# Experimental Entanglement of Temporal Orders

Giulia Rubino1∗, Lee A. Rozema1, Francesco Massa1, Mateus Araújo2,
Magdalena Zych3, Časlav Brukner1,4, Philip Walther1∗ 1Vienna Center for
Quantum Science and Technology (VCQ), Faculty of Physics, University of
Vienna, Boltzmanngasse 5, Vienna A-1090, Austria
2Institute for Theoretical Physics, University of Cologne, Germany
3Centre for Engineered Quantum Systems, School of Mathematics and Physics, The
University of Queensland, St Lucia, QLD 4072, Australia
4Institute for Quantum Optics & Quantum Information (IQOQI), Austrian Academy
of Sciences, Boltzmanngasse 3, Vienna A-1090, Austria

∗To whom correspondence should be addressed;
E-mail<EMAIL_ADDRESS>(G.R<EMAIL_ADDRESS>(P.W.)

(September 2, 2024)

###### Abstract

The study of causal relations, a cornerstone of physics, has recently been
applied to the quantum realm, leading to the discovery that not all quantum
processes have a definite causal structure. Here, we present the first theory-
independent experimental demonstration of entangled temporal orders, resulting
in a process with an indefinite causal structure. While such processes have
previously been observed, these observations relied on the assumption that
experimental operations and systems are described by quantum theory. This
opens a ‘loophole’ wherein the observed process can be explained by an
underlying theory with a definite causal structure. To circumvent this, we
build a model attempting to describe our experimental data using a large class
of general probabilistic theories that are local and have a definite temporal
order. We then experimentally invalidate this model by violating a Bell
inequality. We therefore conclude that nature is incompatible with theories
requiring a local definite temporal order.

Bell’s theorem revolutionized the foundations of physics, proving that quantum
mechanics cannot be described by a local-realist theory, and paving the way
for modern quantum information Bell1964 ; RevModPhys.86.419 . Over the past
decades, the theorem has been violated with many different physical systems
thereby entangling different observables (such as spin PhysRevD.14.2543 ;
Nature.409.791 ; Nature.526.682 , polarization PhysRevLett.28.938 ;
PhysRevLett.49.1804 ; PhysRevLett.115.250402 ; PhysRevLett.115.250401 ,
position PhysRevLett.92.210403 , and energy PhysRevA.47.R2472 ;
PhysRevLett.64.2495 ) of two or more particles. However, since there is no
observable corresponding to a measurement of the temporal order between
events, this theorem had never been applied to causal structures.

Typically, in all of our well-established theories, it is assumed that the
order between events is pre-defined, precluding the possibility of observing
situations where the causal order is genuinely indefinite. Nevertheless, it
was recently realized that quantum mechanics predicts the existence of
processes that are neither causally ordered nor a probabilistic mixture of
causally ordered processes. In other words, these processes cannot be
understood as one-way-signalling quantum channels, quantum states, or any
convex mixture of them PhysRevA.88.022318 ; NatComm.3.012316 ; NatPhys.10.259
. More precisely, a quantum process is called causally separable if it can be
decomposed as a convex combination of causally ordered processes, otherwise it
is causally non-separable. (Note that the term ‘temporal’ order is here used
to refer to events which cannot be used to receive signals — in particular, to
unitary operations — whereas ‘causal’ order refers to more general operations
which allow for the exchange of information.) Recently, a method for
certifying causal separability, based on ‘causal witnesses’, was developed
NewJournPhys.17.102001 ; 1367-2630-18-9-093020 ; ScienRep.6.26018 , and used
to experimentally demonstrate that a certain process — a quantum SWITCH
PhysRevA.86.040301  — is causally non-separable Rubinoe1602589 .

In the quantum SWITCH, a qubit is transmitted between two parties, and the
order in which the parties receive it is entangled with a second system, which
can result in a superposition of temporal orders. The existence of such a
superposition has been experimentally demonstrated NatCommun.6 ;
Rubinoe1602589 . However, the certification of this ‘indefiniteness’ of
temporal orders was theory-dependent, requiring the assumption that the system
under investigation and the applied operations were described by quantum
theory. In more detail, Ref. Rubinoe1602589  reported the measurement of a
value for a causal witness that could not be explained by any model making the
following three assumptions: there was a definite causal order between the
parties, each party acted only once, and the quantum description of their
operations was the correct one. Under these conditions, it was concluded that
the causal order was indefinite in the experiment. Nevertheless, the
experimental results could also potentially have been explained in a causal
manner by a different description (i.e., not by quantum theory) of their
experiment. Thus, the structure of an indefinite causal order has not yet been
probed without assuming the validity of the quantum formalism.

