial state to be
$|\psi\rangle=|g\rangle|\phi\rangle|0\rangle$, which describes the tensor
product of the spacetime, field, and matter degrees of freedom.

Generalizing our classical mass configuration to include quantum
superpositions of different relative configurations between the source and
matter, consider the initial state to be
$|\psi\rangle=|\phi\rangle|0\rangle(|g_{1}(x_{1})\rangle+|g_{2}(x_{2})\rangle)/\sqrt{2}$.
We reiterate that the mass configuration can be conceptualised as being
quantum-controlled by an ancillary system that can be prepared and measured in
appropriate states. We assume that the qubit can be held static at a fixed
distance from the mass configuration in each amplitude of the superposition.
Moreover, the wavefunctions associated with the different branches of the
superposition can be understood to be effectively indistinguishable–that is,
the qubit can be held static outside the mass configuration without obtaining
which-way information about its position relative to the mass. The total
Hilbert space of the relevant systems is given by
$\mathcal{H}=\mathcal{H}_{\mathrm{S}}\otimes\mathcal{H}_{\mathrm{F}}\otimes\mathcal{H}_{\mathrm{M}}$
where $\mathcal{H}_{\mathrm{S}}$, $\mathcal{H}_{\mathrm{M}}$, and
$\mathcal{H}_{\mathrm{F}}$ are respectively associated with the spacetime,
field, and matter DoFs.

Let us assume the following simple form of the interaction Hamiltonian, which
couples all degrees of freedom in the interaction:

$\displaystyle\hat{H}_{\mathrm{int.}}$
$\displaystyle=\hat{H}_{1}(x_{1})\otimes|g_{1}(x_{1})\rangle\langle
g_{1}(x_{1})|\vphantom{\frac{1}{2}}$
$\displaystyle+\hat{H}_{2}(x_{2})\otimes|g_{2}(x_{2})\rangle\langle
g_{2}(x_{2})|.$ (31)

The Hamiltonian Eq. (IV.1) couples the field and qubit with the respective
spacetime amplitudes $|g_{i}\rangle$. For distinguishable mass configurations
when the assigned states $|g_{1}\rangle$, $|g_{2}\rangle$ are mutually
orthogonal, the time evolution operator

$\displaystyle\hat{U}$
$\displaystyle=\exp\left(-i\int\mathrm{d}t\frac{\mathrm{d}\tau}{\mathrm{d}t}\hat{H}_{\mathrm{int.}}\right),$
(32)

can be written generally as

$\displaystyle\hat{U}$
$\displaystyle=\hat{U}_{1}(x_{1})\otimes|g_{1}(x_{1})\rangle\langle
g_{1}(x_{1})|\vphantom{\frac{1}{2}}$
$\displaystyle+\hat{U}_{2}(x_{2})\otimes|g_{2}(x_{2})\rangle\langle
g_{2}(x_{2})|,$ (33)

where $\hat{U}_{1}(x_{1})$ ($\hat{U}_{2}(x_{2})$) is the respective unitary
describing the interaction for a given state $|g_{1}(x_{1})\rangle$
($|g_{2}(x_{2})\rangle$) of the mass configuration (serving here as a
control).

After the interaction and upon measuring the control in the state
$|\eta\rangle=(|g_{1}\rangle+|g_{2}\rangle)/\sqrt{2}$, the final state of the
remaining DoFs (here, the field and the internal state of the qubit) is given
by

$\displaystyle\langle\eta|\hat{U}|\psi\rangle$
$\displaystyle=\frac{1}{2}\big{(}\hat{U}_{1}+\hat{U}_{2}\big{)}|\phi\rangle|0\rangle,$
(34)

i.e. Eq. (10) with $|\phi\rangle$ now specifically describing the product
state of the field and particle. Again, we emphasise that the interpretation
of this interaction is that of quantum matter interacting with a field
quantized on a background in a superposition of geometries $g_{1}$, $g_{2}$.

