# Causal orders, quantum circuits and spacetime: distinguishing between
definite and superposed causal orders

Nikola Paunković<EMAIL_ADDRESS>Instituto de
Telecomunicações and Departamento de Matemática, Instituto Superior Técnico,
Universidade de Lisboa, Avenida Rovisco Pais 1049-001, Lisboa, Portugal  Marko
Vojinović<EMAIL_ADDRESS>Institute of Physics, University of Belgrade,
Pregrevica 118, 11080 Belgrade, Serbia

###### Abstract

We study the notion of causal orders for the cases of (classical and quantum)
circuits and spacetime events. We show that every circuit can be immersed into
a classical spacetime, preserving the compatibility between the two causal
structures. Using the process matrix formalism, we analyse the realisations of
the quantum switch using 4 and 3 spacetime events in classical spacetimes with
fixed causal orders, and the realisation of a gravitational switch with only 2
spacetime events that features superpositions of different gravitational field
configurations and their respective causal orders. We show that the current
quantum switch experimental implementations do not feature superpositions of
causal orders between spacetime events, and that these superpositions can only
occur in the context of superposed gravitational fields. We also discuss a
recently introduced operational notion of an event, which does allow for
superpositions of respective causal orders in flat spacetime quantum switch
implementations. We construct two observables that can distinguish between the
quantum switch realisations in classical spacetimes, and gravitational switch
implementations in superposed spacetimes. Finally, we discuss our results in
the light of the modern relational approach to physics.

##  1 Introduction

The notion of causality is one of the most prominent in science, and also in
philosophy of Nature. Its treatment separates Aristotelian from the modern
physics, and its formal meaning within the latter is likely to have played a
significant role, over the past centuries since Galileo, in forming our
current everyday understanding of the notion of causality. While in Newtonian
physics the cause-effect relations were encompassed by a rather simple linear
and absolute time, Einstein’s analysis of causal relations was pivotal in the
formulation of the theory of relativity. But it was quantum mechanics (QM)
that, through the EPR argument [2], further formalised by Bell [3], showed how
quantum nonlocality, rooted in the superposition principle of QM,
revolutionised our everyday notion of causality. Finally, strong theoretical
evidence that, when combining the two fundamental theories of the modern
physics, one is to expect explicit dynamical nonlocal effects in quantum
gravity (QG), shows that our basic understanding of causality and causal
orders might be crucial in the development of new physics.

Recently, causal orders were, mainly within the quantum information community,
discussed in the context of controlled operations. In particular, it was
argued that the quantum switch, a specific controlled operation introduced in
[4], exhibits superpositions of causal orders, not only in the context of
quantised gravity, where genuine superpositions of different states of gravity
are present, but also in the experimental realisations performed in classical
spacetimes with fixed causal structure [5, 6, 7]. Note that the notion of
causal order discussed in these papers is different from the causal order of
the underlying spacetime structure. We discuss in detail the relation between
the two.

In this paper, we analyse the notion of causal orders in the context of
classical and quantum circuits, and relate it to the spacetime causal
structures. We prove that each circuit can be realised in a classical
spacetime, preserving the fixed causal relations of the former, with respect
to the causal relations between spacetime events of the latter (see the next
section for the details of the theorem). Further, we analyse possible
realisations of the quantum switch, showing that those performed in everyday
labs do not feature superpositions of causal orders between spacetime events
(consistent with our theorem), but rather standard non-relativistic quantum
mechanical (coherent) superpositions of different evolutions of a system. On
the other hand, we argue that genuine superpositions of different causal
orders are indeed to be expected within the QG scenario, where superpositions
of different states of the gravitational field, with their corresponding
causal orders, are manifestly allowed (Hardy was one of the first to discuss
the notion of superpositions of causal orders in the context of QG [8]). In
addition, we explicitly construct two distinct observables that can
distinguish between the realisations of the quantum switch in classical
spacetimes, and implementations of the gravitational switch in superposed
spacetimes. This way, we show that the two notions of causal orders, namely
one discussed in [5, 6, 7] and the other discussed in this paper, can be
experimentally distinguished, in contrast to the opposite claim present in the
literature [5]. Finally, we discuss our results in the context of the
relational approach to physics.

The layout of the paper is as follows. In Section 2, we introduce the notion
of causal order for circuits, and prove the Theorem of the circuit immersion
in classical spacetimes. Section 3 is devoted to the analysis of the quantum
switch implementations in classical spacetimes that do not feature
superpositions of spacetime causal orders, as well as implementations in the
context of QG. In Section 4, we compare the quantum switch implementations
discussed, and introduce observables that can distinguish between those that
feature superpositions of spacetime causal orders, and those that do not.
Section 5 is devoted to the discussion of the superpositions of causal orders
in the context of the relational approach to physics. Finally, in Section 6,
we present and discuss the results, provide some final remarks, and list
possible future research directions.

##  2 Causal orders

We begin by discussing circuits and their realisations in (classical)
spacetimes with well defined fixed causal orders. Given a directional acyclic
graph $G=(I,E)$, where $I$ is the set of graph nodes, and
$E=\\{(u,v)\,|\,u,v\in I\\}$ is the set of its directed edges (arrows pointing
from $u$ to $v$ representing the wires of the circuit), a circuit ${\cal C}$
over the set of operations ${\cal G}$ is a pair ${\cal C}=(G,g)$, where the
mapping $g:I\rightarrow{\cal G}$ assigns operations to each node. Depending on
the type of the operations from ${\cal G}$, we will call the circuit classical
(if the operations are, say, classical logic gates), or quantum (if the
operations are, say, unitaries, measurements, etc.).

The fact that $G$ is directional and acyclic allows one to define a partial
order $\prec_{I}$ over the set $I$ as

$u\prec_{I}v\quad\stackrel{{\scriptstyle\text{def}}}{{\Longleftrightarrow}}\hphantom{mmmmmmmm}$
$\Big{(}\exists\ n\in\mathbb{N}\ \wedge\ \\{u\equiv
u_{1},u_{2},\dots,u_{n}\equiv v\\}\subset I\Big{)}$ $\Big{(}\forall\
i\in\\{1,2,\dots,n\\!-\\!1\\}\Big{)}\ (u_{i},u_{i+1})\in E\,,$ (1)

representing the causal relation between the graph nodes. Next, we define the
set of gates of the circuit ${\cal C}$ as ${\cal G}_{\cal
C}=\\{g_{u}\equiv(u,g(u))\,|\,u\in I\\}$. The induced causal order between the
circuit gates $\prec_{{\cal C}}$ is by definition given as

$g_{u}\prec_{{\cal
C}}g_{v}\quad\stackrel{{\scriptstyle\text{def}}}{{\Longleftrightarrow}}\quad
u\prec_{I}v\,.$ (2)

Moreover, since there exists a canonical bijection between $I$ and ${\cal
G}_{\cal C}$, the order relations $\prec_{I}$ and $\prec_{{\cal C}}$ are
isomorphic.

Finally, we can introduce the set ${\cal M}$ of all spacetime events, which is
assumed to be a traditional $4D$ manifold. On this spacetime manifold we
assume to have a gravitational field, described in a standard way, using a
metric tensor $g_{\mu\nu}$. The metric is assumed to be of Minkowski
signature, such that the metric-induced light cone structure determines a
partial order relation between nearby events, denoted $\prec_{{\cal M}}^{g}$
(or simply $\prec_{{\cal M}}$ when the choice of the metric is implicit). Note
that the causal order over the spacetime events is not an intrinsic property
of the spacetime manifold itself, but rather determined by the metric, i.e.,
the configuration of the gravitational field living on the manifold.

One might pose a question if, given a formal circuit ${\cal C}$ with gates
${\cal G}_{\cal C}$, it is possible to realise it in a lab — if it is possible
to “immerse” it into spacetime. More precisely, given an arbitrary spacetime
manifold ${\cal M}$, our goal is to study if there exists an order-preserving
map ${\cal P}:{\cal G}_{\cal C}\to{\cal M}$, i.e., if the partial order
relations satisfy

$g_{u}\prec_{{\cal C}}g_{v}\;\;\Longrightarrow\;\;{\cal P}(g_{u})\prec_{{\cal
M}}{\cal P}(g_{v})\,,$ (3)

for every $g_{u},g_{v}\in{\cal G}_{\cal C}$. To that end, we formulate the
following theorem (the proof is given in Appendix A).

Theorem. Any circuit ${\cal C}$ can be immersed into a globally hyperbolic
spacetime manifold ${\cal M}$, such that its relation of partial order
$\prec_{{\cal C}}$ is preserved by the relation of spacetime events
$\prec_{{\cal M}}$.

Regarding the physical interpretation of the Theorem, note that it assigns a
spacetime point to each gate in a circuit, as opposed to a point in $3D$
space. Since each spatially localised apparatus may perform the same operation
more than once, at different moments in time, it may then correspond to
several different gates of the circuit, and thus several different nodes of
the graph, instead of just one. In other words, a single piece of experimental
equipment does not always correspond to a single gate of a circuit.

In addition to the above comment, note that in reality each operation actually
takes place in some finite volume of both space and time. However, in
theoretical arguments it is convenient to approximate this finite spacetime
volume with a single point, ignoring the size and time of activity of the
device performing the operation. We adopt this approximation throughout this
paper.

Circuits are seen as operations acting upon certain inputs to obtain the
corresponding outputs. Usually, the initial/final states (which include
instructions, measurement results, etc.) are depicted by the wires. But in our
approach, the input state is prepared by the “initial gate” ${\cal I}$, while
the output state is obtained by the “final gate” ${\cal F}$. This way, the
circuit ${\cal C}$ is seen as an operation ${\cal O}_{\cal C}$ acting from
${\cal I}$ to ${\cal F}$.

Note that, given a circuit ${\cal C}$, the corresponding overall operation
${\cal O}_{\cal C}$ (as well as the input and the output gates $\mathcal{I}$
and $\mathcal{F}$) is uniquely defined. The opposite is not the case: given
the operation ${\cal O}$, one can design different circuits ${\cal C},{\cal
C}^{\prime},\dots$ that achieve it. To see this, let us consider the simplest
case of the operation which satisfies ${\cal O}={\cal O}_{2}\circ{\cal
O}_{1}$, where $\circ$ represents the composition of operations. This
operation can be trivially achieved by the two circuits: (i) ${\cal C}$, which
consists of three nodes — node $i$ whose gate $\mathcal{I}$ prepares the input
state, node $o$ that applies the gate $g_{o}={\cal O}$, and node $f$ whose
gate $\mathcal{F}$ outputs either the quantum state, the classical outcome(s),
or the combination of the two; (ii) ${\cal C}_{12}$, which consists of four
nodes — nodes $i$ and $f$ that perform the same operations as before, and two
intermediate nodes $o_{1}$ and $o_{2}$ that perform $g_{o_{1}}={\cal O}_{1}$
and $g_{o_{2}}={\cal O}_{2}$, respectively. For simplicity, here and elsewhere
in the text, by ${\cal O}$ we denote both the operation and the gate that
implements it. The two situations are depicted in the following diagrams (see
Figure 1).

$i$$\mathcal{I}=g_{i}$$o$${\cal
O}=g_{o}$$f$$\mathcal{F}=g_{f}$$i$$\mathcal{I}$$o_{1}$${\cal
O}_{1}$$o_{2}$${\cal O}_{2}$$f$$\mathcal{F}$ Figure 1: Implementing operation
${\cal O}$ with a single gate (upper diagram), and by two consecutive gates
${\cal O}_{1}$ and ${\cal O}_{2}$ (lower diagram).

Finally, in recent literature one can find a notion of an event which is
different from the notion of a spacetime point [4, 5, 6, 7, 9, 10, 11, 12].
Namely, one can talk about events as interactions between the quantum system
under consideration and the apparatus in the lab. This is motivated by the
operational approach to physics, where the interactions between objects are
taken as fundamental. Then, one can introduce the relation of partial order,
which reflects the causal relationships between such events. Of course, in
general, this causal order does not need to coincide with the spacetime causal
order. Throughout this paper, if not explicitly stated otherwise, by causal
order we mean the order between the spacetime points, which due to our Theorem
can also be regarded as the order between the circuit gates. We discuss the
difference between the two notions of causal orders in Section 4.

##  3  Quantum switch

The most prominent feature of quantum systems is that they can be found in
coherent superpositions of states. This allows for applying the so-called
control operations. For simplicity, let us assume that operations ${\cal O}$
are unitaries, denoted as $U$. Given a control system $C$ in a superposition
${|{\varphi}\rangle}_{C}=a{|{0}\rangle}_{C}+b{|{1}\rangle}_{C}$ (with $\langle
0|1\rangle_{C}=0$), the control operation

$U_{CT}={|{0}\rangle}_{C}{\langle{0}|}\otimes
U_{0}+{|{1}\rangle}_{C}{\langle{1}|}\otimes U_{1}$ (4)

transforms the initial product state
${|{\Psi_{i}}\rangle}_{CT}={|{\varphi}\rangle}_{C}\otimes{|{\psi_{i}}\rangle}_{T}$
between the control and the target systems into the final entangled state
${|{\Psi_{f}}\rangle}_{CT}=a{|{0}\rangle}_{C}\otimes
U_{0}{|{\psi_{i}}\rangle}_{T}+b{|{1}\rangle}_{C}\otimes
U_{1}{|{\psi_{i}}\rangle}_{T}$. A simple realisation of such operation by a
circuit consisting of three gates is shown below (see Figure 2).

$i$$\mathcal{I}$$o$$\quad U_{CT}$$f$$\mathcal{F}$$w_{C}$$w_{T}$$w_{C}$$w_{T}$
Figure 2: Controlled operation $U_{CT}$. Applying operation $U_{b}$ on a
system in the wire $w_{T}$ controlled by the state ${|{b}\rangle}$ on a system
in the wire $w_{C}$, with $b=0,1$.

Here, the first node and the corresponding gate prepares the initial
superposition of the control system, the second implements $U_{CT}$, and the
third is either an identity, a measurement on the two systems, or a
combination (say, a measurement of the target qubit, while leaving the control
intact). In order to allow for the description of quantum superpositions, we
introduce the notion of a vacuum in the analysis of quantum circuits, as is
done for example in [13] (for technical details, see Appendix B).

As noted above, given the operation, many different circuits can achieve it.
Indeed, in standard optical implementations of the above controlled operation
(4), the control qubit is spanned by two spatial modes of a photon, while the
target one is its polarisation degree of freedom. The initial superposition
state of the control qubit is prepared by a beam splitter, while the two
operations $U_{0}$ and $U_{1}$ are implemented locally in Alice’s and Bob’s
laboratories. Note that, since the control qubit is achieved by the means of
two spatial modes of a single photon, while the target qubit is, being the
photon’s polarisation, “attached to” the control, the target is formally
achieved by two degrees of freedom (two wires), one assigned to Alice
($T_{A}$), and the other to Bob ($T_{B}$). Thus, in such a realization, the
control degree of freedom is redundant in the circuit diagram and can be
omitted. Nevertheless, since we will later discuss the case of the
gravitational quantum switch, in which the gravitational degree of freedom
plays the role of the control, here we keep its corresponding wire and gate in
the diagram, as presented below (see Figure 3).

$i$$\mathcal{I}$$a$$U_{0}$$b$$U_{1}$$t_{A}$$T_{A}$$t_{B}$$T_{B}$$c_{f}$$T_{C}$$w_{T_{A}}$$w_{T_{B}}$$C$$\mathcal{F}$
Figure 3: Implementation of the controlled operation using the spatial degree
of freedom as a control.

The final gate $\mathcal{F}$ consists of three “elementary gates”, represented
by the circuit nodes $t_{A}$ and $t_{B}$ for the two target wires, and the
node $c_{f}$ for the final control wire.