In addition to theory-dependent causal witnesses, there are also theory-
independent ways of certifying indefinite causal orders via ‘causal
inequalities’ NatComm.3.012316 ; 1367-2630-18-1-013008 . These inequalities
only require one to measure the probabilities of outcomes for different
parties in the process under study. Any probabilities that show signaling in
only one direction — which can be interpreted as an influence from the past to
the future — or that is a convex mixture of those which allow signaling only
in one direction (from $A$ to $B$ or from $B$ to $A$) satisfy causal
inequalities. It can be shown that the quantum SWITCH satisfies all such
causal inequalities (see Refs. NewJournPhys.17.102001 ; 1367-2630-18-9-093020
or Suppl. Material for details). Currently, it is not known how to realize a
process which violates a causal inequality. The question then arises if it is
at all possible to prove the existence of an indefinite causal order in a
theory-independent manner. Here, we provide an affirmative answer to this
question by experimentally violating a Bell inequality for temporal orders,
thereby demonstrating that the order of events in our experiment cannot be
pre-defined using a large class of generalized probabilistic theories
satisfying our assumptions (i.e., locality of states and locality of
laboratory operations).

In our work, we generalize a Bell’s inequality for temporal orders mag , and
then experimentally violate it. The inequality is derived under ‘theory-
independent’ (i.e., it is valid for a large class of so-called ‘generalized
probabilistic theories’) yet ‘device-dependent’ assumptions (i.e., assumptions
which depend on the internal functionality of experimental devices). Our
experimental data show a violation of the Bell inequality, thus demonstrating
that Nature is incompatible with a large class of theories that assume the
local order between events to be pre-defined.

##  I No-Go Theorem for Definite Temporal Orders

We now present a no-go theorem for definite temporal orders that applies to a
large class of generalized probabilistic theories (GPTs) for which the order
of local events is assumed to be predefined. GPTs are a general framework
encompassing all operational theories – including classical probability theory
and quantum theory as special cases – that specify a set of laboratory devices
that can be applied on physical systems, and assigns probabilities to
experimental outcomes GPT1 ; GPT2 ; GPT3 ; 1367-2630-13-6-063001 . This no-go
theorem was previously derived in the context of gravity mag . Our derivation
uses an assumption about the initial state of the systems weaker than that in
Ref. mag  (we consider Bell-local states rather than separable states, which
are a subset of Bell-local states) and a different notion of locality. (The
relation between the assumptions and implications of the current work and
those of Ref. mag  are analyzed in Methods - Section III.-VI..)

We first define what we mean by a causal order in a GPT. Consider a system in
the state $\omega\in\Omega$ of a GPT state space $\Omega$ and imagine two
parties, Alice and Bob, who perform some operations on this state. For
example, suppose that the operation in Alice’s laboratory is given by a
transformation $\mathcal{A}$ and that in Bob’s laboratory is given by a
transformation $\mathcal{B}$. Alice’s and Bob’s operations are said to undergo
a process that is ‘causally separable’ in GPTs whenever Alice’s operation
happens before Bob’s ($\mathcal{A}\preceq\mathcal{B}$), Bob’s operation
happens before Alice’s ($\mathcal{B}\preceq\mathcal{A}$), or there is a convex
mixture of these two cases:

$\mathcal{S}(\omega)=\zeta\,\mathcal{B}\bigl{(}\mathcal{A}(\omega)\bigr{)}+(1-\zeta)\,\mathcal{A}\bigl{(}\mathcal{B}(\omega)\bigr{)}.$
(1)

where $0\leqslant\zeta\leqslant 1$ is the probability with which one or the
other order is chosen and $\mathcal{Y}\bigl{(}\mathcal{X}(\cdot)\bigr{)}$ is a
composition of operations $\mathcal{X}$ and $\mathcal{Y}$. While in the
current work we limit our analysis to the case of only $N=2$ parties, an
analogue relation can be established for $N>2$ parties, giving rise to a
classical mixture of all possible permutations among the $N$ parties, or to a
dynamical causal order, where the causal order between operations may depend
on operations performed beforehand Abbott2017genuinely . If a process cannot
be written in the form of Eq. 1, it is called a ‘causally non-separable
process’.