An alternative interpretation is to consider the scenario from the “quantum
coordinates” of the mass configuration, in which there will only be a single,
fixed background with the detector interacting with the field in a
superposition of coordinate parametrizations. As argued above, the coordinates
assigned to the joint state of the control states with the field
$|g_{1}\rangle|\phi\rangle$, $|g_{2}\rangle|\phi\rangle$ can be related using
an operator $\hat{U}(\delta x)$ that connect the coordinates parametrizing the
two relative configurations of the mass with respect to the matter DoFs. In
general, we can rewrite the relative position states in terms of the
coordinate relabelling $|g_{2}\rangle|\phi\rangle=\hat{U}(\delta
x)|g_{1}\rangle|\phi\rangle=\hat{U}^{g}(\delta
x)|g_{1}\rangle\otimes\hat{U}^{\phi}(\delta x)|\phi\rangle$. The
$\hat{U}(\delta x)$ operator can be considered as enacting a coordinate
transformation on the entire system. For clarity, we have distinguished
$\hat{U}^{g}(\delta x)$, which acts on the level of the metric while
$\hat{U}^{\phi}(\delta x)$ acts on the coordinates of the field operator. That
is, $\hat{\phi}(x_{2})=\hat{U}^{\phi\dagger}(\delta
x)\hat{\phi}(x_{1})\hat{U}^{\phi}(\delta x)$, or at the level of the unitary,
$\hat{U}^{\phi}(x_{2})=\hat{U}^{\phi\dagger}(\delta
x)\hat{U}^{\phi}(x_{1})\hat{U}^{\phi}(\delta x)$. The state of the system is
now

$\displaystyle\hat{U}|\tilde{\psi}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\hat{U}\big{(}\hat{I}+\hat{U}(\delta
x)\big{)}|g_{1}\rangle|\phi\rangle|0\rangle,$ (35)

where again, we have used the label $|\tilde{\psi}\rangle\equiv|\psi\rangle$
to clarify that though this is an equivalent physical situation to that
described by Eq. (34), the change of the representation of the state from
$|\psi\rangle$ to $|\tilde{\psi}\rangle$ gives this situation a different
interpretation. There is now a single classical background $g_{1}(x_{1})$ upon
which the field and quantum matter evolve in some modified “superposition” of
coordinate parametrizations. After projecting onto the control in the basis
$|\tilde{\eta}\rangle=(\hat{I}+\hat{U}(\delta x))|g_{1}\rangle/\sqrt{2}$,

$\displaystyle\langle\tilde{\eta}|\hat{U}|\tilde{\psi}\rangle$
$\displaystyle\>\>\>=\frac{1}{2}\langle
g_{1}|\big{(}\hat{I}+\hat{U}^{\dagger}(\delta
x)\big{)}\hat{U}_{1}\big{(}\hat{I}+\hat{U}(\delta
x)\big{)}|g_{1}\rangle|0\rangle|\phi\rangle.$ (36)

The right-hand side simply expresses that the dynamics occur on a single,
fixed background with a relative configuration state described by
$|g_{1}(x)\rangle$. The unitary $\hat{U}$ now acts only on this configuration
and thus reduces to $\hat{U}_{1}$ (since
$\hat{U}|g_{1}\rangle\equiv\hat{U}_{1}|g_{1}\rangle$). Recognising that
$\langle g_{1}|\hat{U}^{\dagger}(\delta x)\hat{U}|g_{1}\rangle=\langle
g_{1}|\hat{U}\hat{U}(\delta x)|g_{1}\rangle=0$, this simplifies to

$\displaystyle\langle\tilde{\eta}|\hat{U}|\tilde{\psi}\rangle$
$\displaystyle\>\>\>=\frac{1}{2}\langle
g_{1}|\big{(}\hat{U}_{1}+\hat{U}^{\dagger}(\delta x)\hat{U}_{1}\hat{U}(\delta
x)\big{)}|g_{1}\rangle|0\rangle|\phi\rangle.$ (37)

Using the fact that $\hat{U}^{\dagger}(\delta x)\hat{U}_{1}\hat{U}(\delta
x)\equiv\hat{U}_{2}$ we obtain

$\displaystyle\langle\tilde{\eta}|\hat{U}|\tilde{\psi}\rangle$
$\displaystyle\equiv\langle\eta|\hat{U}|\psi\rangle.$ (38)

This states that physical observables–which could be the transition amplitude
between different initial states of the field, or qubit–is invariant between
the two representations, $|\psi\rangle$, $|\tilde{\psi}\rangle$.