An important instance of controlled operations is the so-called quantum
switch, for which the two controlled operations are given by $U_{0}=UV$ and
$U_{1}=VU$, where $U$ and $V$ are two arbitrary unitaries [4]. Having two
pairs of equipment, one applying $U$ and the other $V$, it is straightforward
to implement the quantum switch through the circuit similar to the one above,
which instead of two gates, one in the node $a$ applying $U_{0}$, and another
in node $b$ applying $U_{1}$, contains four gates placed in the nodes $a_{U}$,
$a_{V}$, $b_{V}$ and $b_{U}$ (see Figure 4).

$i$$\mathcal{I}$$a_{U}$$U$$b_{V}$$V$$a_{V}$$V$$b_{U}$$U$$t_{A}$$T_{A}$$t_{B}$$T_{B}$$c_{f}$$T_{C}$$w_{T_{A}}$$w_{T_{B}}$$C$$\mathcal{F}$
Figure 4: The quantum switch.

The question arises, is it possible to achieve the same using less resources,
say, using only two such pieces of equipment, located in two different points
(regions) of 3D space? Indeed, it is possible to do so, and recently a number
of implementations of the quantum switch were performed in flat Minkowski
spacetime [5, 6, 7]. Nevertheless, such implementations still correspond to
circuits that implement $U_{0}$ and $U_{1}$ by four, rather than two gates.
The difference is that, when immersing it in a flat spacetime, the two pairs
of gates are now distinguished only by the temporal, rather than all four
spacetime coordinates. Thus, one cannot talk of superpositions of causal
orders between spacetime events in such implementations, as flat (indeed, any
globally hyperbolic) spacetime has a manifestly fixed causal order. To
implement $U_{0}$ and $U_{1}$ of the quantum switch by a circuit that consists
of two gates only (and thus two corresponding spacetime points), one needs a
superposition of gravitational fields with different (incompatible) causal
orders. In the following two subsections, we analyse in more detail the
“4-event” and the “3-event” implementations of the quantum switch, while the
“2-event” case is discussed in the last subsection (the numbers 4, 3 and 2
refer to the numbers of spacetime events corresponding to distinct gates used
to achieve $U_{0}$ and $U_{1}$). A detailed mathematical description using the
process matrix formalism [9], is presented in the Appendices C, D and E.

Following the previously mentioned distinction between the spacetime event and
the operational notion of the event, the $4$-event and $3$-event quantum
switch implementations will have a description within the operational approach
that is different from the spacetime description. In particular, in such
approach these two implementations of quantum switch would feature only $2$
operationally defined events, and thus the superposition of the corresponding
causal orders.

###  3.1 4-event process

The realisations of the quantum switch are performed in table-top experiments
in the gravitational field of the Earth, and can be for all practical purposes
considered as being performed in flat Minkowski spacetime. In such
experiments, Alice performs the unitary $U$ in her localised laboratory, and
Bob performs $V$ in his separate localised laboratory, such that both are
stationary with respect to each other and the Earth. The operations are
applied on a single particle that arrives from the beam splitter, in a
superposition of trajectories towards Alice and Bob, and, upon the exchange
between the two agents, is finally recombined on the same beam splitter (for
simplicity, we chose one beam splitter, but the whole analysis equally holds
for two spatially separated beam splitters), and then measured. Below, we
present a spacetime diagram of this experimental realisation of the quantum
switch, which also represents a circuit of the implementation scheme (see
Figure 5).

spacetime$t_{i}$$t_{1}$$t_{2}$$t_{f}$AliceBob$\begin{subarray}{c}\text{\tiny
beam}\\\ \text{\tiny splitter}\end{subarray}$${\cal I}$${\cal
F}$$A$$B^{\prime}$$B$$A^{\prime}$ Figure 5: Spacetime diagram, as well as the
circuit representation, of the $4$-event implementation of the quantum switch.

Black horizontal lines represent world lines for Alice and Bob, as well as the
global time coordinate line at the bottom. The black vertical line represents
global space coordinate line. Quantum gates are represented by big dots. The
composite gate $\mathcal{I}$ consists of the two preparation gates and the
initial beam splitter gate, while $\mathcal{F}$ consists of the final beam
splitter gate and the target gates that perform the final measurements (for
details, see Appendix C). For simplicity, from now on we omit writing the
labels of the nodes and keep only the labels of the corresponding circuit
gates. The two histories of the particle exchanged between Alice and Bob,
representing Alice’s and Bob’s wires, are full lines coloured in blue and red,
respectively.

From the diagram we can see that in the blue history we have the following
chain of gates

$\mathcal{I}\prec_{{\cal C}}A\prec_{{\cal C}}B^{\prime}\prec_{{\cal
C}}\mathcal{F}\,,$ (5)

while for the red history we have

$\mathcal{I}\prec_{{\cal C}}B\prec_{{\cal C}}A^{\prime}\prec_{{\cal
C}}\mathcal{F}\,.$ (6)

In total, there are four spacetime events involving Alice’s and Bob’s actions
on the particle (gates), namely $A$, $B$, $A^{\prime}$ and $B^{\prime}$. Thus,
we call the above diagram the “$4$-event diagram”. This setup was already
discussed in the literature (see the very end of the Supplementary Notes of
[14]).

In order to compare the cases of the quantum and the gravitational switches,
it would be interesting to analyse the two examples within recently introduced
powerful process matrix formalism [9]. To do so, one needs to formulate the
formalism involving the vacuum state (see Appendix B for details). The
straightforward application of the formalism to the $4$-event case is in full
accord with the experimental results, as demonstrated in Appendix C.

###  3.2 3-event process

One can imagine that instead of two, one of the agents implements only one
gate. For example, by conveniently choosing the velocity of the particle along
its trajectory between Alice and Bob, we can identify Bob’s two gates,

$B\equiv B^{\prime}\,.$ (7)

We thus arrive to the new spacetime diagram and the associated circuit, called
the “$3$-event diagram” (see Figure 6).

spacetime$t_{i}$$t_{1}$$t_{B}$$t_{2}$$t_{f}$AliceBob$\begin{subarray}{c}\text{\tiny
beam}\\\ \text{\tiny splitter}\end{subarray}$${\cal I}$${\cal
F}$$A$$B$$A^{\prime}$ Figure 6: Spacetime diagram, as well as the circuit
representation, of the $3$-event implementation of the quantum switch.

Now, the obvious question is the following — can we, in addition to (7),
impose also that

$A\equiv A^{\prime}\,,$ (8)

i.e., also identify Alice’s gates into a single spacetime event? In flat
Minkowski spacetime, the answer is negative. Namely, by simply looking at the
$3$-event diagram one can see that the trajectory of the particle between
Alice and Bob would require either superluminal speed, or backwards-in-time
trajectory in at least one history (note that the diagram assumes that light
propagates along the lines that form the $45^{\circ}$ angle with the
coordinate axes). This is also seen directly from inequalities (5) and (6):
identifying both $A\equiv A^{\prime}$ and $B\equiv B^{\prime}$ would lead to
requiring that both $A\prec_{{\cal C}}B$ and $B\prec_{{\cal C}}A$ are
satisfied, i.e., $A\equiv B$. As guaranteed by our Theorem from Section 2, in
a curved spacetime it is also impossible to make both identifications (7) and
(8), at least if spacetime were globally hyperbolic. Finally, as in the
$4$-event case, here also the process matrix formalism is consistent with the
experimental results, see Appendix D.

###  3.3  2-event process — gravitational switch

Despite the conclusion of the previous subsection, within the framework of
quantum gravity one is allowed to construct superpositions of different
gravitational field configurations, leading to superpositions of different
causal structures for the spacetime manifold. The assumption of superpositions
of different gravitational field configurations is common to all models of QG.
Other than that, we will not have any additional assumptions, and thus our
approach does not depend on any particular QG model.

In what follows, for the sake of concreteness, we assume the “traditional”
approach to the formulation of the QG formalism. Namely, we assume that there
exists a smooth $4D$ manifold, called spacetime, and denoted as ${\cal M}$.
Quantum fields, including the gravitational field, live on top of ${\cal M}$.
The gravitational field is described either via the metric or via some other
degrees of freedom (for example, tetrads and spin connection), such that the
metric is a function of these. We call this kind of construction “traditional”
because it represents a minimal deviation from the mathematical structure of
quantum field theory (QFT) in flat Minkowski spacetime, in the sense of
preserving the underlying manifold structure. A QG model implementing this
approach is, for example, the asymptotic safety framework [15]. Of course, we
do not aim to provide a full-fledged model of QG, but rather to only specify
the status of the manifold structure within it. As an alternative, in
Subsection 5.2, we will discuss the relational framework of QG in which the
manifold structure does not exist a priori, but is emergent from relational
properties of quantum fields themselves. Finally, note that the discussion of
the flat-spacetime cases in the previous sections implicitly assumes the
traditional point of view on spacetime manifold. Nevertheless, it has to be
compatible with the semiclassical limit of any viable QG model.

As a consequence of the superposition of causal structures in QG, it is
possible to achieve a gravitational switch, which implements the same quantum
switch as described above, with a circuit consisting (in addition to the
initial and final gates $\mathcal{I}$ and $\mathcal{F}$) of only two gates:
the Alice’s gate $A$ that applies $U$, and Bob’s gate $B$ that applies $V$.
Superposing two gravity-matter states, such that in the first the spacetime
geometry (described by the metric tensor $g_{0}$) establishes the causal
structure

$\mathcal{I}\prec_{{\cal M}}^{g_{0}}A\prec_{{\cal M}}^{g_{0}}B\prec_{{\cal
M}}^{g_{0}}\mathcal{F}\,,$ (9)

while in the second (described by the metric $g_{1}$) it is

$\mathcal{I}\prec_{{\cal M}}^{g_{1}}B\prec_{{\cal M}}^{g_{1}}A\prec_{{\cal
M}}^{g_{1}}\mathcal{F}\,,$ (10)

the overall circuit applies operations $U_{0}=UV$ and $U_{1}=VU$, conditioned
on the state of gravity. As a side note, it is clear from (9) and (10) that
superpositions of the spacetime causal orders can occur only in the framework
of quantum gravity.

Such a switch was previously introduced by Zych et al. [16], in the context of
two spacetimes which are solutions of the Einstein equations. In their
proposal, the beam splitter acted only on the gravitational degree of freedom
(and the accompanied source, the planet), while leaving the rest of the
matter, in particular the particle, Alice and Bob, unaffected. Upon the final
beam splitter recombination, the matter is left in an incoherent mixture of
two states proportional to $\,\\{\,U\,,\,V\,\\}\,{|{\Psi}\rangle}$ and
$\,[\,U\,,\,V\,]\,{|{\Psi}\rangle}$. Subsequently, the mass (along with its
gravitational degrees of freedom) is being measured in the superposition
basis. Upon a post-selection conditioned on the outcome of the measurement,
the matter is again in a pure state.

Another way to obtain a genuine superposition of two different causal orders
is by using a spatially delocalised beam splitter, that acts on both
gravitational and matter fields. This can be depicted by the following
$2$-event diagram (see Figure 7).

particleAliceBob$A$$B$$\mathcal{I}$$\mathcal{F}$ Figure 7: Spacetime diagram
of the $2$-event implementation of the gravitational switch. Note that
formally this is not a circuit diagram, as the control wire, implemented by
the state of the gravitational field in the yellow region, is missing.

The yellow region in this diagram represents a compact piece of spacetime
where the gravitational field is in a superposition of the two distinct
states, and plays the role of the control degree of freedom. Along the
boundary of that region, both gravitational configurations smoothly join into
a single configuration outside. The boundary of the yellow region thus acts as
a beam splitter for anything that enters, and again (in the recombining role)
for anything that exits. Therefore, all worldlines (namely, of Alice, Bob and
the particle) are doubled inside the yellow region. The blue and red colours
represent their spacetime trajectories in two different gravitational field
backgrounds, respectively.

We model our gravitational switch such that the overall output state is the
product between the state of the gravitational field and the state of the
particle. The state of the particle is of the form $(\alpha UV+\beta
VU){|{\Psi}\rangle}$, obtained without performing final selective measurement.
In particular, in order to compare it with the other quantum switch
realizations, we choose either $\,\\{\,U\,,\,V\,\\}\,{|{\Psi}\rangle}$ or
$\,[\,U\,,\,V\,]\,{|{\Psi}\rangle}$. In order to achieve this, the
gravitational switch should act upon all degrees of freedom, both
gravitational and matter. Note that our gravitational switch does require
certain fine tuning, in the sense that the whole, delocalised beam splitter,
that acts nontrivially on the whole joint gravity-matter system, is designed
for the particular pair of operations applied by Alice and Bob: only for those
operations, the beam splitter will output the product state between gravity
and matter. Otherwise, the output will be the entangled gravity-matter state,
like in the cases of the optical quantum switch and the gravitational switch
introduced by Zych et al. (before the final selective measurement). Still, the
process matrix describing the gravitational switch itself is independent of
the choice of the gate operations of the agents. See Appendix E for details.

The question whether this kind of diagram is admissible in some theory of
quantum gravity is nontrivial, and model dependent, on several grounds. First,
it is impossible to construct this diagram by superposing two classical
configurations of gravitational field, such that each configuration satisfies
Einstein equations. The reason is simple — assuming that the gravitational
field is specified outside the yellow region, Einstein equations have a unique
solution (up to diffeomorphism symmetry) for the compact yellow region, given
such a boundary condition. Therefore, one cannot have two different solutions
to superpose inside. The only two options are to either superpose one on-shell
and one off-shell configuration of gravity, or two off-shell configurations.
This scenario can arguably be considered within the path integral framework
for quantum gravity.

Second, the question of the particle trajectory is nontrivial. Namely, given
one gravitational configuration in which the particle has the spacetime causal
structure (9), corresponding to the blue history, it is not obvious that there
can exist another gravitational configuration (with the same boundary
conditions at the edge of the yellow region), in which the particle has the
spacetime causal structure (10), corresponding to the red history. Even if one
admits arbitrary off-shell configurations of gravity, it may turn out that the
order of events inside the yellow region must be fixed by the boundary
conditions. The only viable way to answer this question is to try and
construct an explicit example of two geometries implementing (9) and (10) for
the same boundary conditions. Numerical investigations are underway to explore
this possibility.

##  4 Distinguishing 2-, 3-, and 4-event realisations of the quantum switch

In a number of both theoretical proposals [4, 9, 10, 11, 12], as well as
experimental realisations [5, 6, 7] of the quantum switch, it is claimed that
they feature genuine superpositions of causal orders. The reason for this is
the introduction of an alternative, operational notion of the event, which
differs from a spacetime point. The motivation for this lies in the claim that
the individual spacetime points $A$ and $A^{\prime}$ (and $B$ and
$B^{\prime}$) do not have an operational meaning. In words of the authors of
[5] (see the Discussion section):

> “The results of the experiment confirm that such [which way] information is
> not available anywhere and that the interpretation of the experiment in
> terms of four, causally-ordered events cannot be given any operational
> meaning. If, on the other hand, one requires events to be defined
> operationally, in terms of measurable interactions with physical systems
> […], then the experiment should be described in terms of only two events — a
> single use of each of the two gates.”

While it is obvious that the mentioned which-way information is not available
in the quantum switch experiment, in what follows we argue that this does not
imply that one cannot give an operational meaning to spacetime points, even in
the context of the quantum switch in classical geometries.

Below, we first present a critical analysis of the arguments behind
introducing the operational notion of event. Then, we show how one can
experimentally, at least in principle, distinguish 2-, 3-, and 4-event
realisations of the quantum switch.