Within the GPT framework, we now consider $\omega$ to be a state of the
following composite system: one system (the control system) governing the
order in which the operations $\mathcal{A}$ and $\mathcal{B}$ are applied, and
one system (the target system) on which the operations are performed. We will
further imagine that there are two parties, S1 and S2, each possessing one
such composite system.

In the Methods - Section III. we prove a no-go theorem, stating that any two-
party system that obeys the following three assumptions cannot violate a Bell
inequality.

>   1. 1.
>
> The initial joint state of the target system is local (i.e., it does not
> violate a Bell inequality).
>
>   2. 2.
>
> The laboratory operations are  local transformations of the target systems
> (i.e., they do not increase the amount of a violation of Bell inequalities
> between the two target systems).
>
>   3. 3.
>
> The order of local operations on the two target systems is pre-defined.
>
>

We will briefly comment on assumption 2. here, and refer to the Results
section for an in-depth analysis of all three assumptions. Assumption 2.
implies that the laboratory operations do not have any ‘global’ effects on the
joint states of the target and control systems in a GPT on the operational
level. This has two implications, and hence we further identify two sub-
assumptions within it. (2a) The laboratory operations cannot increase the
‘non-classical correlations’ (i.e., non-local correlations) between the target
systems of parties S$1$ and S$2$. In other words there is no non-local
interaction between the two systems. (2b) Within a single party S$i$, the
laboratory operations act only on the target system, and hence they cannot
increase the ‘non-classical correlations’ between the control and the target
systems (i.e., they do not ‘couple’ the two systems.)

In the next section, we will present a quantum mechanical process that
violates this no-go theorem. Hence, at least one of the assumptions must not
hold for this process. In the Results section, we will analyze the
experimental data to corroborate assumptions 1. and 2. using GPTs. This
implies that the order of operations within each system $\text{S}i$ is
indefinite.

###  I.1 Entangled quantum SWITCH

To understand a single quantum SWITCH, first imagine two parties, Alice and
Bob, who are in two closed laboratories; i.e., their only interaction with the
outside environment is through an input and output system. They each perform a
fixed operation on the same qubit (the ‘target’ qubit). The target qubit can
be first sent to Alice and then Bob, or vice versa. In a quantum SWITCH, one
controls the order of the operations on the target qubit with the state of a
second, a ‘control’, qubit. For example, if the control qubit is in the state
$\ket{0}^{C}$, the target qubit is sent first to Alice and then to Bob, and
vice versa if the control qubit is in the state $\ket{1}^{C}$. When the
control qubit is prepared in the state
$\bigl{(}{\ket{0}^{C}+\ket{1}^{C}}\bigr{)}/\ \sqrt{2}$, the resulting process
has been shown to be causally non-separable within quantum mechanics
NewJournPhys.17.102001 ; Rubinoe1602589 ; PhysRevA.88.022318 ;
PhysRevA.86.040301 .

Now consider two quantum SWITCHes (S1 and S2), each containing an Alice and a
Bob. S1 and S2 are prepared in a state where their control qubits are
entangled, but their target qubits are in a product state (see Fig. 1):

$\ket{0}_{1}^{T}\otimes\ket{0}_{2}^{T}\otimes\left(\dfrac{\ket{0}_{1}^{C}\otimes\ket{0}_{2}^{C}-\ket{1}^{C}_{1}\otimes\ket{1}^{C}_{2}}{\sqrt{2}}\right)$
(2)

The superscripts $C$ and $T$ refer to the control and target qubits within one
SWITCH, respectively, while the subscripts $1$ and $2$ refer to SWITCH S1 and
S2. Since we will attempt to observe a Bell violation with the target qubits,
which are in a separable state, this initial condition satisfies assumption 2.
in quantum theory.