###  IV.2 General Superposition States

In this section, we generalise the results shown in Sec. II.2 to generic
superposition states of the metric:

$\displaystyle|\psi\rangle$
$\displaystyle=\beta|g_{1}(x_{1})\rangle+\sum_{i\neq
1}\alpha_{i}|g_{i}(x_{i})\rangle,$ (39)

where $|\beta|^{2}+\sum_{i}|\alpha_{i}|^{2}=1$ and $x_{i}$ parametrise the
coordinate system of the $i$th spacetime amplitude. Again, we assume that the
spacetimes are related by some diffeomorphism allowing for the states
$|g_{i}(x_{i})\rangle$ to be related to our chosen fixed background metric
state $|g_{1}(x_{1})\rangle$, via the unitary
$|g_{i}(x_{i})\rangle=\hat{U}(\delta x_{i})|g_{1}(x_{1})\rangle$. As before,
we describe some quantum DoFs in the state $|\phi\rangle$ interacting with the
spacetime superposition through the time-evolution governed by $\hat{U}$,
before performing a general measurement of the system in the state
$|\Omega\rangle$:

$\displaystyle\langle\Omega|\hat{U}|\psi\rangle$
$\displaystyle=\beta\langle\Omega|\hat{U}|g_{1}(x_{1})\rangle+\sum_{i}\alpha_{i}\langle\Omega|\hat{U}|g_{i}(x_{i})\rangle|\phi\rangle.$
(40)

Following the same procedure as before, we find that the dynamics can be
expressed in such a way as to occur on a single spacetime metric
$|g_{1}(x_{1})\rangle$ while the other DoFs are in a superposition of
coordinate transformations and the measurement is performed in some modified
basis:

$\displaystyle\langle\Omega|\hat{U}|\psi\rangle=$
$\displaystyle\>\>\>\beta\langle\Omega|\hat{U}|g_{1}(x)\rangle+\sum_{i}\alpha_{i}\langle\delta
x^{-1}\Omega|\hat{U}|g_{i}(x_{i})\rangle|\delta x^{-1}\phi\rangle.$ (41)

Let us now generalise the argument to the scenario where the state undergoes a
complete evolution through an “interferometric” setup (i.e. where the state of
the control or spacetime is measured in an appropriate basis where
interference can be witnessed). For an initial state of the spacetime and
matter DoFs (in the state $|\phi\rangle$) given by

$\displaystyle|\psi\rangle$
$\displaystyle=\left(\beta|g_{1}(x_{1})\rangle+\sum_{i\neq
1}\alpha_{i}|g_{i}(x_{i})\rangle\right)|\phi\rangle,$ (42)

one can always express the time-evolved state as some modified dynamics on a
classical background metric $g_{1}(x_{1})$:

$\displaystyle\hat{U}|\tilde{\psi}\rangle$
$\displaystyle=\hat{U}\left(\beta+\sum_{i\neq 1}\alpha_{i}\hat{U}(\delta
x_{i})\right)|g_{1}(x_{1})\rangle|\phi\rangle,$ (43)

where $\hat{U}(\delta
x_{i})|g_{1}(x_{1})\rangle|\phi\rangle=\hat{U}^{g}(\delta
x_{i})|g_{1}(x_{1})\rangle\otimes\hat{U}^{\phi}(\delta x)|\phi\rangle$ and we
have used the notation $|\psi\rangle\equiv|\tilde{\psi}\rangle$ as usual to
distinguish the different representations of the same physical situation.
“Measuring” the spacetime (i.e. the control) state in the modified basis given
by