It is the operational approach to understanding spacetime, applied within the
framework of relationalism (see Section 5 for a detailed discussion of the
relation between the two frameworks), that is arguably the main argument for
introducing the alternative notion of an event. This new notion of an event
gives rise to the superposition of respective causal orders in the
realisations of the quantum switch even in classical spacetimes. Assuming that
the smooth (classical) spacetime is an emergent phenomenon, in the operational
approach one considers “closed laboratories” [9] as the primal entities within
which one can locally apply standard quantum mechanics, while their
connections form the relations from which the spacetime emerges. Indeed, it
seems that the process matrix formalism was developed precisely with this idea
in mind: to be a mathematical tool in analysing the emergence of the spacetime
through the relations between the closed laboratories. We would like to note
that, as shown in Appendices C, D and E, the mentioned formalism is also fully
applicable within the standard formulation of quantum mechanics in classical
Minkowski spacetime.

Given that in the case of coherent superpositions of the two paths (a particle
first goes to Alice, then to Bob, and vice versa) it is not possible to know
which of the two has actually been taken, one may conclude that one cannot
distinguish between spacetime events $A$ and $A^{\prime}$, and that the two
are operationally given by the single action of a spatially localised
laboratory. However, this point of view is at odds with our understanding of
the ordinary double slit experiment. Namely, by exchanging the roles of time
and space, and following the above logic, applied to the case of the standard
double slit experiment, one could analogously conclude that, since in order to
obtain the interference pattern at the screen one must not (and thus cannot)
learn which slit the particle went through, the two slits represent one and
the same operational “lab”, and one operational point (region) in space.

Let us explain our argument in slightly more detail. Consider first the
optical quantum switch. Here, a particle passes through Alice’s lab, described
by the two spacetime points, $(x_{A},t)$ and $(x_{A},t^{\prime})$. Any attempt
to distinguish the times $t$ and $t^{\prime}$ at which the particle passes
through Alice’s lab would destroy the superposition. Consider now the standard
double slit experiment. Here, a particle passes through the two slits,
described by the two spacetime points, $(x_{L},t)$ and $(x_{R},t)$. Any
attempt to distinguish the positions of the slits $x_{L}$ and $x_{R}$ through
which the particle passes would destroy the superposition. Note that by
exchanging the roles of space and time, the descriptions of the above two
situations are essentially identical.

According to the operational approach, as a consequence of the above, one
should describe Alice’s actions in the optical quantum switch in terms of only
one operational event. Thus, analogously, one should also describe the
particle passing through the slits in terms of only one operational event.
However, such interpretation of the double slit experiment is, to the best of
our knowledge, absent from the literature.

Note also that the 3-event realisation of the quantum switch offers a natural
alternative interpretation of this phenomenon, as a well known time double
slit experiment [17]. Indeed, the two events (gates) $A$ and $A^{\prime}$ play
the role of the two time-like slits, while the event (gate) B separates the
two in the same way the closed shutter separates the two time-like slits in
the time double slit experiment. This comes as no surprise: quantum
superpositions are in general accompanied by the interference effects, and the
quantum switch is, as already emphasised in Section 3, just another instance
of a superposition of two different states of the standard quantum mechanics
in Minkowski spacetime.

The operational interpretation of identifying the events $A$ and $A^{\prime}$
in the current experimental realisations of the quantum switch indeed seems to
be a tempting proposal. Nevertheless, we would like to point out that in fact
it does not resolve any open problem. In addition, being similar to Mach’s
ideas, it too may be at odds with the theory of general relativity (GR), see
Subsection 5.1 for a detailed discussion.

###  4.1 Distinguishing by decohering the particle

In the above quote from [5], the authors claim that in order to directly
distinguish points $A$ and $A^{\prime}$ (as well as $B$ and $B^{\prime}$), one
must destroy the superposition in the apparatus. Conversely, being unable to
distinguish those points in any experiment that maintains superposition and
realises the quantum switch, one cannot give them operational meaning.
Therefore, those spacetime points are redundant in the theory, and each pair
should be replaced by a single operational event. In this subsection, we
discuss this type of an argument. In the next, we give an explicit example of
an observable that does distinguish such spacetime points without obtaining
the which way information.

Let us study one concrete way of distinguishing the mentioned pairs of points,
which decoheres the particle. For simplicity, we will analyse the 4- and
2-event cases only. To this end, we will introduce a third agent, Alice’s and
Bob’s Friend. At each run of the quantum switch experiment, Alice will,
independently and at random, decide whether just to apply her operation onto
the particle, or in addition to that, send a photon to Friend. The same holds
for Bob. In 25% of the cases, both agents just perform their respective
operations, thus performing the quantum switch. Next, in the 25% of the cases,
both agents decide, in addition to applying their respective operations, to
send the photons to Friend, who detects them in his spatially localised lab.
The remaining 50% of the cases are essentially the same as the previous ones,
so for simplicity we omit their analysis.

First, we present the spacetime diagram of the 4-event quantum switch for the
case when the agents decide to send the photons to Friend (see Figure 8).

spacetime$t_{i}$$t_{1}$$t_{2}$$t_{f}$FriendAliceBob$\mathcal{I}$$\mathcal{F}$$A$$B^{\prime}$$F_{A}$$F_{B^{\prime}}$$B$$A^{\prime}$$F_{B}$$F_{A^{\prime}}$
Figure 8: Distinguishing spacetime points by decohering the particle in the
$4$-event quantum switch. The dotted (dashed) lines represent photons sent by
Alice (Bob) to Friend.

The photons coming from Alice are dotted, while the photons coming from Bob
are dashed. By knowing the geometry of the whole experiment, Friend would be
able to measure, in a generic setup, four different times of the photon
arrivals: $t_{A}$ and $t_{A^{\prime}}$ for spacetime points $F_{A}$ and
$F_{A^{\prime}}$, and two more for the photons sent by Bob.

On the other hand, in the case of the 2-event gravitational switch
realisation, Friend would detect only two times of the photons’ arrival.
Below, we extend the diagram of the gravitational switch we introduced in
Section 3.3, by adding the photons sent to Friend. In order to indicate the
fact that the events $A$ and $A^{\prime}$, etc., are in this setup indeed
identified, we write the tilde over the corresponding letters $A$, $B$ and $F$
(see Figure 9).

Friend
particleAliceBob$\tilde{A}$$\tilde{B}$$\tilde{F}_{A}$$\tilde{F}_{B}$$\mathcal{I}$$\mathcal{F}$
Figure 9: Distinguishing spacetime points by decohering the particle in the
$2$-event gravitational switch. The dotted (dashed) lines represent photons
sent by Alice (Bob) to Friend.

Clearly, the two situations are experimentally distinguishable.

Nevertheless, as noted in [5], one might argue that, since the photons sent to
Friend in the $4$-event case decohere the particle in the switch, this
situation does not correspond to the experiment in which the coherence is
maintained. Therefore, in the latter, the pair of spacetime events $A$ and
$A^{\prime}$ still ought to be substituted with a single operational event
(and analogously for $B$ and $B^{\prime}$).

However, even if instead of spacetime points one decides to talk about
operational events, such a framework should still be consistent with the
experimentally tested theories, GR in particular. According to GR, in flat
spacetime (or in any classical configuration of the gravitational field),
regardless of whether we decohere the particle or not, both experiments
feature four spacetime points, such that $A$ and $B$ (as well as $A^{\prime}$
and $B^{\prime}$) can be considered to be simultaneous (see Figure 8).
Therefore, the time of execution of both experiments is $\delta
t=t_{2}-t_{1}+C$, where $C\equiv(t_{1}-t_{i})+(t_{f}-t_{2})$. Note that the
time period $t_{2}-t_{1}$ represents the travel time of the particle from one
laboratory to the other, and is therefore strictly positive.

From the operational point of view, the decohered version of the experiment
also features four operational events, and is thus manifestly consistent with
the GR description. Note that a decohered version of the switch still features
only two events per run: in a classical mixture between “Alice’s event before
Bob’s event” and “Bob’s event before Alice’s event” each run features just two
events, and the duration of the overall experiment in each run is the time
between the two events of that run (plus the above constant $C$).

On the other hand, if the coherence were maintained, the operational point of
view features only two operational events, one per laboratory. Then, the total
time of execution of the experiment ought to be $\delta\tau=0+C$, which is
clearly different from the GR prediction. The total time of execution of the
quantum switch experiment is a measurable quantity. This means that one can
easily determine whether this time is $\delta\tau$ or $\delta t$. The former
outcome invalidates GR, which would necessitate the formulation of an
alternative theory. Note that in this case, a sheer decision to either
decohere a particle or not would allow agents to influence the time flow in
their labs. Moreover, it raises the question of the time flow in nearby labs
isolated from the experiment during its execution. The latter outcome poses
the problem of the precise formulation of an operational theory such that the
experiment which features only two operational events lasts precisely the same
time as the experiment which features four operational events.

###  4.2 Distinguishing without decohering the particle

In addition to the above argument, supported by the experimental setup
presented in the previous subsection, by erasing the which way information it
is possible for Friend to distinguish the 4-event and the 2-event realisations
even when the “full” quantum switch is executed. For that, one needs to supply
Friend with a photon non-demolition measurement. This is in principle possible
to construct, although in practice a bit challenging. It thus might be
technically easier to use some particles other than photons for sending
signals to Friend.

By agreeing in advance of the particular experimental setup, Friend would be
able to predict the distinct times of arrival of the photons, $t_{F_{A}}$,
$t_{F_{A^{\prime}}}$, $t_{F_{B}}$ and $t_{F_{B^{\prime}}}$ in the 4-event
case, and $t_{\tilde{F}_{A}}$, $t_{\tilde{F}_{B}}$ in the 2-event case, thus
defining the states of the two photons that arrive to his lab:
${|{F_{A},F_{B^{\prime}}}\rangle}$ , ${|{F_{A^{\prime}},F_{B}}\rangle}$, and
${|{\tilde{F}_{A},\tilde{F}_{B}}\rangle}$, respectively. Let us define
$\mathcal{H}_{A\prec B^{\prime}}=\mathop{\rm
span}\nolimits\\{{|{F_{A},F_{B^{\prime}}}\rangle}\\}$, $\mathcal{H}_{B\prec
A^{\prime}}=\mathop{\rm span}\nolimits\\{{|{F_{A^{\prime}},F_{B}}\rangle}\\}$,
and $\mathcal{H}_{A\prec B\land B\prec A}=\mathop{\rm
span}\nolimits\\{{|{\tilde{F}_{A},\tilde{F}_{B}}\rangle}\\}$. Then, the
relevant Hilbert space of the two photons is

$\begin{array}[]{rl}\mathcal{H}_{ph}=\mathcal{H}_{A\prec
B^{\prime}}\oplus\mathcal{H}_{B\prec A^{\prime}}\oplus\mathcal{H}_{A\prec
B\land B\prec A}\,.\end{array}$ (11)

Let us define $P_{<}$, $P_{>}$ and $P_{=}$ as orthogonal projectors onto
$\mathcal{H}_{A\prec B^{\prime}}$, $\mathcal{H}_{B\prec A^{\prime}}$ and
$\mathcal{H}_{A\prec B\land B\prec A}$, respectively. One can then define a
dichotomic photon non-demolition orthogonal observable performed by Friend on
the two photons in his laboratory:

$M=1\cdot(P_{<}+P_{>})+0\cdot P_{=}\,.$ (12)

Provided that the experimental setup is either that of the 4-event, or the
2-event type, such measurement would not change the state of the experimental
setup (the interferometer, the particle in it, and the photons in the Friend’s
apparatus), while still leaking the information to Friend (via the measurement
outcome) about the type of the quantum switch realisation. Finally, by
performing the quantum erasing procedure [18, 19], the which way information
is lost, and the final state of the particle is restored to a coherent
superposition.

Let us examine this more formally. Let the two states of the particle in the
quantum switch be ${|{R}\rangle}$ and ${|{B}\rangle}$, corresponding to the
red and the blue trajectory, respectively. After $\mathcal{I}$, the state of
the particle in the quantum switch is
$\frac{1}{\sqrt{2}}({|{R}\rangle}+{|{B}\rangle})$. As the particle passes
through Alice’s and Bob’s labs, two photons are emitted, which arrive at the
Friend’s lab. The overall state of the particle and the two photons in the
2-event quantum switch is then

$\frac{1}{\sqrt{2}}\Big{(}{|{R}\rangle}+{|{B}\rangle}\Big{)}{|{\tilde{F}_{A},\tilde{F}_{B}}\rangle}.$
(13)

The particle in the quantum switch is in superposition of the two paths, and
it stays so upon measuring $M$ and obtaining the result 0.

On the other hand, the overall state of the particle and the two photons in
the 4-event quantum switch is, upon the photons’ arrival in the Friend’s lab,
given by

$\frac{1}{\sqrt{2}}\Big{(}{|{R}\rangle}{|{F_{A^{\prime}},F_{B}}\rangle}+{|{B}\rangle}{|{F_{A},F_{B^{\prime}}}\rangle}\Big{)}\hphantom{aaaaaa}$
(14)
$\hphantom{aa}=\frac{1}{2\sqrt{2}}\Big{[}\Big{(}{|{R}\rangle}+{|{B}\rangle}\Big{)}\Big{(}{|{F_{A^{\prime}},F_{B}}\rangle}+{|{F_{A},F_{B^{\prime}}}\rangle}\Big{)}$
$\hphantom{aaaa}+\Big{(}{|{R}\rangle}-{|{B}\rangle}\Big{)}\Big{(}{|{F_{A^{\prime}},F_{B}}\rangle}-{|{F_{A},F_{B^{\prime}}}\rangle}\Big{)}\Big{]}.$

The particle is now decohered by the two photons, and it remains so upon
measuring $M$ and obtaining 1 as the result. Therefore, to erase the which way
information, Friend has to perform an additional measurement in the basis

${|{\pm}\rangle}=\frac{1}{\sqrt{2}}\Big{(}{|{F_{A^{\prime}},F_{B}}\rangle}\pm{|{F_{A},F_{B^{\prime}}}\rangle}\Big{)},$
(15)

thus collapsing the state of the particle in one of the two pure states

$\frac{1}{\sqrt{2}}\Big{(}{|{R}\rangle}\pm{|{B}\rangle}\Big{)}.$ (16)

Knowing the outcome of the measurement of $M$, Friend can post-select the
output of the particle coming out of the quantum switch. Alternatively, in the
case of obtaining the ${|{-}\rangle}$ result, Friend can change the relative
phase between the two of the particle’s superposed states.

###  4.3 Other types of gravitational switches

It is important to note that the framework of QG also allows for the
construction of $3$\- and $4$-event switches, in addition to the $2$-event
one. This is straightforward to see, for example by immersing the above $3$\-
or $4$-event spacetime diagram into a superposition of different geometries.

Moreover, all of these gravitational switches may give different outcomes when
measuring the observable $M$, given by (12), followed by the quantum erasing
procedure (15). The criteria to necessarily obtain the outcome $0$ are: (i)
that the photons in red and blue histories meet at the boundary of the yellow
region, and (ii) from that point on they recombine into a single photon
history. Depending on the details of their construction, all gravitational
switches either may or may not satisfy the criteria (i) and (ii). On the other
hand, no quantum switch realisations in classical spacetimes with definite
causal order could ever yield result $0$. Finally, we note that even though
some of $2$-event gravitational switches may give the outcome $1$ when
measuring $M$, it does not necessarily mean that there exist no other
observable that could distinguish them from the $4$-event quantum switches in
a classical geometry. This is a matter for further research.

Detailed graphical visualisations of various gravitational switches are
presented in the Appendix F.