Figure 1: Entangled quantum SWITCH. Our work is based on two quantum SWITCHes
(SWITCH S1 and SWITCH S2). In each quantum SWITCH, there are two parties,
Alice (${U}_{i_{\text{A}}}$) and Bob (${U}_{i_{\text{B}}}$). A target qubit is
first sent to one party and then to the other. The order in which the qubit is
sent to the two parties is governed by the state of an additional qubit: if
the state of the control qubit is $\ket{0}_{i}$, the target qubit is sent
first to Alice and then Bob (Panel a) ), and vice versa if the control qubit
is in the state $\ket{1}_{i}$ (Panel b) ). In our work, we entangle the
control qubits (Panel c)). In this case, the order in which the target qubit
in SWITCH S1 passes through ${U}_{1_{\text{A}}}$ and ${U}_{1_{\text{B}}}$ is
entangled with the order in which the target qubit in SWITCH S2 passes through
${U}_{2_{\text{A}}}$ and ${U}_{2_{\text{B}}}$. The control qubits are measured
in the basis $\\{\ket{+}_{i},\ket{-}_{i}\\}$ (labelled $O$). If the orders
inside the two SWITCHes are entangled, it will be possible to violate a Bell
inequality by measuring the target qubits after the SWITCHes (BM). This is
possible even if the target qubits start in a separable state and only local
operations are applied within each SWITCH.

Given this input state and the action of an individual SWITCH, it is
straightforward to calculate the output of the entangled quantum SWITCH system

$\displaystyle\dfrac{1}{\sqrt{2}}\Bigl{(}{U}_{1_{\text{B}}}{U}_{1_{\text{A}}}\ket{0}_{1}^{T}\Bigr{)}\otimes\ket{0}_{1}^{C}\otimes\Bigl{(}{U}_{2_{\text{B}}}{U}_{2_{\text{A}}}\ket{0}_{2}^{T}\Bigr{)}\otimes\ket{0}_{2}^{C}$
(3)
$\displaystyle-\dfrac{1}{\sqrt{2}}\Bigl{(}{U}_{1_{\text{A}}}{U}_{1_{\text{B}}}\ket{0}_{1}^{T}\Bigr{)}\otimes\ket{1}_{1}^{C}\otimes\Bigl{(}{U}_{2_{\text{A}}}{U}_{2_{\text{B}}}\ket{0}_{2}^{T}\Bigr{)}\otimes\ket{1}_{2}^{C}$

where ${U}_{i_{\text{A}}}$ and ${U}_{i_{\text{B}}}$ $(i=1,2)$ are the
unitaries performed by the two parties Alice and Bob inside each SWITCH S$i$.

Next, we measure the two control qubits in the basis
$\\{\ket{+}\bra{+},\ket{-}\bra{-}\\}$. If we observe both of the control
qubits in the same state (either $\ket{+}_{1}^{C}\ket{+}_{2}^{C}$ or
$\ket{-}_{1}^{C}\ket{-}_{2}^{C}$), the target qubits will be in the (in
general) unnormalised state

$\dfrac{1}{\sqrt{2}}\bigl{(}{U}_{1_{\text{B}}}{U}_{1_{\text{A}}}\ket{0}^{T}_{1}\otimes{U}_{2_{\text{B}}}{U}_{2_{\text{A}}}\ket{0}^{T}_{2}-{U}_{1_{\text{A}}}{U}_{1_{\text{B}}}\ket{0}^{T}_{1}\otimes{U}_{2_{\text{A}}}{U}_{2_{\text{B}}}\ket{0}^{T}_{2}\bigr{)},$
(4)

while if we find the control qubits in orthogonal states (either
$\ket{+}_{1}^{C}\ket{-}_{2}^{C}$ or $\ket{-}_{1}^{C}\ket{+}_{2}^{C}$) the sign
between the two terms in the superposition in the equation above is ‘+’. In
general, depending on the choice of the unitaries in the two SWITCHes, the
target qubits will be left either in a separable or an entangled state. In
particular, if we choose the gates

${U}_{1_{\text{A}}}={U}_{2_{\text{A}}}=\sigma_{z}\qquad{U}_{1_{\text{B}}}={U}_{2_{\text{B}}}=\dfrac{\mathbb{1}+i\sigma_{x}}{\sqrt{2}},$
(5)

where $\sigma_{x}$ ans $\sigma_{z}$ are the Pauli operators, the state of the
target qubits becomes

$\dfrac{1}{\sqrt{2}}\bigl{(}\ket{l}_{1}^{T}\ket{l}_{2}^{T}-\ket{r}_{1}^{T}\ket{r}_{2}^{T}\bigr{)},$
(6)

where $\ket{r}=\bigl{(}\ket{0}-i\ket{1}\bigr{)}/\ \sqrt{2}$ and
$\ket{l}=\bigl{(}\ket{0}+i\ket{1}\bigr{)}/\ \sqrt{2}$. This is a maximally
entangled state and, as a result, one can now violate a Bell inequality
maximally on the target qubits.