$\displaystyle|\tilde{\eta}\rangle$
$\displaystyle=\left(\beta+\sum_{i}\alpha_{i}\hat{U}(\delta
x_{i})\right)|g_{1}(x_{1})\rangle$ (44)

yields,

$\displaystyle\langle\tilde{\eta}|\hat{U}|\psi\rangle$ $\displaystyle=\langle
g(x)|\left(|\beta|^{2}\hat{U}+\sum_{i,j\neq
1}\alpha_{i}^{\star}\alpha_{j}\hat{U}^{\dagger}(x_{i})\hat{U}\hat{U}(x_{j})\right)|g_{1}(x_{1})\rangle|\phi\rangle,$
$\displaystyle\equiv\left(|\beta|^{2}\hat{U}_{1}(x)+\sum_{i\neq
1}|\alpha_{i}|^{2}\hat{U}_{i}(x_{i})\right)|\phi\rangle.$ (45)

This is of course the same result one would obtain if preparing and measuring
the spacetime in a superposition of states that are mutually related to each
other via some diffeomorphism encoded within $\hat{U}_{ij}$. The
diffeomorphism invariance of the two scenarios shows that one can always map
such a superposition of geometries–something that is not currently encompassed
within the paradigms of general relativity and quantum theory–onto a single
spacetime metric with modified dynamics.

###  IV.3 Unruh-deWitt Model

In this section, we apply a specific interaction model to calculate the
coherence of a generic spacetime superposition coupled to a quantum field and
matter DoF modelled as a qubit. We utilise the Unruh-deWitt model (UdW) used
widely in relativistic quantum information and analogue gravity settings.

For simplicity, let us consider a superposition of two spacetime states
$|g_{1}\rangle$, $|g_{2}\rangle$, such that the initial (product) state of all
DoFs is given by
$|\psi\rangle=(|g_{1}\rangle+|g_{2}\rangle|\phi\rangle|0\rangle/\sqrt{2}$ as
utilised throughout Results. We work in the interaction picture, with the
interaction between all DoFs described by the Hamiltonian,

$\displaystyle\hat{H}_{\mathrm{int.}}$
$\displaystyle=\lambda\hat{\sigma}(\tau)\eta(\tau)\big{(}\hat{\phi}(x_{1})\otimes|g_{1}\rangle\langle
g_{1}|$ $\displaystyle\qquad+\hat{\phi}(x_{2})\otimes|g_{2}\rangle\langle
g_{2}|\big{)}.$ (46)

Here, $\lambda\ll 1$ is a weak coupling constant,
$\hat{\sigma}=|1\rangle\langle 0|e^{iE\tau}+\mathrm{H.c}$ is the SU(2) ladder
operator between the internal states of the qubit $\\{|0\rangle,|1\rangle\\}$,
$x_{1}$ and $x_{2}$ describe the coordinates parametrising the field operator
on the different branches of the superposition, and $\eta(\tau)$ is a time-
dependent switching function. Note that we have chosen to factor
$\hat{\sigma}(\tau)$, $\eta(\tau)$ from the control, which assumes that there
exists a common coordinate system describing the two spacetimes. One could,
however, describe the two spacetimes using different coordinate systems. The
time-evolution operator can be expanded in the Dyson series,