##  5 Relational approach to physics

In the light of the operational framework, which suggests the substitution of
the spacetime events $A$ and $A^{\prime}$ with a single operational event (and
analogously for $B$ and $B^{\prime}$), it is important to comment on one
different but related approach to understanding spacetime, called
relationalism. Note that by this promotion of operational events as
fundamental entities that ought to replace and play the role of the spacetime
events, effectively means the identification of $A$ with $A^{\prime}$, and $B$
with $B^{\prime}$. In this section, we first present a historical review of
the relational approach to physics. Then, we discuss the operational framework
within the context of the modern approach to relationalism.

###  5.1 Mach principle and the history of relationalism

The idea of relationalism is an old one, it traces back at least as far as
Decartes, and is very important in human thought, in particular in the history
of physics. It was brought back to science by Mach in the second half of the
XIX century (for an overview and history of the Mach principle and the
relational approach to space, from its origins in ancient Greece, see for
example [20] and the references therein). Based on the Leibniz ideas of a
relational world, Mach formulated his famous Mach principle, an intuitively
reasonable approach in analysing physics, and space(time) relations in
particular. One of the main characteristics of the Mach principle is that (see
[21], page 17):

> “Space as such plays no role in physics; it is merely an abstraction from
> the totality of spatial relations between material objects.”

The same formulation can be found in [22], slightly re-phrased as “Mach7: If
you take away all matter, there is no more space.” It is interesting to note
that the authors attribute this formulation to A. S. Eddington [23], page 164.

As discussed at the beginning of Section 4, in the operational approach one
attributes the ultimate existence to the “closed laboratories” only, while
their mutual relations, epitomised by the process matrix, are then giving rise
to higher level emergent entities. This clearly shows striking similarities
between the Mach’s and the operational approaches to space(time).

Mach’s ideas were crucial for Einstein in formulating the theory of
relativity. And while many of Mach’s predictions were indeed realised in the
new theory, some of them were not. Mach’s idea that the matter is the basic
entity, and that by abstracting the relations between the objects the space
emerges, led him to the following statement: if the matter in the universe
were finite and had 3D rotational symmetry, it would be impossible to
determine its angular momentum (indeed, even talking about it would have no
meaning). This is a plausible idea. Nevertheless, it does not hold in general
relativity (GR), where one can find two solutions of the Einstein equations
for the isolated black hole (the stationary Schwarzschild solution and the
rotating Kerr solution [24]). Moreover, while according to the Mach principle
the matter completely determines the space, this is not the case in GR: not
only that there exists a solution for the gravitational field in the absence
of matter (when the stress-energy tensor $T$ is identically zero), but the
solution is not unique, as it depends on the boundary conditions as well
(i.e., flat Minkowski spacetime is not the only solution — gravitational waves
being a possible alternative [24]). This also holds for the general $T\neq 0$
case, as there too boundary conditions play an important role. Thus, matter
does not fully determine the inertia, as should according to Mach principle,
which states that the inertia of a massive body is given solely in terms of
its relations with the other massive bodies.

Motivated by giving the ultimate reality to material objects only (closed
laboratories in the case of the operational approach), Mach formulated the
above list of claims. Nevertheless, they were later shown not to hold in GR.
Provided the similarities between the Mach ideas and the operational approach,
the latter might face similar problems as well. We thus believe that
introducing the operationalist notion of an event should be accompanied by
more elaborate proposals of new physical hypotheses and theories. We hope that
our discussion may serve as a small step towards achieving this goal.

###  5.2 Modern approach to relationalism

In contrast to the historical approach to relationalism and Mach’s ideas, that
sounded plausible at the time but ultimately failed with the development of
GR, the more elaborate modern approach to relationalism is epitomised in the
words of Carlo Rovelli (see Section 2.3 of [25]):

> “The world is made up of fields. Physically, these do not live on spacetime.
> They live, so to say, on one another. No more fields on spacetime, just
> fields on fields.”

In particular, the modern relational approach to spacetime defines a
particular spacetime point by the physical processes that are “happening at
that point”. More technically, given an ordered set of classical fields
$\phi\equiv(\phi_{1},\dots,\phi_{n})$ used to describe physics in a given
classical theoretical framework, one traditionally starts from some spacetime
point $\tilde{x}$ and evaluates the fields at that point,
$\tilde{\phi}_{i}=\phi_{i}(\tilde{x})$, obtaining an $n$-tuple of numbers
$(\tilde{\phi}_{1},\dots,\tilde{\phi}_{n})$. The idea of relationalism does
the opposite — one starts from $n$-tuples of field values, and then defines a
spacetime point using an $n$-tuple,
$\tilde{x}\equiv(\tilde{\phi}_{1},\dots,\tilde{\phi}_{n})$, so that the same
equation $\tilde{\phi}_{i}=\phi_{i}(\tilde{x})$ holds. The question of how to
operationally relate values of different fields, and assign and distribute
them into $n$-tuples, is a matter of a separate study [26]. In this work, we
assume that this problem is already solved. Moreover, note that fields $\phi$
need not be observable, due to potential gauge symmetries (for example, the
electromagnetic potential $A_{\mu}$ and the metric $g_{\mu\nu}$). To that end,
we introduce an ordered set of gauge invariant functions ${\cal
O}(\phi)\equiv({\cal O}_{1}(\phi),\dots,{\cal O}_{m}(\phi))$, where $m\geq n$
(for example, the electromagnetic field strength $F_{\mu\nu}$ and the
curvature $R^{\lambda}{}_{\mu\nu\rho}$), and define a spacetime point as an
$m$-tuple of their values $\tilde{\cal O}$.

Unless the physical system features some global symmetry, each $m$-tuple
$\tilde{\cal O}$ defines a unique point in spacetime. Note that, in the
context of GR, the absence of global symmetries is actually the generic case.
Thus, the essential feature of this definition is that it does not make sense
to say that the same $m$-tuple of field strengths can occur in two different
spacetime points, since “both” spacetime points in question are defined in
terms of the one and the same $m$-tuple, and therefore represent a single
point.

Moving from classical to the quantum framework, where no system has
predetermined physical properties independent of observation, one needs to
talk about observables. Given an ordered set of quantum fields
$\phi\equiv(\phi_{1},\dots,\phi_{n})$, one constructs one specific complete
set of compatible observables ${\cal O}\equiv({\cal O}_{1}(\phi),\dots.{\cal
O}_{m}(\phi))$, where

  * •

compatible means that all observables mutually commute, $\,[\,{\cal
O}_{i}\,,\,{\cal O}_{j}\,]\,=0$ for every $i$ and $j$, while

  * •

complete means that the eigenspaces common for all these observables are
nondegenerate, i.e., they are one-dimensional subspaces of the total Hilbert
space.

Here, by “specific” we mean the set of observables which depend only on fields
$\phi$, but not on their conjugated momenta. This fixes the coordinate
representation, such that each common eigenvector corresponds to one classical
configuration of fields. The outcomes of the measurements of these observables
can then be grouped into $m$-tuples and used to define individual spacetime
points, as in the classical case above, thus giving rise to an emergent
classical spacetime. On the other hand, if the state is not an eigenvector of
${\cal O}$, one cannot speak of a single classical configuration of fields,
and thus the notion of emergent spacetime and its points ceases to make sense
globally, according to the relational approach. At most, one could speak of a
superposition of classical configurations and corresponding emergent
spacetimes, but without any natural way to relate spacetime points across
different branches in the superposition. Nevertheless, this does not mean that
establishing such a relation is impossible for certain subregions of
spacetime. Indeed, the whole non-yellow “outside” part of the gravitational
switch picture from Subsection 3.3 represents a subregion with a locally
classical configuration and thus well defined spacetime points.

In order to better appreciate the relational definition of spacetime points
given above, it is instructive to look at the realisation of spacetime in the
context of a relational quantum gravity model, such as a spinfoam model in the
Loop Quantum Gravity (LQG) framework [25, 27]. There, the spacetime is “built”
out of the spin foam — a lattice-like structure with vertices, edges and
faces, each labeled by the eigenvalues of particular field operators that
“live” on these structures, depicted as follows (see Figure 10):

$j_{f}$$j_{f^{\prime}}$$\iota_{\epsilon}$$\iota_{\epsilon^{\prime}}$ Figure
10: A piece of a spin foam diagram. The field $j$ labels the faces
$f,f^{\prime},\dots$, while the field $\iota$ labels the edges
$\epsilon,\epsilon^{\prime},\dots$, of the diagram.

For example, the area operator, which is a function of the gravitational
field, has eigenvalues determined by a half-integer label $j\in\mathbb{N}/2$,
and each face of the spin foam carries one such label, specifying the area of
the surface dual to that face. In particular, the spectrum of the area
operator is given as

$A(j)=8\pi\gamma l_{p}^{2}\sum_{f}\sqrt{j_{f}(j_{f}+1)}\,,$ (17)

where $l_{p}$ is the Planck length, $\gamma$ is the Barbero-Immirzi parameter,
while the sum goes over all faces $f$ of the spin foam that intersect the
surface whose area we are interested in, see [25, 27] for details. All other
physical observables similarly provide appropriate labels for each vertex,
edge and face of the spin foam. Since edges and faces meet at vertices, a
given vertex carries labels of all observables of all edges and faces that are
connected to that vertex. These observables form the complete set of
compatible observables ${\cal O}$, and their eigenvalues label each vertex,
determining the identity of that vertex. In other words, each labeled vertex
of a spin foam defines a “spacetime point”, and if two vertices have
completely identical properties in the sense of their labels and their
connectedness to neighbouring objects, they actually represent the one and the
same vertex.

At first sight, it is tempting to apply the ideas of relational spacetime to
the case of the quantum switch, as follows. At the spacetime event $A$, Alice
interacts with the particle as it enters and exits her lab, while at the
spacetime event $A^{\prime}$ Alice also interacts (in exactly the same way)
with the same particle. The idea of relational spacetime then might suggest
that one should define the spacetime events $A$ and $A^{\prime}$ by the
physical event of interaction between Alice and the particle. Since this
interaction is the same in both cases, one ought to identify the two points,
$A\equiv A^{\prime}$, and claim that both of these correspond to the same
spacetime event, defined by the interaction between Alice and the particle.
The same argument applies to Bob, and events $B$ and $B^{\prime}$.

Unfortunately, this argument is not fully in line with relationalism. The
reason lies in the fact that the interaction between Alice and the particle
(and also between Bob and the particle) does not meet the criteria given in
the above relational definition of a spacetime point. Namely, neither Alice,
nor Bob, performs a measurement of a complete set of compatible observables
${\cal O}$. The mentioned interaction with the particle is merely a subset of
this. In particular, the interaction of Alice with the particle does not
uniquely fix the state of, say, the gravitational field, or the
electromagnetic field, or the Higgs field, etc. Therefore, it may happen that
the measurement outcomes of the whole set of observables ${\cal O}$ at
spacetime events $A$ and $A^{\prime}$ are still mutually distinct, thereby
defining the events $A$ and $A^{\prime}$ as two distinguishable spacetime
points. In order to be certain that $A$ and $A^{\prime}$ are really the same
spacetime event, Alice would need to measure the complete set of observables
${\cal O}$, and convince herself that the results of all those measurements at
$A$ and at $A^{\prime}$ are identical. The mere interaction with the particle
is not enough to achieve this, and the experimental setups such as [5, 6, 7]
obviously fall short of accounting for the state of all other possible
physical fields that Alice and Bob can interact with, in addition to the
interaction with the particle.

We see that, when applied to the case of the quantum switch in classical
gravitational field, the relational framework is at odds with the operational
approach — the former distinguishes $A$ and $A^{\prime}$ while the latter
regards them as identical. This is because the matter fields of the particle
are in a superposition of two classical configurations. Similarly, in the case
of the $2$-event gravitational switch introduced in Subsection 3.3, the
overall state of gravity and matter is a superposition of two distinct
classical configurations. Therefore, within the relational framework, it is
not possible to talk about a single emergent spacetime, nor to compare the
points that belong to different branches. This is different from the
operational approach, which aims to identify points from different branches.
It is also different from the traditional approach, since the latter
postulates a unique classical spacetime manifold.

Note that, if understood as an interpretation, relational framework ought to
have all experimental predictions the same as those from the traditional
approach. Thus, the observable constructed in Subsection 4.2 should
distinguish the quantum from the gravitational switch, in the same way as in
the traditional approach. On the other hand, potential new physics formulated
based on the relational framework might, or might not, feature different
experimental predictions.

It is important to emphasise that, as discussed in Subsection 4.3, various
realisations of the quantum switch are possible by superposing different
causal orders in the framework of QG. In particular, regarding the $2$-event
realisations, one can consider the following diagram (see Figure 11):

particleAliceBob$A$$B$$\mathcal{I}$$\mathcal{F}$ Figure 11: Spacetime diagram
of a version of a $2$-event gravitational switch, in which Alice and Bob
perform their respective operations in the regions of spacetime with a single
gravitational configuration.

This diagram features two classical spacetime subregions surrounding Alice’s
and Bob’s laboratories. As such, Alice and Bob can measure the complete set of
compatible observables within their laboratories, without obtaining which-way
information and destroying the superposition. Therefore, even from the
relational point of view, this represents an implementation of a $2$-event
gravitational switch. Note that in this case Alice and Bob do not even need
Friend in order to verify the $2$-event nature of their gravitational switch.

It is interesting to observe that this realisation of the quantum switch
implements the operational idea of a $2$-event quantum switch, in terms of
closed laboratories. However, to achieve such an implementation, it is
necessary to have a genuine superposition of metric-induced spacetime causal
orders in the yellow region of spacetime, which does not feature in
experimental realisations [5, 6, 7].

##  6 Conclusions

In this paper, we analysed the notion of causal orders both in classical and
quantum worlds, with the emphasis on the latter. We defined the notion of the
causal order for the case of (classical and quantum) circuits, in terms of
partial ordering between the nodes of the circuit’s underlying graph that
defines the cause-effect structure. We discussed the possibility of
implementing an abstract circuit in the real world, showing that it is always
possible to do so for the case of a globally hyperbolic (classical) spacetime,
in which the circuit’s causal order is preserved by the metric-induced
relation between the spacetime events.

The superposition principle of quantum mechanics offers the possibility of
controlled operations, in particular the quantum switch, whose experimental
realisations have been claimed to present genuine superpositions of causal
orders. Within the process matrix formalism, we have analysed the $4$\- and
$3$-event realisations of the quantum switch in classical spacetimes with
fixed spacetime causal orders, and the 2-event realisation of a gravitational
switch that features superpositions of different gravitational field
configurations and their respective spacetime causal orders. To that end, we
have extended the process matrix formalism, by introducing the notion of a
vacuum state. Our analysis shows that the process matrix formalism can explain
the quantum switch realisations within the standard physics, and is thus
consistent with it.

Thus, as a consequence of our Theorem, and the analysis of the quantum switch
implementations, we argued that, in contrast to the gravitational switch, the
current experimental implementations do not feature superpositions of
spacetime causal orders, and that they are variants of the time double slit
experiment. Moreover, by explicitly constructing two different observables,
presented in Sections 4.1 and 4.2, respectively, we showed that it is possible
to experimentally distinguish between different realisations of the quantum
switch.

Finally, in Section 5, we analysed the relation among the traditional QFT
approach to QG (used throughout this paper), the operational point of view,
and the relational framework of QG. On the example of the quantum switch, we
showed that the operational viewpoint, while consistent with the approach
advocated by Mach, is nevertheless at odds with the modern relational
framework. On the other hand, the traditional QFT approach and the relational
framework may or may not be compatible, depending on the concrete realisation
of the quantum switch. In particular, for the specific realisation of the
gravitational switch given in Subsection 5.2, the two frameworks are
compatible in the prediction that Alice and Bob can locally (without the help
of Friend) verify that the switch is implemented on $2$ events.