In the language of quantum mechanics, this shows that the indefiniteness of
the temporal orders in the two SWITCHes can be displayed via the entanglement
between their target qubits. In other words, such entanglement has not been
‘generated’, but rather ‘transferred’ from the control qubits by means of the
indefinite temporal order of the unitaries applied.

Outside of quantum mechanics, the presence of non-classical correlations can
be determined through a violation of a Bell inequality. In our case, violating
a Bell inequality also violates the no-go theorem for temporal orders, proving
that no underlying GPTs where 1., 2. and 3. hold can explain the experimental
data. By demonstrating without assuming validity of quantum mechanics that
assumptions 1. and 2. hold, the presence of an indefinite temporal order is
shown theory-independently.

###  I.2 Experimental scheme

We create a quantum SWITCH with entangled control qubits using a photonic set-
up. Let us first consider a single quantum SWITCH. Each quantum SWITCH applies
gates on a target qubit, where the gates’ order depends on the state of a
control qubit. Experimentally, we encode the control qubit in a path degree of
freedom (DOF), and the target qubit in the polarization DOF of a single
photon. The photon is initially placed in a superposition of two paths (as
explained in Fig. 2 and the Methods - Section I.). These paths are labeled
$0_{1}$ and $1_{1}$ for SWITCH S1 and $0_{2}$ and $1_{2}$ for SWITCH S2 in
Fig. 2. The two paths are then routed through a two-loop Mach-Zehnder
interferometer NatCommun.6 ; Rubinoe1602589 . The $0_{i}$ paths lead the
photons through a set of gates acting on the polarization DOF in the order
${U}_{i_{\text{A}}}\preceq{U}_{i_{\text{B}}}$. While the paths $1_{i}$ guide
the photons through the gates in the opposite order
${U}_{i_{\text{B}}}\preceq{U}_{i_{\text{A}}}$. To generate the maximally
entangled state between the target qubits (Eq. (6) ), we need to implement the
non-commuting gates ${U}_{i_{\text{A}}}=\sigma_{z}$ and
${U}_{i_{\text{B}}}=(\mathbb{1}+i\sigma_{x})/\sqrt{2}$, which we do with
waveplates. In particular, a half-waveplate (HWP) at $0^{\circ}$ for
$\sigma_{z}$ and a sequence of quarter-waveplate (QWP) and HWP both at
$45^{\circ}$ for $(\mathbb{1}+i\sigma_{x})/\sqrt{2}$). After this, the two
paths are recombined on a 50/50 beamsplitter (BS) — which projects the path
DOF in the basis $\\{\ket{+}\bra{+},\ket{-}\bra{-}\\}$. The path lengths and
the relative phases are set by means of a piezo-driven trombone-arm delay
line. At the two outputs of each interferometer, QWPs, HWPs and polarizing
beam splitters (PBSs) are used to perform arbitrary polarization measurements
on the target qubits.

To entangle the two quantum SWITCHes, we first entangle the path DOFs of the
two photons. As explained in the Methods - Section I., we generate path-
entangled photon pairs that are separable in their polarization DOF:

$\displaystyle\ket{\Phi^{-}}_{1,2}^{\text{path}}\otimes(\ket{H}_{1}\ket{H}_{2})^{\text{polarization}}=$
(7)
$\displaystyle=\left(\dfrac{\ket{0}_{1}\ket{0}_{2}-\ket{1}_{1}\ket{1}_{2}}{\sqrt{2}}\right)^{\text{path}}\otimes(\ket{H}_{1}\ket{H}_{2})^{\text{polarization}}.$

Each photon is thus delocalized over two paths. The two photons are then sent
to their respective SWITCHes, and, since the control qubits began in an
entangled state, the order in which the gates act on the two target qubits
becomes entangled.