$\displaystyle\hat{U}$
$\displaystyle=\hat{I}+\hat{U}^{(1)}+\hat{U}^{(2)}+\mathcal{O}(\lambda^{3}),$
(47)

up to second-order in the weak coupling $\lambda$, where

$\displaystyle\hat{U}^{(1)}$
$\displaystyle=-i\int_{-\infty}^{+\infty}\mathrm{d}\tau\>\hat{H}_{\mathrm{int.}}(\tau),$
$\displaystyle\hat{U}^{(2)}$
$\displaystyle=\int_{-\infty}^{+\infty}\mathrm{d}\tau\int_{-\infty}^{\tau}\mathrm{d}\tau^{\prime}\>\hat{H}_{\mathrm{int.}}(\tau)\hat{H}_{\mathrm{int.}}(\tau^{\prime}),$
(48)

are the first- and second-order terms. Now,

$\displaystyle\hat{U}$ $\displaystyle=\hat{U}_{1}\otimes|g_{1}\rangle\langle
g_{1}|+\hat{U}_{2}\otimes|g_{2}\rangle\langle g_{2}|,$ (49)

where

$\displaystyle\hat{U}_{1}$
$\displaystyle=\hat{I}+\hat{U}_{1}^{(1)}+\hat{U}_{1}^{(2)}+\mathcal{O}(\lambda^{3}),\vphantom{\frac{1}{2}}$
$\displaystyle\hat{U}_{2}$
$\displaystyle=\hat{I}+\hat{U}_{2}^{(1)}+\hat{U}_{2}^{(2)}+\mathcal{O}(\lambda^{3}),\vphantom{\frac{1}{2}}$
(50)

are the components of the time-evolution that occur on $|g_{1}\rangle$,
$|g_{2}\rangle$ respectively. The time-evolved state is

$\displaystyle\hat{\rho}$
$\displaystyle=\hat{U}|\psi\rangle\langle\psi|\hat{U}^{\dagger},\vphantom{\frac{1}{2}}$
$\displaystyle=\frac{1}{2}\Big{(}\hat{U}_{1}|g_{1}\rangle\langle
g_{1}|\otimes\hat{\rho}_{\mathrm{FD}}\hat{U}_{1}^{\dagger}+\hat{U}_{1}|g_{1}\rangle\langle
g_{2}|\otimes\hat{\rho}_{\mathrm{FD}}\hat{U}_{2}^{\dagger}\vphantom{\frac{1}{2}}$
$\displaystyle+\hat{U}_{2}|g_{2}\rangle\langle
g_{1}|\otimes\hat{\rho}_{\mathrm{FD}}\hat{U}_{1}^{\dagger}\vphantom{\frac{1}{2}}+\hat{U}_{2}|g_{2}\rangle\langle
g_{2}|\otimes\hat{\rho}_{\mathrm{FD}}\hat{U}_{2}^{\dagger}\Big{)}\vphantom{\frac{1}{2}}$
(51)

where $\hat{\rho}_{\mathrm{FD}}$ is the density matrix of the field-detector
subsystem. The state of the spacetime after the interaction, and upon tracing
out the field and qubit DoFs is:

$\displaystyle\mathrm{Tr}_{\mathrm{FD}}\hat{\rho}$
$\displaystyle=\frac{|g_{1}\rangle\langle
g_{1}|}{2}+\frac{|g_{1}\rangle\langle g_{2}|}{2}$
$\displaystyle+\frac{|g_{2}\rangle\langle
g_{1}|}{2}\left(1+L_{12}-\frac{P_{1}+P_{2}}{2}\right)$
$\displaystyle+\frac{|g_{2}\rangle\langle
g_{2}|}{2}\left(1+L_{12}-\frac{P_{1}+P_{2}}{2}\right).$ (52)

Closing the superposition in the basis
$|\eta\rangle=(|g_{1}\rangle+\exp(i\phi)|g_{2}\rangle)/\sqrt{2}$ where $\phi$
is some controllable phase, we obtain

$\displaystyle\langle\eta|\mathrm{Tr}_{\mathrm{FM}}\hat{\rho}|\eta\rangle$
$\displaystyle\>\>\>=\frac{1}{2}\left(1+\left(1+L_{12}-\frac{P_{1}+P_{2}}{2}\right)\cos(\delta\phi)\right).$
(53)

The visibility is the coefficient of the $\cos(\delta\phi)$ term, namely

$\displaystyle V$ $\displaystyle=1+L_{12}-\frac{1}{2}\left(P_{1}+P_{2}\right)$
(54)

as stated in the main text. The integral forms of $P_{x}$ and $L_{12}$ are
given by