In a recent work [28], the authors report on a violation of the causal
inequality [9] in flat Minkowski spacetime with a definite causal order. To
achieve it, they consider laboratories that are localised in space only, while
delocalised in time. Therefore, their alternative notion of a “closed
laboratory”, and that considered in [9], do not coincide, this way manifestly
violating the conditions necessary for the causal inequality to hold. For the
same reason, the scenario considered in [28] falls out of the scope of the
current work as well. Additionally, in another recent work [29], the author
discusses the quantum switch in terms of the time-delocalised quantum
subsystems and operations, and generalises it to more complex quantum circuits
and processes. The results of these two papers deserve further analysis and
remain to be a subject of future research.

Exploring possible generalisations of our Theorem, as suggested at the end of
Appendix A, presents a straightforward future line of research. Also, one
could further analyse the process matrix formalism, in particular by exploring
the situations in which the operational approach interpretation fails to
describe the known processes. Or, to search for the opposite — the instances
of physical processes that cannot be explained by the process matrix
formalism, when applied within the standard physics. In order to show that the
process matrix formalism is perfectly suitable for describing the quantum
switch implementations within the standard physics, we formulated its version
that features the vacuum state. One can thus further study possible
generalisations of this formalism and its applications to the cases that go
beyond simple non-relativistic mechanics. Finally, motivated by our analysis
and discussion from Subsections 4.1 and 5.1, one can try to formulate
alternative theories that would be consistent with the experimentally tested
known physics (GR in particular), while at the same time substituting the
spacetime events $A$ and $A^{\prime}$, from the quantum switch realisations in
classical spacetimes, with a single operational event (and analogously for $B$
and $B^{\prime}$).

Acknowledgments

The authors would like to thank Časlav Brukner, Esteban Castro-Ruiz and
Flaminia Giacomini for helpful comments and discussions.

NP acknowledges the support of SQIG — Security and Quantum Information Group,
the Instituto de Telecomunicações (IT) Research Unit, ref. UIDB/50008/2020,
funded by Fundação para a Ciência e Tecnologia (FCT), and the FCT projects
QuantumMining POCI-01-0145-FEDER-031826 and Predict PTDC/CCI-CIF/29877/2017,
supported by the European Regional Development Fund (FEDER), through the
Competitiveness and Internationalisation Operational Programme (COMPETE 2020),
and by the Regional Operational Program of Lisbon, as well as the FCT Estímulo
ao Emprego Científico grant no. CEECIND/04594/2017/CP1393/CT0006. The
bilateral scientific cooperation between Portugal and Serbia through the
project “Noise and measurement errors in multi-party quantum security
protocols”, no. 451-03-01765/2014-09/04 supported by the FCT, Portugal, and
the Ministry of Education, Science and Technological Development of the
Republic of Serbia is also acknowledged.

MV was supported by the project ON 171031 of the Ministry of Education,
Science and Technological Development of the Republic of Serbia, and the
bilateral scientific cooperation between Austria and Serbia through the
project “Causality in Quantum Mechanics and Quantum Gravity — 2018-2019”, no.
451-03-02141/2017-09/02 supported by the Austrian Academy of Sciences (ÖAW),
Austria, and the Ministry of Education, Science and Technological Development
of the Republic of Serbia.

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##  Appendix A Proof of the Theorem

Here we give an explicit constructive proof of the Theorem from the main text.

Given the graph $G$, we begin the proof by partitioning its set of nodes $I$
into disjoint subsets, in the following way. Since the graph is finite, we
introduce the subset $M_{1}\subset I$ which consists of all minimal nodes of
the graph G:

$M_{1}=\\{u\in I\,|\,(\neg\exists v\in I)\,v\prec_{I}u\\}\,.$ (18)

Since all nodes in $M_{1}$ are minimal, there is no order relation $\prec_{I}$
between any two of them. Therefore, we can intuitively understand them as
“simultaneous”. As a next step, we remove these nodes and the corresponding
edges from $G$, reducing it to a subgraph $G_{2}=(I_{2},E_{2})$, where

$I_{2}=I\backslash N_{1}\,,\qquad E_{2}=\\{(u,v)\,|\,u,v\in I_{2}\,,\,(u,v)\in
E\\}\,.$ (19)

Then we repeat the construction for the graph $G_{2}$, obtaining the new
minimal set $M_{2}$, and the next subgraph $G_{3}$, in an analogous way. Since
the graph $G$ is finite, after a certain finite number of steps we will
exhaust all nodes in $I$, ending up with a partition of “simultaneous” subsets
$M_{1},\dots,M_{m}$ ($m\in\mathbb{N}$), such that

$(\forall i\neq j)\quad M_{i}\cap
M_{j}=\emptyset\,,\qquad\bigcup_{i=1}^{m}M_{i}=I\,.$ (20)

Once we have partitioned the set of nodes $I$ into subsets, we turn to the
construction of the immersing map $P:I\to{\cal M}$, in the following way.
Since spacetime is globally hyperbolic, we can write ${\cal
M}=\Sigma\times\mathbb{R}$, where $\Sigma$ is a spatial $3$-dimensional
hypersurface, and $\mathbb{R}$ is timelike. Without loss of generality, one
can then introduce a foliation of spacetime into a family of such
hypersurfaces, denoted $\Sigma_{t}$ and labeled by a parameter
$t\in\mathbb{R}$. Start from some initial parameter $t_{1}$, and choose a
compact subset $S_{t_{1}}\subset\Sigma_{t_{1}}$. Denoting the number of
elements in the partition $M_{i}$ as $\|M_{i}\|$, we pick in an arbitrary way
the set of $\|M_{1}\|$ points $\vec{x}_{k}\in S_{t_{1}}$ (here,
$k=1,\dots,\|M_{1}\|$), and define the map $P$ to assign a node from $M_{1}$
to each point $\vec{x}_{k}$ in a one-to-one fashion:

$P(u_{k})=(t_{1},\vec{x}_{k})\in{\cal M}\,,\qquad k=1,\dots,\|M_{1}\|\,.$ (21)

Once this assignment has been defined, construct a future-pointing light cone
from each spacetime point $(t_{1},\vec{x}_{k})$. Then we find a new
hypersurface, $\Sigma_{t_{2}}$, which contains a common intersection with all
constructed light cones, and denote this intersection
$S_{t_{2}}\subset\Sigma_{t_{2}}$. In this way, by construction, all points
$(t_{1},\vec{x}_{k})$ are in the past of all points in $S_{t_{2}}$,

$(t_{1},\vec{x}_{k})\prec_{{\cal M}}S_{t_{2}}\,,\qquad k=1,\dots,\|M_{1}\|\,.$
(22)

Now extend the definition of $P$ such that it assigns the nodes from the next
partition, $M_{2}$, to a randomly chosen set of points in $S_{t_{2}}$ in a
similar way as before, then construct a set of light cones from them, and
repeat the construction for all partitions $M_{i}$. Constructed in this way,
the map $P$ ensures that for every pair of nodes $u,v\in I$, we have

$u\prec_{I}v\quad\Longrightarrow\quad P(u)\prec_{{\cal M}}P(v)\,,\qquad\forall
u,v\in I\,.$ (23)

Once we have constructed the map $P:I\to{\cal M}$ satisfying (23), using the
definition (2), it induces the map ${\cal P}:{\cal G}_{\cal C}\to{\cal M}$,
which satisfies the required statement (3).

This completes the proof. $\Box$

Note that, while the causal order $\prec_{{\cal M}}$ indeed preserves the
causal order $\prec_{{\cal C}}$, it is “stronger” in the sense that it may
introduce additional relations between the images of nodes, which do not hold
in the graph itself. Indeed, the construction of the map $P$ in the above
proof is such that each image of a node from some given partition $M_{i}$ is
in the causal past of all images from the previous partition $M_{i-1}$, which
is not necessarily the case for the nodes themselves. One might study if the
causal orders over the set of nodes and over the set of its images can be
equivalent, i.e., if the opposite implication from equation (23) also holds
(in this case the immersion $P$ is called an embedding of $G$ into ${\cal
M}$). Whether such an embedding exists for all hyperbolic spacetimes, or at
least for some, is an open question.

Next, one could also discuss the generalisation of the above theorem to the
case of countably infinite graphs $G$. However, for our purposes, the
existence of a partially ordered map ${\cal P}$ over the set of finite graphs
will suffice.

Regarding the proof itself, one can formulate an alternative (and simpler)
approach to the proof of the theorem. Namely, one can first prove that every
circuit can be immersed into the flat Minkowski spacetime. Then, knowing that
the sufficiently small neighbourhood of every spacetime point in an arbitrary
manifold ${\cal M}$ can be well approximated with its tangent space, one can
always immerse the whole circuit into this small neighbourhood. However, this
implies that the geometric size of the circuit can be considered negligible
compared to the curvature scale of the manifold, which may render such
implementation practically unfeasible. Moreover, this alternative approach
does not cover the cases where one actually wants the scale of the circuit to
be comparable to the curvature scale. Specifically, if one wishes to employ
the circuit to study gravitational phenomena, its gates must be distributed
across spacetime precisely in a way that is sensitive to curvature. Therefore,
the construction of the map $P$ used in the proof of the theorem is more
general than the construction in this alternative approach.

Finally, given the construction in the proof, the gates of the set of minimal
nodes $M_{1}$ define the initial gate ${\cal I}$, the set of maximal nodes
$M_{m}$ define the final gate ${\cal F}$, while the the gates of the remaining
intermediary sets of nodes $M_{2},\dots M_{m-1}$ define the operation ${\cal
O}_{\cal C}$. This is illustrated in the diagram below (see Figure 12).

time$i_{1}$$i_{2}$$\vphantom{i_{3}}\dots$$i_{\|M_{1}\|}$${\cal
I}$$f_{1}$$f_{2}$$\vphantom{f_{3}}\dots$$f_{\|M_{m}\|}$${\cal
F}$$\cdots$$\cdots$${\cal O}_{\cal C}$ Figure 12: The spacetime diagram of the
circuit ${\cal C}$, with the initial gate ${\cal I}$, the operation gate
${\cal O}_{{\cal C}}$, and the final gate ${\cal F}$.

##  Appendix B Qutrit states, operators and bases

The notion of a qubit can be generalised from a $2$-dimensional Hilbert space
to a $d$-dimensional Hilbert space. The generalised object is called “qudit”
in $d$ dimensions [30]. Since we are interested in describing ordinary
$2$-dimensional qubits with an additional vacuum state, it is natural to
consider qudits in $d=3$, called “qutrits”. We introduce the following
notation for the basis states of a qutrit in ${\cal H}_{3}=\mathbb{C}^{3}$:

${|{0}\rangle}\equiv\left[\begin{array}[]{c}1\\\ 0\\\ 0\\\
\end{array}\right]\,,\qquad{|{1}\rangle}\equiv\left[\begin{array}[]{c}0\\\
1\\\ 0\\\
\end{array}\right]\,,\qquad{|{v}\rangle}\equiv\left[\begin{array}[]{c}0\\\
0\\\ 1\\\ \end{array}\right]\,.$ (24)

The states ${|{0}\rangle}$ and ${|{1}\rangle}$ will be understood as the usual
computational basis for a $2$-dimensional qubit, while the state
${|{v}\rangle}$ will represent the vacuum, i.e., the “absence of a qubit”. In
cases when we take sums over the basis vectors, we will assume that the vacuum
state carries the index $2$, i.e., ${|{v}\rangle}\equiv{|{2}\rangle}$, so that
we can write

$\sum_{i=0}^{2}{|{i}\rangle}={|{0}\rangle}+{|{1}\rangle}+{|{v}\rangle}\,,\qquad\text{
and }\qquad\sum_{i=0}^{1}{|{i}\rangle}={|{0}\rangle}+{|{1}\rangle}\,.$ (25)

Using this notation, we write the unnormalised maximally correlated states for
the qutrit and the qubit as

${|{\mathbbm{1}}\rangle\\!\rangle}=\sum_{i=0}^{2}{|{i}\rangle}{|{i}\rangle}={|{0}\rangle}{|{0}\rangle}+{|{1}\rangle}{|{1}\rangle}+{|{v}\rangle}{|{v}\rangle}\in{\cal
H}_{3}\otimes{\cal
H}_{3}\,,\qquad{|{1}\rangle\\!\rangle}=\sum_{i=0}^{1}{|{i}\rangle}{|{i}\rangle}={|{0}\rangle}{|{0}\rangle}+{|{1}\rangle}{|{1}\rangle}\in{\cal
H}_{2}\otimes{\cal H}_{2}\,,$ (26)

so that

${|{\mathbbm{1}}\rangle\\!\rangle}={|{1}\rangle\\!\rangle}+{|{v}\rangle}{|{v}\rangle}\,.$
(27)

One can also introduce the standard Hilbert-Schmidt basis in the space ${\cal
L}({\cal H}_{3})$ of linear operators on ${\cal H}_{3}$. This basis consists
of $9$ matrices $3\times 3$, labeled as $\lambda_{0},\dots,\lambda_{8}$, as
follows:

  * •

the three symmetric matrices

$\lambda_{1}=\sqrt{\frac{3}{2}}\left[\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\
0&0&0\\\
\end{array}\right]\,,\qquad\lambda_{2}=\sqrt{\frac{3}{2}}\left[\begin{array}[]{ccc}0&0&1\\\
0&0&0\\\ 1&0&0\\\
\end{array}\right]\,,\qquad\lambda_{3}=\sqrt{\frac{3}{2}}\left[\begin{array}[]{ccc}0&0&0\\\
0&0&1\\\ 0&1&0\\\ \end{array}\right]\,,$ (28)

  * •

the three antisymmetric matrices

$\lambda_{4}=\sqrt{\frac{3}{2}}\left[\begin{array}[]{ccc}0&-i&0\\\ i&0&0\\\
0&0&0\\\
\end{array}\right]\,,\qquad\lambda_{5}=\sqrt{\frac{3}{2}}\left[\begin{array}[]{ccc}0&0&-i\\\
0&0&0\\\ i&0&0\\\
\end{array}\right]\,,\qquad\lambda_{6}=\sqrt{\frac{3}{2}}\left[\begin{array}[]{ccc}0&0&0\\\
0&0&-i\\\ 0&i&0\\\ \end{array}\right]\,,$ (29)

  * •

and the three diagonal matrices

$\lambda_{7}=\sqrt{\frac{3}{2}}\left[\begin{array}[]{ccc}1&0&0\\\ 0&-1&0\\\
0&0&0\\\
\end{array}\right]\,,\qquad\lambda_{8}=\frac{1}{\sqrt{2}}\left[\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&-2\\\
\end{array}\right]\,,\qquad\lambda_{0}=\left[\begin{array}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&1\\\ \end{array}\right]\,.$ (30)

The matrix $\lambda_{0}$ is the unit matrix, while
$\lambda_{1},\dots,\lambda_{8}$ are self-adjoint, traceless, and orthogonal
with respect to the standard scalar product:

$\lambda_{i}^{{\dagger}}=\lambda_{i}\,,\qquad\mathop{\rm
Tr}\nolimits\lambda_{i}=0\,,\qquad\mathop{\rm
Tr}\nolimits\lambda_{i}^{{\dagger}}\lambda_{j}=3\delta_{ij}\,,\qquad
i=1,\dots,8\,.$ (31)

They represent the generators of the $SU(3)$ group, and are known as the Gell-
Mann matrices (up to a normalisation factor $\sqrt{3/2}$).