Figure 2: Experimental implementation of an entangled quantum SWITCH. Each
SWITCH is composed of a two-loop Mach-Zehnder interferometer. The
interferometers start in the photon-pair source, wherein photon 1 and photon 2
are placed in superposition of the paths $0_{1}$ and $1_{1}$, and $0_{2}$ and
$1_{2}$, respectively (see the Methods - Section I.). (For simplicity we have
drawn these paths as fibers, however the photons are transmitted via free-
space from the source to the experiment.) These paths are routed such that
path $0_{i}$ sees gate ${U}_{i_{\text{A}}}$ and then gate
${U}_{i_{\text{B}}}$, and vice versa for the path $1_{i}$. Each gate, acting
on the polarization degree of freedom, is made up of waveplates (as described
in the main text). The paths $0_{i}$ and $1_{i}$ are then combined on a beam
splitter (BS). In SWITCH S1 (SWITCH S2), the photon is detected after the
polarization measurement at $M_{1}$ or $M_{2}$ ($M_{3}$ or $M_{4}$). Together
with the BS (which applies a Hadamard gate to the qubit encoded in the path
DOF), detecting the photon at $M_{1}$ or $M_{2}$ ($M_{3}$ or $M_{4}$) projects
the path qubit on $\ket{+}$ or $\ket{-}$, respectively. Furthermore, within
each measurement $M_{i}$, the polarization qubit can be measured in any basis
by a combination of a quarter-waveplate (QWP), half-waveplate (HWP) and
polarizing beam splitter (PBS).

##  II Results

Our goal is to demonstrate that the order of application of the gates within
the two quantum SWITCHes is genuinely indefinite without assuming that the
laboratory operations and the states of the systems are described by quantum
theory. We can arrive at this conclusion in two steps. We will first argue
that assumptions 1. and 2. are satisfied in our experiment using both quantum
theory and a large class of GPTs. We will then show experimental data that
violates a Bell inequality. From this we can conclude that assumption 3. does
not hold in our experiment, and, therefore, that the local operations within
the two SWITCHes must have been applied in an indefinite temporal order.

Assumption 1. of our no-go theorem says that the joint target state (shared
between system S1 and S2) does not initially violate a Bell inequality. Within
quantum theory, one can show this by demonstrating that the state is
separable; this can be done using quantum state tomography, for example. To
this end, we performed tomography on the target states before the SWITCHes.
The resulting density matrix is shown in Fig. 3, Panels a) and b). For our
experiment, the target was nominally prepared in $\ket{HH}$; our measured
state has a fidelity of $0.935\pm 0.004$ with $\ket{HH}$. Furthermore, the
concurrence of the estimated state is $0.001\pm 0.010$, indicating that,
within experimental error, the initial target state is separable, in agreement
with assumption 1.. The error bars are computed using a Monte Carlo simulation
of our experiment; the dominant contribution comes from errors in setting the
WPs and cross-talk in the polarizing BSs.

To show that a given bipartite state does not violate a Bell inequality within
a GPT, one can demonstrate that the joint state is a product state. This test
can be carried out by performing a set of ‘fiducial’ measurements (defined in
Methods - Section III.) GPT1 ; GPT2 ; GPT3 ; 1367-2630-13-6-063001 , and
analyzing the resulting probabilities directly. If the probabilities for
outcome pairs on a bipartite-state are equal to the product of the two
marginalized probabilities of each subsystem, then the state is a product
state (see Methods - Section III. for more details). We experimentally
performed this characterization for a wide range of measurements; the results
are presented in Suppl. Tabs. 1 and 2. The first four columns are the measured
joint probabilities, the second four columns are the products of the
marginalized probabilities. The excellent agreement between the two sets
indicates that the target joint state is indeed a product state, and cannot
violate a Bell inequality. We further quantify this by calculating the root-
mean-square (RMS) difference between the two distributions, resulting in an
average difference of $0.6\cdot 10^{-2}\pm 2.7\cdot 10^{-2}$. This proves, in
a theory-independent manner, that assumption 1. of our no-go theorem holds.

The second assumption of our no-go theorem says that the laboratory operations
are local transformations acting on the target states. As discussed earlier,
this has two implications. First, it implies that the laboratory operations
performed in the two SWITCHes cannot transform the joint state of the target
systems of S$1$ and S$2$ from a local state to a non-local one (2a). In the
language of quantum mechanics, this means that no entangling operation is
applied between the two target systems. This could be ensured by performing
the operations with a space-like separation in which case the condition would
be guaranteed in any theory obeying relativistic locality. However, in our
experiment, we make the (well-justified) device-dependent assumption that the
laboratory operations are local transformations within S$1$ and S$2$ in GPTs,
since the transformations of the systems take place at spatially separated
parts of the optical table.