$\displaystyle P_{i}$
$\displaystyle=\lambda^{2}\int_{-\infty}^{+\infty}\mathrm{d}\tau\mathrm{d}\tau^{\prime}\chi(\tau)\bar{\chi}(\tau^{\prime})W(x_{i}(\tau),x_{i}^{\prime}(\tau^{\prime})),$
(55) $\displaystyle L_{ij}$
$\displaystyle=\lambda^{2}\int_{-\infty}^{+\infty}\mathrm{d}\tau\mathrm{d}\tau^{\prime}\>\chi(\tau)\bar{\chi}(\tau^{\prime})W(x_{i}(\tau),x_{j}(\tau^{\prime})),$
(56)

where $\chi(\tau)=\eta(\tau)\exp(-i\Omega\tau)$, and the Wightman functions
are two-point field correlation functions,

$\displaystyle W(x_{i}(\tau),x_{j}(\tau^{\prime}))$
$\displaystyle=\langle\phi|\hat{\phi}(x_{i})\hat{\phi}(x_{j}^{\prime})|\phi\rangle.$
(57)

The field operators in Eq. (57) are pulled back to the worldline of the qubit
as parametrised by the coordinates $x_{i}$, $x_{j}^{\prime}$, which in the
case $i\neq j$, describe the different spacetime metrics in superposition,
$|g_{i}\rangle$, $|g_{j}\rangle$.

###  IV.4 Dark Matter Scattering

In this section, we highlight the utility of our framework by applying it to
another case of decoherence, namely of general relativistic sources of gravity
(such as a dark matter clump) by scattering particles. This scenario has been
considered recently in the context of dark matter detection [18, 19]. The
authors of [18, 19] consider, in analogy with Eq. (21), a dark matter (DM)
“Schrödinger-cat state” that interacts with its environment $|\chi\rangle$,
leading to an entangled state of the DM and environment:

$\displaystyle|\psi\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|\mathrm{DM}_{1}\rangle|\chi_{1}\rangle+|\mathrm{DM}_{2}\rangle|\chi_{2}\rangle\right).$
(58)

The superposition of DM configurations is tacitly defined with respect to some
coordinate system through which relative distances between the involved
systems may be understood. We have demonstrated that the choice of relative
quantum coordinates affects whether decoherence appears to manifest or not.
Here, we emphasise that decoherence can be attributed to either of the
subsystems involved in the scattering process, implying that it cannot be
fundamentally associated with either subsystem. Instead, only physically
relevant DoF that decoheres in such a scenario is the relative distance
between (in this case) the dark matter and the scattering particles.

The rate of decoherence can be computed through the overlap of the scattering
particle states, $\langle\psi_{1}|\psi_{2}\rangle$ after interaction with the
DM states, $|\mathrm{DM}_{1}\rangle$, $|\mathrm{DM}_{2}\rangle$. The
computation of the overlap will in general, depend on the convolution of the
wavefunctions of the scattered states, given by [18],

$\displaystyle S_{12}$
$\displaystyle=\int\mathrm{d}^{2}b\int\mathrm{d}^{3}x\>\psi_{s1}^{\star}(t,x)\psi_{s2}(t,x),$
(59)

where $b$ is the impact parameter of the scattering process, while
$\psi_{s1,2}$ denotes the scattered part of the respective wavefunctions.
Importantly, Eq. (59) only depends on the wavepacket parameters and the
relative distance between the two branches of the DM superposition, which we
denote $\delta x_{12}$. Taking into consideration the relativity of the
subsystems (the DM distribution and the spacetime generated by it, and the
wavefunction of the scattered particle), this means that such a scenario is
physically equivalent to that of a single “fixed” DM source from which a
particle, in a superposition of relative distances from the source, is
scattered. The decoherence in this case is not fundamental to either the
scattering particle or the DM; both are equivalent situations that depend only
on one’s choice of coordinates.

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