If we denote ${\cal H}_{v}$ as the $1$-dimensional vacuum-spanned subspace of
${\cal H}_{3}$, one can see that ${\cal L}({\cal H}_{2})\oplus{\cal L}({\cal
H}_{v})\subset{\cal L}({\cal H}_{3})$. In particular, if we denote the
standard Pauli matrices as $\sigma_{x},\sigma_{y},\sigma_{z}$ and the unit
$2\times 2$ matrix as $I_{2}$, they form the basis in ${\cal L}({\cal
H}_{2})$, and the qubit basis can thus be constructed as

$\sqrt{\frac{2}{3}}\lambda_{1}=\left[\begin{array}[]{cc|c}\lx@intercol\hfil\hbox{\multirowsetup$\sigma_{x}$}\hfil\lx@intercol\vrule\lx@intercol&0\\\
&&0\\\ \hline\cr 0&0&0\\\
\end{array}\right]\,,\qquad\sqrt{\frac{2}{3}}\lambda_{4}=\left[\begin{array}[]{cc|c}\lx@intercol\hfil\hbox{\multirowsetup$\sigma_{y}$}\hfil\lx@intercol\vrule\lx@intercol&0\\\
&&0\\\ \hline\cr 0&0&0\\\
\end{array}\right]\,,\qquad\sqrt{\frac{2}{3}}\lambda_{7}=\left[\begin{array}[]{cc|c}\lx@intercol\hfil\hbox{\multirowsetup$\sigma_{z}$}\hfil\lx@intercol\vrule\lx@intercol&0\\\
&&0\\\ \hline\cr 0&0&0\\\ \end{array}\right]\,,$ (32)

along with

$\frac{2}{3}\lambda_{0}+\frac{\sqrt{2}}{3}\lambda_{8}=\left[\begin{array}[]{cc|c}\lx@intercol\hfil\hbox{\multirowsetup$I_{2}$}\hfil\lx@intercol\vrule\lx@intercol&0\\\
&&0\\\ \hline\cr 0&0&0\\\ \end{array}\right]\,.$ (33)

Also, the vacuum space ${\cal L}({\cal H}_{v})$ is one-dimensional, and the
basis is

$\frac{1}{3}\lambda_{0}-\frac{\sqrt{2}}{3}\lambda_{8}=\left[\begin{array}[]{cc|c}0&0&0\\\
0&0&0\\\ \hline\cr 0&0&1\\\ \end{array}\right]\,.$ (34)

##  Appendix C Process matrix evaluation

Let us give an explicit step by step evaluation of the probability
distribution for the $4$-event process discussed in the text, using the
process matrix formalism. The complete spacetime diagram of the process is
given as (see Figure 13):

spacetime$t_{i}$$t_{1}$$t_{2}$$t_{f}$AliceBob$\begin{subarray}{c}\text{\tiny
beam}\\\ \text{\tiny
splitter}\end{subarray}$$S_{i}$$S_{f}$$A$$B^{\prime}$$B$$A^{\prime}$$P_{A}$$P_{B}$$\mathcal{I}$$T_{A}$$T_{B}$$\mathcal{F}$
Figure 13: Spacetime diagram of the $4$-event implementation of the quantum
switch. The internal structures of the composite gates ${\cal I}$ and ${\cal
F}$ are explicitly depicted.

The composite event $\mathcal{I}$ consists of the two preparation events
$P_{A}$, $P_{B}$, and the initial beam splitting event $S_{i}$, while
$\mathcal{F}$ consists of the recombination event $S_{f}$ and the measurement
events $T_{A}$ and $T_{B}$.

The corresponding circuit diagram is obtained from the above one by promoting
each event of interaction to a gate, and the propagation of each particle to a
channel. This leads to the following circuit diagram (see Figure 14):

$P_{A}$$P_{B}$$\;\;S^{i}$$A$$B$$A^{\prime}$$B^{\prime}$$\;\;S^{f}$$T_{A}$$T_{B}$$\mathcal{I}$$\mathcal{F}$
Figure 14: Circuit diagram of the $4$-event implementation of the quantum
switch. The internal structures of the composite gates ${\cal I}$ and ${\cal
F}$ are explicitly depicted.

Its structure is in one-to-one correspondence with the spacetime diagram for
the $4$-event process, where the preparation and measurement spacetime events
$\mathcal{I}$ and $\mathcal{F}$ have been split into three sub-gates each, for
clarity.

The operations on each of the gates are given as follows. The preparation gate
$P_{A}$ maps from the input Hilbert space $P_{A_{I}}$ to the output Hilbert
space $P_{A_{O}}$, and analogously for gate $P_{B}$. The input spaces are
trivial, $\dim P_{A_{I}}=\dim P_{B_{I}}=1$, while each output space is spanned
by vectors ${|{0}\rangle}$, ${|{1}\rangle}$ and ${|{v}\rangle}$. Here,
${|{0}\rangle}$ and ${|{1}\rangle}$ represent the two orthogonal qubit states
(say, vertical and horizontal polarisations along certain axis in 3D space),
while ${|{v}\rangle}$ is the vacuum state, representing the absence of
particles in the corresponding arm of the interferometer. The operations
performed at these gates, $P_{A}={|{\Psi}\rangle}$ and $P_{B}={|{v}\rangle}$,
specify the initial conditions for the rest of the circuit diagram, and are
described by the Choi-Jamiołkowski (CJ) states as

${|{P_{A}^{\ast}}\rangle\\!\rangle}{}^{P_{A_{I}}P_{A_{O}}}={|{\Psi^{\ast}}\rangle}{}^{P_{A_{O}}}\,,\qquad{|{P_{B}^{\ast}}\rangle\\!\rangle}{}^{P_{B_{I}}P_{B_{O}}}={|{v}\rangle}{}^{P_{B_{O}}}\,.$
(35)

Here, $\ast$ denotes the complex conjugation.

Analogously, the target gates $T_{A}$ and $T_{B}$ facilitate the final
measurement outcomes of the circuit diagram. The input spaces $T_{A_{I}}$ and
$T_{B_{I}}$ are three-dimensional, spanned over the two qubit states and the
vacuum, while the output spaces are one-dimensional. The operations performed
at these gates, $T_{\alpha}={\langle{\alpha}|}$ and
$T_{\beta}={\langle{\beta}|}$, read out the measurement results
$\alpha,\beta\in\\{0,1,v\\}$. The corresponding CJ states are given as

${|{T_{\alpha}^{\ast}}\rangle\\!\rangle}^{T_{A_{I}}T_{A_{O}}}={|{\alpha}\rangle}^{T_{A_{I}}}\,,\qquad{|{T_{\beta}^{\ast}}\rangle\\!\rangle}^{T_{B_{I}}T_{B_{O}}}={|{\beta}\rangle}^{T_{B_{I}}}\,.$
(36)

The gates $A$, $A^{\prime}$, $B$ and $B^{\prime}$ perform the unitaries $U$
and $V$. The input and output spaces $A_{I}$ and $A_{O}$ of the Alice’s gate
$A$ are both spanned by vectors ${|{0}\rangle}$, ${|{1}\rangle}$ and
${|{v}\rangle}$, and analogously for the input and output spaces of the
remaining three gates. Assuming that in her (spatially) local laboratory Alice
performs the unitary $U$ on the particle’s internal degree of freedom, the
induced operation between the three-dimensional spaces $A_{I}$ and $A_{O}$
that include the vacuum states is given by

$\tilde{U}^{A_{O}A_{I}}=U^{A_{O}A_{I}}P_{01}^{A_{I}A_{I}}+I^{A_{O}A_{I}}P_{v}^{A_{I}A_{I}}\,,$
(37)

where
$P_{01}^{A_{I}A_{I}}={|{0}\rangle}^{A_{I}}{\langle{0}|}^{A_{I}}+{|{1}\rangle}^{A_{I}}{\langle{1}|}^{A_{I}}$,
$P_{v}^{A_{I}A_{I}}={|{v}\rangle}^{A_{I}}{\langle{v}|}^{A_{I}}$, and
$I^{A_{O}A_{I}}$ represents the identity map between the Hilbert spaces
$A_{O}$ and $A_{I}$. The analogous construction also holds for the gate
$A^{\prime}$, so the respective CJ states for the gates $A$ and $A^{\prime}$
are then given by:

$\begin{array}[]{rl}{|{\tilde{U}^{\ast}}\rangle\\!\rangle}^{A_{I}A_{O}}&=\left[I^{A_{I}A_{I}}\otimes(\tilde{U}^{\ast})^{A_{O}A_{I}}\right]{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{I}A_{I}}\,,\vphantom{\displaystyle\int}\\\
{|{\tilde{U}^{\ast}}\rangle\\!\rangle}^{A_{I}^{\prime}A_{O}^{\prime}}&=\left[I^{A_{I}^{\prime}A_{I}^{\prime}}\otimes(\tilde{U}^{\ast})^{A_{O}^{\prime}A_{I}^{\prime}}\right]{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{I}^{\prime}A_{I}^{\prime}}\,.\vphantom{\displaystyle\int}\end{array}$
(38)

Here, the “transport vector” is given by (for details of the process matrix
formalism for the case of three-dimensional spaces — qutrits, see Appendix B):

${|{\mathbbm{1}}\rangle\\!\rangle}={|{0}\rangle}{|{0}\rangle}+{|{1}\rangle}{|{1}\rangle}+{|{v}\rangle}{|{v}\rangle}\,.$
(39)

Bob performs $V$ in his (spatially) local laboratory, and therefore the CJ
states for the gates $B$ and $B^{\prime}$ are given as:

$\begin{array}[]{rl}{|{\tilde{V}^{\ast}}\rangle\\!\rangle}^{B_{I}B_{O}}&=\left[I^{B_{I}B_{I}}\otimes(\tilde{V}^{\ast})^{B_{O}B_{I}}\right]{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{I}B_{I}}\,,\vphantom{\displaystyle\int}\\\
{|{\tilde{V}^{\ast}}\rangle\\!\rangle}^{B_{I}^{\prime}B_{O}^{\prime}}&=\left[I^{B_{I}^{\prime}B_{I}^{\prime}}\otimes(\tilde{V}^{\ast})^{B_{O}^{\prime}B_{I}^{\prime}}\right]{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{I}^{\prime}B_{I}^{\prime}}\,.\vphantom{\displaystyle\int}\end{array}$
(40)

The gates $S^{i}$ and $S^{f}$ act as beam splitters, i.e., they both perform
the same Hadamard operation $H$, given as follows. The beam splitter input and
output spaces consist of the Alice’s and Bob’s factor spaces. For the case of
the Alice’s input space, we have $S_{A_{I}}=\mathop{\rm
span}\nolimits\\{{|{0}\rangle}^{S_{A_{I}}},{|{1}\rangle}^{S_{A_{I}}},{|{v}\rangle}^{S_{A_{I}}}\\}$,
and analogously for the output space, as well as for Bob’s factor spaces. The
overall input and output beam splitter spaces are therefore defined as
$S_{I}=S_{(AB)_{I}}=S_{A_{I}}\otimes S_{B_{I}}$ and
$S_{O}=S_{(AB)_{O}}=S_{A_{O}}\otimes S_{B_{O}}$. Finally, the unitary matrix
associated to gate $S$ representing the action of the balanced Hadamard beam
splitter is given by:

$\begin{array}[]{rl}H^{S_{O}S_{I}}=&\displaystyle\frac{1}{\sqrt{2}}\left({|{0}\rangle}^{S_{A_{O}}}{|{v}\rangle}^{S_{B_{O}}}+{|{v}\rangle}^{S_{A_{O}}}{|{0}\rangle}^{S_{B_{O}}}\right){\langle{0}|}^{S_{A_{I}}}{\langle{v}|}^{S_{B_{I}}}\\\
&+\displaystyle\frac{1}{\sqrt{2}}\left({|{1}\rangle}^{S_{A_{O}}}{|{v}\rangle}^{S_{B_{O}}}+{|{v}\rangle}^{S_{A_{O}}}{|{1}\rangle}^{S_{B_{O}}}\right){\langle{1}|}^{S_{A_{I}}}{\langle{v}|}^{S_{B_{I}}}\\\
&+\displaystyle\frac{1}{\sqrt{2}}\left({|{0}\rangle}^{S_{A_{O}}}{|{v}\rangle}^{S_{B_{O}}}-{|{v}\rangle}^{S_{A_{O}}}{|{0}\rangle}^{S_{B_{O}}}\right){\langle{v}|}^{S_{A_{I}}}{\langle{0}|}^{S_{B_{I}}}\\\
&+\displaystyle\frac{1}{\sqrt{2}}\left({|{1}\rangle}^{S_{A_{O}}}{|{v}\rangle}^{S_{B_{O}}}-{|{v}\rangle}^{S_{A_{O}}}{|{1}\rangle}^{S_{B_{O}}}\right){\langle{v}|}^{S_{A_{I}}}{\langle{1}|}^{S_{B_{I}}}\,.\end{array}$
(41)

The beam splitter acts such that the system coming from the Alice’s side comes
into an equal superposition of the two output spatial modes coming to Alice
and Bob, with zero relative phase, while the system coming from the Bob’s side
(blue line) comes into an equal superposition of the two output spatial modes
with relative phase $\pi$. Thus, in the output space the correlation between
the Alice’s and Bob’s vacuum state is the opposite as in the input case. The
corresponding CJ state is then

${|{H^{\ast}}\rangle\\!\rangle}^{S_{I}S_{O}}=\left[I^{S_{I}S_{I}}\otimes(H^{\ast})^{S_{O}S_{I}}\right]{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{I}S_{I}}\,,$
(42)

where the transport vector ${|{\mathbbm{1}}\rangle\\!\rangle}$ for the beam
splitter, when projected to a single-particle subspace, is given by

${|{\mathbbm{1}}\rangle\\!\rangle}={|{0v}\rangle}{|{0v}\rangle}+{|{1v}\rangle}{|{1v}\rangle}+{|{v0}\rangle}{|{v0}\rangle}+{|{v1}\rangle}{|{v1}\rangle}\,.$
(43)

Note that the full transport vector contains nine terms instead of the above
four, but for the purpose of this paper, we do not need those five additional
terms.

The process vector encodes the wires between the gates, and it is being
constructed by taking the tensor product over appropriate transport vectors
${|{\mathbbm{1}}\rangle\\!\rangle}$ for Alice’s and Bob’s qutrits, see
equations (26) and (27), such that each transport vector corresponds to one
wire in the circuit diagram, connecting the output of the source gate to the
input of the target gate. The process vector is thus given as:

$\begin{array}[]{lcl}{|{W_{\text{4-event}}}\rangle\\!\rangle}&=&\displaystyle\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{P_{A_{O}}S^{i}_{A_{I}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{P_{B_{O}}S^{i}_{B_{I}}}}_{\text{initial}}\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{i}_{A_{O}}A_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}B^{\prime}_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B^{\prime}_{O}S^{f}_{B_{I}}}}_{\text{blue}}\\\
&&\displaystyle\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{i}_{B_{O}}B_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}A^{\prime}_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A^{\prime}_{O}S^{f}_{A_{I}}}}_{\text{red}}\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{f}_{A_{O}}T_{A_{I}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{f}_{B_{O}}T_{B_{I}}}}_{\text{final}}\,.\\\
\end{array}$ (44)

One can now evaluate the probability distribution

$p(\alpha,\beta)=\left\|{\cal
M}(\alpha,\beta)\vphantom{A^{A}_{A}}\right\|^{2}\,,$ (45)

where the probability amplitude ${\cal M}(\alpha,\beta)$ is constructed by
acting with the tensor product of all gate operations (35), (42), (38), (40),
(42) and (36), on the process vector (44). Since each of the gate operations
acts in its own part of the total Hilbert space, the order of application of
these operations is immaterial, and we are free to choose the most convenient
one.

To see what happens when the operations (35) of the preparation gates act on
the process vector, let us evaluate the action of
${|{P_{A}^{\ast}}\rangle\\!\rangle}^{P_{A_{I}}P_{A_{O}}}$ on
${|{\mathbbm{1}}\rangle\\!\rangle}^{P_{A_{O}}S^{i}_{A_{I}}}$:

${\langle\\!\langle{P_{A}^{\ast}}|}^{P_{A_{I}}P_{A_{O}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{P_{A_{O}}S^{i}_{A_{I}}}={\langle{\Psi^{\ast}}|}^{P_{A_{O}}}\sum_{k=0}^{2}{|{k}\rangle}^{P_{A_{O}}}{|{k}\rangle}^{S^{i}_{A_{I}}}=\sum_{k=0}^{2}\Big{(}\langle\Psi|k\rangle\Big{)}^{\ast}{|{k}\rangle}^{S^{i}_{A_{I}}}={|{\Psi}\rangle}^{S^{i}_{A_{I}}}\,.$
(46)

An analogous calculation can be performed for
${|{P_{B}^{\ast}}\rangle\\!\rangle}^{P_{B_{I}}P_{B_{O}}}$, so the action of
both preparation operations (35) on the process vector (44) evaluates to:

$\begin{array}[]{l}\displaystyle\left({\langle\\!\langle{P_{A}^{\ast}}|}^{P_{A_{I}}P_{A_{O}}}\otimes{\langle\\!\langle{P_{B}^{\ast}}|}^{P_{B_{I}}P_{B_{O}}}\right){|{W_{\text{4-event}}}\rangle\\!\rangle}=\vphantom{\displaystyle\frac{1}{\sqrt{2}}}\\\
\hphantom{mmmm}\displaystyle{|{\Psi}\rangle}^{S^{i}_{A_{I}}}{|{v}\rangle}^{S^{i}_{B_{I}}}\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{i}_{A_{O}}A_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}B^{\prime}_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B^{\prime}_{O}S^{f}_{B_{I}}}}_{\text{blue}}\\\
\hphantom{mmmm}\displaystyle\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{i}_{B_{O}}B_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}A^{\prime}_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A^{\prime}_{O}S^{f}_{A_{I}}}}_{\text{red}}\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{f}_{A_{O}}T_{A_{I}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{f}_{B_{O}}T_{B_{I}}}}_{\text{final}}\,.\\\
\end{array}$ (47)

Next one acts with the initial Hadamard operation (42) on this process vector,
transforming it into

$\begin{array}[]{l}\displaystyle\left({\langle\\!\langle{P_{A}^{\ast}}|}^{P_{A_{I}}P_{A_{O}}}\otimes{\langle\\!\langle{P_{B}^{\ast}}|}^{P_{B_{I}}P_{B_{O}}}\otimes{\langle\\!\langle{S^{\ast}}|}^{S^{i}_{(AB)_{I}}S^{i}_{(AB)_{O}}}\right){|{W_{\text{4-event}}}\rangle\\!\rangle}=\vphantom{\displaystyle\frac{1}{\sqrt{2}}}\\\
\hphantom{mmmm}\displaystyle\frac{1}{\sqrt{2}}\left({|{\Psi}\rangle}^{A_{I}}{|{v}\rangle}^{B_{I}}+{|{v}\rangle}^{A_{I}}{|{\Psi}\rangle}^{B_{I}}\right)\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}B^{\prime}_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B^{\prime}_{O}S^{f}_{B_{I}}}}_{\text{blue}}\\\
\hphantom{mmmm}\displaystyle\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}A^{\prime}_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A^{\prime}_{O}S^{f}_{A_{I}}}}_{\text{red}}\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{f}_{A_{O}}T_{A_{I}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{f}_{B_{O}}T_{B_{I}}}}_{\text{final}}\equiv{|{W_{{QS}_{4}}}\rangle\\!\rangle}\,.\\\
\end{array}$ (48)

The resulting process vector is the outcome of the action of the composite
gate $\mathcal{I}$ on (44), before the actions of Alice and Bob (note that
often in the literature this is taken as the initial process vector in the
analysis):

${|{W_{{QS}_{4}}}\rangle\\!\rangle}=\frac{1}{\sqrt{2}}\left({|{\Psi}\rangle}^{A_{I}}{|{v}\rangle}^{B_{I}}+{|{v}\rangle}^{A_{I}}{|{\Psi}\rangle}^{B_{I}}\right){|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}B_{I}^{\prime}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}A_{I}^{\prime}}{|{\mathbbm{1}}\rangle\\!\rangle}^{(A_{O}^{\prime}B_{O}^{\prime})S_{(AB)_{I}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{(AB)_{O}}T_{(AB)_{I}}}\,.$
(49)

Continuing the computation, the action of the remaining gate operations (38),
(40), (36) and (42) on the process vector (49) gives us the probability
amplitude,

${\cal
M}(\alpha,\beta)\equiv\hphantom{mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm}$
$\left({\langle\\!\langle{\tilde{U}^{\ast}}|}^{A_{I}A_{O}}\otimes{\langle\\!\langle{\tilde{U}^{\ast}}|}^{A_{I}^{\prime}A_{O}^{\prime}}\otimes{\langle\\!\langle{\tilde{V}^{\ast}}|}^{B_{I}B_{O}}\otimes{\langle\\!\langle{\tilde{V}^{\ast}}|}^{B_{I}^{\prime}B_{O}^{\prime}}\otimes{\langle\\!\langle{H^{\ast}}|}^{S_{I}S_{O}}\otimes{\langle\\!\langle{T_{\alpha}^{\ast}}|}^{T_{A_{I}}T_{A_{O}}}\otimes{\langle\\!\langle{T_{\beta}^{\ast}}|}^{T_{B_{I}}T_{B_{O}}}\right){|{W_{{QS}_{4}}}\rangle\\!\rangle}\,.$
(50)

Let us now calculate the action of
${\langle\\!\langle{\tilde{U}^{\ast}}|}^{A_{I}A_{O}}$ on (49):

${\langle\\!\langle{\tilde{U}^{\ast}}|}^{A_{I}A_{O}}{|{W_{{QS}_{4}}}\rangle\\!\rangle}={\langle\\!\langle{\mathbbm{1}}|}^{A_{I}A_{I}}\left[I^{A_{I}A_{I}}\otimes(\tilde{U}^{T})^{A_{O}A_{I}}\right]{|{W_{{QS}_{4}}}\rangle\\!\rangle}\,.$
(51)

Looking at the structure of the process vector, one sees that the resulting
new process vector will have the form

${\langle\\!\langle{\tilde{U}^{\ast}}|}^{A_{I}A_{O}}{|{W_{{QS}_{4}}}\rangle\\!\rangle}=\frac{1}{\sqrt{2}}\left({|{X}\rangle}^{B_{I}^{\prime}}{|{v}\rangle}^{B_{I}}+{|{Y}\rangle}^{B_{I}^{\prime}}{|{\Psi}\rangle}^{B_{I}}\right){|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}A_{I}^{\prime}}{|{\mathbbm{1}}\rangle\\!\rangle}^{(A_{O}^{\prime}B_{O}^{\prime})S_{(AB)_{I}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{(AB)_{O}}T_{(AB)_{I}}}\,,$
(52)

where ${|{X}\rangle}^{B_{I}^{\prime}}$ and ${|{Y}\rangle}^{B_{I}^{\prime}}$
are shorthands for the expressions

${|{X}\rangle}^{B_{I}^{\prime}}\equiv{\langle\\!\langle{\mathbbm{1}}|}^{A_{I}A_{I}}\left[I^{A_{I}A_{I}}\otimes(\tilde{U}^{T})^{A_{O}A_{I}}\right]{|{\Psi}\rangle}^{A_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}B_{I}^{\prime}}$
(53)

and

${|{Y}\rangle}^{B_{I}^{\prime}}\equiv{\langle\\!\langle{\mathbbm{1}}|}^{A_{I}A_{I}}\left[I^{A_{I}A_{I}}\otimes(\tilde{U}^{T})^{A_{O}A_{I}}\right]{|{v}\rangle}^{A_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}B_{I}^{\prime}}\,,$
(54)

which need to be evaluated. The explicit computation of the first expression
goes as follows:

$\begin{array}[]{lcl}{|{X}\rangle}^{B_{I}^{\prime}}&=&\displaystyle{\langle\\!\langle{\mathbbm{1}}|}^{A_{I}A_{I}}\left[I^{A_{I}A_{I}}\otimes(\tilde{U}^{T})^{A_{O}A_{I}}\right]{|{\Psi}\rangle}^{A_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}B_{I}^{\prime}}\vphantom{\displaystyle\sum_{k=0}^{2}}\\\
&=&\displaystyle\sum_{k=0}^{2}{\langle{k}|}^{A_{I}}{\langle{k}|}^{A_{I}}\left[I^{A_{I}A_{I}}\otimes(\tilde{U}^{T})^{A_{O}A_{I}}\right]{|{\Psi}\rangle}^{A_{I}}\sum_{m=0}^{2}{|{m}\rangle}^{A_{O}}{|{m}\rangle}^{B_{I}^{\prime}}\\\
&=&\displaystyle\sum_{m=0}^{2}\left[\sum_{k=0}^{2}\left({\langle{k}|}^{A_{I}}I^{A_{I}A_{I}}{|{\Psi}\rangle}^{A_{I}}\right)\left({\langle{k}|}^{A_{I}}(\tilde{U}^{T})^{A_{O}A_{I}}{|{m}\rangle}^{A_{O}}\right)\right]{|{m}\rangle}^{B_{I}^{\prime}}\\\
&=&\displaystyle\sum_{m=0}^{2}\left[\sum_{k=0}^{2}\langle
k|\Psi\rangle^{A_{I}}\;{\langle{m}|}^{A_{O}}\tilde{U}^{A_{O}A_{I}}{|{k}\rangle}^{A_{I}}\right]{|{m}\rangle}^{B_{I}^{\prime}}\\\
&=&\displaystyle\sum_{m=0}^{2}\left[{\langle{m}|}^{A_{O}}\tilde{U}^{A_{O}A_{I}}{|{\Psi}\rangle}^{A_{I}}\right]{|{m}\rangle}^{B_{I}^{\prime}}\,.\\\
\end{array}$ (55)

Using (37), the coefficient in the brackets can be evaluated as

${\langle{m}|}^{A_{O}}\tilde{U}^{A_{O}A_{I}}{|{\Psi}\rangle}^{A_{I}}={\langle{m}|}^{A_{O}}\left(U^{A_{O}A_{I}}P_{01}^{A_{I}A_{I}}+I^{A_{O}A_{I}}P_{v}^{A_{I}A_{I}}\right){|{\Psi}\rangle}^{A_{I}}={\langle{m}|}U{|{\Psi}\rangle}\,,$
(56)

since $P_{01}^{A_{I}A_{I}}{|{\Psi}\rangle}^{A_{I}}={|{\Psi}\rangle}^{A_{I}}$
and $P_{v}^{A_{I}A_{I}}{|{\Psi}\rangle}^{A_{I}}=0$. Thus, we have

${|{X}\rangle}^{B_{I}^{\prime}}=\sum_{m=0}^{2}{\langle{m}|}U{|{\Psi}\rangle}\;{|{m}\rangle}^{B_{I}^{\prime}}=U{|{\Psi}\rangle}^{B_{I}^{\prime}}\equiv{|{U\Psi}\rangle}^{B_{I}^{\prime}}\,.$
(57)

The computation of ${|{Y}\rangle}^{B_{I}^{\prime}}$ proceeds in an analogous
way to (55), and the result is

${|{Y}\rangle}^{B_{I}^{\prime}}=\sum_{m=0}^{2}\left[{\langle{m}|}^{A_{O}}\tilde{U}^{A_{O}A_{I}}{|{v}\rangle}^{A_{I}}\right]{|{m}\rangle}^{B_{I}^{\prime}}\,.$
(58)

Again, using (37), the coefficient in the brackets can be evaluated as

${\langle{m}|}^{A_{O}}\tilde{U}^{A_{O}A_{I}}{|{v}\rangle}^{A_{I}}={\langle{m}|}^{A_{O}}\left(U^{A_{O}A_{I}}P_{01}^{A_{I}A_{I}}+I^{A_{O}A_{I}}P_{v}^{A_{I}A_{I}}\right){|{v}\rangle}^{A_{I}}=\langle
m|v\rangle=\delta_{mv}\,,$ (59)

since $P_{01}^{A_{I}A_{I}}{|{v}\rangle}^{A_{I}}=0$ and
$P_{v}^{A_{I}A_{I}}{|{v}\rangle}^{A_{I}}={|{v}\rangle}^{A_{I}}$. Thus, we have

${|{Y}\rangle}^{B_{I}^{\prime}}=\sum_{m=0}^{2}\delta_{mv}\;{|{m}\rangle}^{B_{I}^{\prime}}={|{v}\rangle}^{B_{I}^{\prime}}\,.$
(60)

Finally, substituting (57) and (60) back into (52), we obtain:

${\langle\\!\langle{\tilde{U}^{\ast}}|}^{A_{I}A_{O}}{|{W_{{QS}_{4}}}\rangle\\!\rangle}=\frac{1}{\sqrt{2}}\left({|{U\Psi}\rangle}^{B_{I}^{\prime}}{|{v}\rangle}^{B_{I}}+{|{v}\rangle}^{B_{I}^{\prime}}{|{\Psi}\rangle}^{B_{I}}\right){|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}A_{I}^{\prime}}{|{\mathbbm{1}}\rangle\\!\rangle}^{(A_{O}^{\prime}B_{O}^{\prime})S_{(AB)_{I}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{(AB)_{O}}T_{(AB)_{I}}}\,.$
(61)

One should note, comparing (61) with (49), that the action of the gate $A$
operation onto the process vector effectively performs the following
transformation,

${|{\Psi}\rangle}^{A_{I}}\to{|{U\Psi}\rangle}^{A_{O}}\to{|{U\Psi}\rangle}^{B_{I}^{\prime}}\,,\qquad{|{v}\rangle}^{A_{I}}\to{|{v}\rangle}^{A_{O}}\to{|{v}\rangle}^{B_{I}^{\prime}}\,,$
(62)

where the transport vector
${|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}B_{I}^{\prime}}$ has been utilised
for “transporting” the state from the output $A_{O}$ of gate $A$ to the input
$B_{I}^{\prime}$ of the gate $B^{\prime}$, in line with the spacetime diagram.
This scheme repeats itself with the action of all remaining gate operations on
(61). In particular, the subsequent action of the gate $B$ operation gives:

$\left({\langle\\!\langle{\tilde{V}^{\ast}}|}^{B_{I}B_{O}}\otimes{\langle\\!\langle{\tilde{U}^{\ast}}|}^{A_{I}A_{O}}\right){|{W_{{QS}_{4}}}\rangle\\!\rangle}=\hphantom{mmmmmmmmmmmmmmmmmmmmmmm}$
$\frac{1}{\sqrt{2}}\left({|{U\Psi}\rangle}^{B_{I}^{\prime}}{|{v}\rangle}^{A_{I}^{\prime}}+{|{v}\rangle}^{B_{I}^{\prime}}{|{V\Psi}\rangle}^{A_{I}^{\prime}}\right){|{\mathbbm{1}}\rangle\\!\rangle}^{(A_{O}^{\prime}B_{O}^{\prime})S_{(AB)_{I}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S_{(AB)_{O}}T_{(AB)_{I}}}\,,$
(63)

which can also be verified with an explicit calculation similar to the above.
Continuing on, the operations at the gates $A^{\prime}$ and $B^{\prime}$ give:

$\begin{array}[]{l}\displaystyle\left({\langle\\!\langle{\tilde{V}^{\ast}}|}^{B_{I}^{\prime}B_{O}^{\prime}}\otimes{\langle\\!\langle{\tilde{U}^{\ast}}|}^{A_{I}^{\prime}A_{O}^{\prime}}\otimes{\langle\\!\langle{\tilde{V}^{\ast}}|}^{B_{I}B_{O}}\otimes{\langle\\!\langle{\tilde{U}^{\ast}}|}^{A_{I}A_{O}}\right){|{W_{{QS}_{4}}}\rangle\\!\rangle}=\vphantom{\displaystyle\frac{1}{\sqrt{2}}}\\\
\hphantom{mmmm}\displaystyle\frac{1}{\sqrt{2}}\left({|{VU\Psi}\rangle}^{S_{B_{I}}}{|{v}\rangle}^{S_{A_{I}}}+{|{v}\rangle}^{S_{B_{I}}}{|{UV\Psi}\rangle}^{S_{A_{I}}}\right){|{\mathbbm{1}}\rangle\\!\rangle}^{S_{(AB)_{O}}T_{(AB)_{I}}}\,.\\\
\end{array}$ (64)

Next, the action of the beam splitter at the gate $S_{f}$ gives

$\begin{array}[]{l}\displaystyle\left({\langle\\!\langle{\tilde{H}^{\ast}}|}^{S_{I}S_{O}}\otimes{\langle\\!\langle{\tilde{V}^{\ast}}|}^{B_{I}^{\prime}B_{O}^{\prime}}\otimes{\langle\\!\langle{\tilde{U}^{\ast}}|}^{A_{I}^{\prime}A_{O}^{\prime}}\otimes{\langle\\!\langle{\tilde{V}^{\ast}}|}^{B_{I}B_{O}}\otimes{\langle\\!\langle{\tilde{U}^{\ast}}|}^{A_{I}A_{O}}\right){|{W_{{QS}_{4}}}\rangle\\!\rangle}=\vphantom{\displaystyle\frac{1}{\sqrt{2}}}\\\
\hphantom{mmmm}\displaystyle\frac{1}{2}\sum_{i=0}^{1}\left({\langle{i}|}\,\\{\,U\,,\,V\,\\}\,{|{\Psi}\rangle}\;{|{i}\rangle}^{T_{A_{I}}}{|{v}\rangle}^{T_{B_{I}}}+{\langle{i}|}\,[\,U\,,\,V\,]\,{|{\Psi}\rangle}\;{|{v}\rangle}^{T_{A_{I}}}{|{i}\rangle}^{T_{B_{I}}}\right)\,.\\\
\end{array}$ (65)

Finally, the action of the operations of the target gates $T_{A}$ and $T_{B}$
gives us the probability amplitude as a function of the measurement outcomes
$\alpha$ and $\beta$,

${\cal M}(\alpha,\beta)\equiv\hphantom{mmmmmmmmmmmmmmmmmmmmmmmmmm}$
$\left({\langle\\!\langle{\tilde{U}^{\ast}}|}^{A_{I}A_{O}}\otimes{\langle\\!\langle{\tilde{U}^{\ast}}|}^{A_{I}^{\prime}A_{O}^{\prime}}\otimes{\langle\\!\langle{\tilde{V}^{\ast}}|}^{B_{I}B_{O}}\otimes{\langle\\!\langle{\tilde{V}^{\ast}}|}^{B_{I}^{\prime}B_{O}^{\prime}}\otimes{\langle\\!\langle{H^{\ast}}|}^{S_{I}S_{O}}\otimes{\langle\\!\langle{T_{\alpha}^{\ast}}|}^{T_{A_{I}}T_{A_{O}}}\otimes{\langle\\!\langle{T_{\beta}^{\ast}}|}^{T_{B_{I}}T_{B_{O}}}\right){|{W_{{QS}_{4}}}\rangle\\!\rangle}$
$=\frac{1}{2}\Big{[}\delta_{\beta
v}{\langle{\alpha}|}\,\\{\,U\,,\,V\,\\}\,{|{\Psi}\rangle}+\delta_{\alpha
v}{\langle{\beta}|}\,[\,U\,,\,V\,]\,{|{\Psi}\rangle}\Big{]}\,.$ (66)

At this point we can employ (45) and calculate the probability distribution,

$p(\alpha,\beta)=\frac{1}{4}\Big{[}\delta_{\beta
v}\left|{\langle{\alpha}|}\,\\{\,U\,,\,V\,\\}\,{|{\Psi}\rangle}\vphantom{A^{A}}\right|^{2}+\delta_{\alpha
v}\left|{\langle{\beta}|}\,[\,U\,,\,V\,]\,{|{\Psi}\rangle}\vphantom{A^{A}}\right|^{2}\Big{]}\,,\\\
$ (67)

where we have used the fact that the vacuum state ${|{v}\rangle}$ is
orthogonal to both $\,\\{\,U\,,\,V\,\\}\,{|{\Psi}\rangle}$ and
$\,[\,U\,,\,V\,]\,{|{\Psi}\rangle}$. In particular, we see that for
$i\in\\{0,1\\}$ we have

$p(i,v)=\frac{1}{4}\left|{\langle{i}|}\,\\{\,U\,,\,V\,\\}\,{|{\Psi}\rangle}\vphantom{A^{A}}\right|^{2}\,,\qquad
p(v,i)=\frac{1}{4}\left|{\langle{i}|}\,[\,U\,,\,V\,]\,{|{\Psi}\rangle}\vphantom{A^{A}}\right|^{2}\,,$
(68)

while all other choices of $\alpha$ and $\beta$ give $p(\alpha,\beta)=0$. The
total probability that Alice will detect a particle is given by the marginal

$p_{A}=\sum_{i=1}^{2}p(i,v)=\frac{1}{2}\left(1+\mathop{\rm
Re}\nolimits{\langle{\Psi}|}U^{\dagger}V^{\dagger}UV{|{\Psi}\rangle}\vphantom{A^{A}}\right)\,,$
(69)

while the corresponding total probability that Bob will detect a particle is

$p_{B}=\sum_{i=1}^{2}p(v,i)=\frac{1}{2}\left(1-\mathop{\rm
Re}\nolimits{\langle{\Psi}|}U^{\dagger}V^{\dagger}UV{|{\Psi}\rangle}\vphantom{A^{A}}\right)\,.$
(70)

As a final point, we can verify that the probability distribution is correctly
normalised. Using the fact that the only nonzero values for the probability
are given in (68), we have

$p_{\text{total}}=\sum_{\alpha=0}^{2}\sum_{\beta=0}^{2}p(\alpha,\beta)=\underbrace{p(0,v)+p(1,v)}_{p_{A}}+\underbrace{p(v,0)+p(v,1)}_{p_{B}}=1\,,$
(71)

as expected.

Instead of recombining the particle on the second beam splitter, one can
consider the case in which the final gate $\mathcal{F}$ consists only of local
measurements performed onto a particle in the Alice’s and Bob’s paths. In this
case, the final gate is equivalent to two target gates $T_{A}$ and $T_{B}$.
Given that all the processes considered are pure, the corresponding process
vector for the 4-event quantum switch implementation is given as (in order to
compare the current with the previous works, we present the process that
starts after $\mathcal{I}$, as was done in, say, [10]):

${|{W_{{QS}_{4}}}\rangle\\!\rangle}=\frac{1}{\sqrt{2}}\left({|{\Psi}\rangle}^{A_{I}}{|{v}\rangle}^{B_{I}}+{|{v}\rangle}^{A_{I}}{|{\Psi}\rangle}^{B_{I}}\right){|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}B_{I}^{\prime}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}A_{I}^{\prime}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}^{\prime}T_{A_{I}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}^{\prime}T_{B_{I}}}.$
(72)

##  Appendix D 3-event process vector

The detailed spacetime diagram for the $3$-event quantum switch is given below
(see Figure 15).

spacetime$t_{i}$$t_{1}$$t_{B}$$t_{2}$$t_{f}$AliceBob$\begin{subarray}{c}\text{\tiny
beam}\\\ \text{\tiny
splitter}\end{subarray}$$S_{i}$$S_{f}$$A$$B$$A^{\prime}$$P_{A}$$P_{B}$$\mathcal{I}$$T_{A}$$T_{B}$$\mathcal{F}$
Figure 15: Spacetime diagram of the $3$-event implementation of the quantum
switch. The internal structures of the composite gates ${\cal I}$ and ${\cal
F}$ are explicitly depicted.

The process vector for this case is obtained from (44) by identifying the
spacetime positions of the gates $B$ and $B^{\prime}$, yet keeping the
corresponding Hilbert spaces in the mathematical description (i.e., keeping
the dimensionality of the problem). Thus, the corresponding circuit is
identical to the $4$-event circuit, and the process vector has the identical
mathematical form as in the case of four gates. In order to emphasise the
physical difference between the two cases, instead of $B_{I/O}$ and
$B_{I/O}^{\prime}$, we write $B_{I_{1}/O_{1}}$ and $B_{I_{2}/O_{2}}$,
respectively:

$\begin{array}[]{lcl}{|{W_{\text{3-event}}}\rangle\\!\rangle}&=&\displaystyle\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{P_{A_{O}}S^{i}_{A_{I}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{P_{B_{O}}S^{i}_{B_{I}}}}_{\text{initial}}\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{i}_{A_{O}}A_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}B_{I_{2}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O_{2}}S^{f}_{B_{I}}}}_{\text{blue}}\\\
&&\displaystyle\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{i}_{B_{O}}B_{I_{1}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O_{1}}A^{\prime}_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A^{\prime}_{O}S^{f}_{A_{I}}}}_{\text{red}}\underbrace{{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{f}_{A_{O}}T_{A_{I}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{S^{f}_{B_{O}}T_{B_{I}}}}_{\text{final}}\,.\\\
\end{array}$ (73)

The final probability distribution is identical to the one for the 4-event
process, given by (67).

##  Appendix E 2-event

In this Appendix we present process vectors for the two gravitational switches
discussed in the main text. First, the process vector of the gravitational
switch without recombination [16], is given by (since the “control” is now
played by gravity, it is thus denoted as $G$, instead of $C$):

${|{W_{{QS}_{2}}}\rangle\\!\rangle}=\frac{1}{\sqrt{2}}\left({|{0}\rangle}^{G}{|{\Psi}\rangle}^{A_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}B_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}T_{B_{I}}}+{|{1}\rangle}^{G}{|{\Psi}\rangle}^{B_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}A_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}T_{A_{I}}}\right){|{\mathbbm{1}}\rangle\\!\rangle}^{GT_{G_{I}}}.$
(74)

It is then straightforward to obtain the process vector for the case of
recombining only the gravity on the final beam splitter $S_{f}$ (a part of a
bigger final gate $\mathcal{F}$), obtaining the analogue of (49), while the
particle is not being recombined. Note that in this case the introduction of
the vacuum state is not necessary, as in each branch of superposition all
gates are acting upon a particle.

In contrast to the above case, the process vector describing the gravitational
2-event quantum switch with the recombination of both gravity and the particle
is given as follows:

$\begin{array}[]{rl}{|{W_{{QS}_{2}}^{(r)}}\rangle\\!\rangle}=\displaystyle\frac{1}{\sqrt{2}}\\!\\!\\!\\!&\left({|{0}\rangle}^{G}{|{\Psi}\rangle}^{A_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}B_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}S_{P_{I}}}+{|{1}\rangle}^{G}{|{\Psi}\rangle}^{B_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}A_{I}}{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}S_{P_{I}}}\right)\\\
&\otimes{|{\mathbbm{1}}\rangle\\!\rangle}^{GS_{G_{I}}}{|{\mathbbm{1}}\rangle\\!\rangle}^{(S_{G_{O}}S_{P_{O}})(T_{G_{I}}T_{P_{I}})}.\end{array}$
(75)

Here, $P$ stands for “the particle” (whose corresponding input space
$S_{P_{I}}$ is isomorphic to the tensor product of Alice’s and Bob’s output
spaces, $S_{P_{I}}\simeq A_{O}\otimes B_{O}$), and $S$ stands for a “beam
splitter” (whose corresponding input space is $S_{I}=S_{G_{I}}\otimes
S_{P_{I}}$, and analogously for the output space).

While defining the spaces $S_{G_{I/O}}$, $S_{P_{I/O}}$, $T_{G_{I}}$,
$T_{P_{I}}$, and the vector
${|{\mathbbm{1}}\rangle\\!\rangle}^{(S_{G_{O}}S_{P_{O}})(T_{G_{I}}T_{P_{O}})}$
is straightforward, it is not so for the “final” recombination vector
${|{U_{BS}^{\ast}}\rangle\\!\rangle}^{(S_{G_{I}}S_{P_{I}})(S_{G_{O}}S_{P_{O}})}$.
Namely, note that in our gravitational switch all degrees of freedom, both
gravitational and matter, are recombined by $U_{BS}$ such that, by acting on
the beam splitter input entangled state,

${|{\Psi_{i}}\rangle}^{S_{G_{I}}S_{P_{I}}}=\frac{1}{\sqrt{2}}\left({|{0}\rangle}^{S_{G_{I}}}\otimes\left[UV{|{\Psi}\rangle}^{S_{P_{I}}}\right]+{|{1}\rangle}^{S_{G_{I}}}\otimes\left[VU{|{\Psi}\rangle}^{S_{P_{I}}}\right]\right),$
(76)

the overall output state is a product one, of the form

${|{\Psi_{o}}\rangle}^{S_{G_{O}}S_{P_{O}}}={|{\Gamma}\rangle}^{S_{G_{O}}}\otimes\left(\alpha
UV+\beta VU\right){|{\Psi}\rangle}^{S_{P_{O}}},$ (77)

where ${|{\Gamma}\rangle}^{S_{G_{O}}}$ is some (not necessarily classical)
state of gravity. The above evolution is, at least in principle, allowed by
the quantum laws, which is all that we need to know regarding the action of
$U_{BS}$ at this point. Its action on the rest of the overall Hilbert space
is, for the purpose of our argument, irrelevant, and can thus be chosen
arbitrarily.

Finally, we would like to note that the same type of the 4-, or 3-event
quantum switches in classical spacetimes can also be defined, resulting in the
same type of the output state as (77), with the gravity degree of freedom
being replaced by any additional matter degree of freedom that plays the role
of the control.

##  Appendix F Various implementations of the gravitational switch

In this Appendix we present a few representative additional constructions of
the gravitational quantum switch. First, we start with a $2$-event switch for
which the requirement (i) from Subsection 4.3 is not satisfied. It is obvious
from the diagram on the left that each of the photon’s superposed trajectories
fail to meet at the boundary region, violating requirement (i), see the left
diagram of Figure 16. Next, we proceed with the $2$-event implementation for
which requirement (i) is satisfied, but the requirement (ii) is not, since the
superposed trajectories fail to recombine. This is depicted on the right
diagram of Figure 16.

Finally, we present a $4$-event implementation for which both requirements (i)
and (ii) are satisfied (see Figure 17).

Of course, other variations are possible as well.

Friend
particleAliceBob$\tilde{A}$$\tilde{B}$$\tilde{F}_{A}$$\tilde{F}_{A}$$\tilde{F}_{B}$$\tilde{F}_{B}$$\mathcal{I}$$\mathcal{F}$

Friend
particleAliceBob$\tilde{A}$$\tilde{B}$$\tilde{F}_{A}$$\tilde{F}_{A}$$\tilde{F}_{B}$$\tilde{F}_{B}$$\mathcal{I}$$\mathcal{F}$

Figure 16: The spacetime diagrams of a $2$-event gravitational switch
implementations, with Friend’s measurements, which fail to distinguish them
from the optical implementation of the quantum switch. Friend
particleAliceBob$A$$A^{\prime}$$B$$B^{\prime}$$F_{AA^{\prime}}$$F_{BB^{\prime}}$$\mathcal{I}$$\mathcal{F}$
Figure 17: The spacetime diagram of a $4$-event gravitational switch
implementation, with Friend’s measurement, which fails to distinguish it from
the $2$-event implementation of the gravitational switch.