The second implication of Assumption 2. is that the laboratory operations do
not increase the non-classical correlations between the control and the target
systems within each SWITCH (2b). This means, in other words, that the
laboratory operations of one party S$i$ do not ‘couple’ the control and the
target systems. Such a coupling would make it possible to transfer the non-
local correlations from the control systems of the two parties S$1$ and S$2$
to their target systems, and therefore a violation of Bell’s inequalities
would be possible even in presence of a definite temporal order. In our
experiment, the laboratory operations (implemented using waveplates) act only
on the state of the target system (the polarization DOF), and there is no
device that directly couples it to the control DOF (the path), i.e., there is
no gate (such as, e.g., a PBS) that could directly ‘swap’ the entanglement.

We can quantify how well our experiment agrees with Assumption (2b) in a
theory independent manner using a similar technique as for Assumption 1.. We
start by preparing the target and control systems of a single SWITCH in a
product state, and choosing a state for the control system such that the order
of operations within each SWITCH is well-defined. (Notice that this is a
device-dependent step, but the overall proof will remain theory-independent as
everything is described in the framework of GPTs.) In this case, the output
state of the two systems remains a product state provided that there is no
coupling between the control and target systems. We analyze this by performing
a set of measurements on the joint control-target system and showing that the
joint probabilities can be described by the product of the marginal
probabilities (see Suppl. Tabs. 3 and 4, and Methods - Section VI. for more
details). The RMS difference between the two distributions is, on average,
$0.045\pm 0.016\pm 0.010$ (where the first error is a statistical error due to
limited photon counts and the second one is a systematic error due to leakage
from the PBSs used to generate the path entanglement. This measurement is
particularly sensitive to this leakage as it can lead to a non-negligible
amplitude in the unwanted causal order). This value is within two st. dev. of
zero, confirming that the probability distribution is consistent with that of
a product state, and hence the laboratory operations do not couple the control
and target systems.

The next step is to perform a Bell test between the target states at the
output of the apparatus. This allows us to experimentally probe a conjunction
of all three assumptions. Since we know from the above measurements that
assumptions 1. and 2. hold for our apparatus, it will not be possible to
violate a Bell inequality on the target states if the order of operations
inside of SWITCHes S1 and S2 is well-defined.

In previous theory-dependent tests Rubinoe1602589 ; NatCommun.6 , the temporal
order of the operations was made indefinite by placing the state of the
control system in a superposition state. As we showed above (Section 1.1), one
can also achieve this indefiniteness by entangling the two control systems.
This has the advantage of allowing us to perform a theory-independent test of
the indefinite temporal order. Experimentally, the control systems are encoded
in path DOF of two photons. We verified the initial entanglement by performing
a Bell measurement on the joint control system before the SWITCH, obtaining a
CHSH parameter of $2.58\pm 0.09$ (see Fig. 3). Thus, we have confirmed that
the joint target system starts in a separable state, while the joint control
system is initially entangled. We then send this joint state into our two
quantum SWITCHes and perform measurements on the output state.

Within quantum mechanics, we first perform polarization-state tomography on
the two-qubit output target state after the SWITCHes, using four equivalent
measurement set-ups (orange and blue boxes in Fig. 2). Since the 50/50 BSs
apply a Hadamard gate on the path qubits, we post-select the control qubits in
the same state (either $\ket{+}_{1}^{O}\ket{+}_{2}^{O}$ or
$\ket{-}_{1}^{O}\ket{-}_{2}^{O}$) by grouping the results of $M_{1}$ with
$M_{3}$ (orange boxes) and $M_{2}$ with $M_{4}$ (blue boxes). The resulting
density matrix is presented in Fig. 4, and it shows a clear presence of
entanglement. The reconstructed state has a fidelity of $0.922\pm 0.005$ with
the ideal one (Eq. (6) ), and a concurrence of $0.95\pm 0.01$. Finally, to
perform a theory-independent measurement, we perform a Bell test (more
specifically, we measure a CHSH inequality PhysRevLett.23.880 ) on the
polarization DOF, obtaining $S_{\text{target}}=2.55\pm 0.08$. This violates
the inequality, and thus also the no-go theorem, by almost 7 standard
deviations. Hence, no GPT satisfying assumptions 1., 2. and 3. is compatible
with the experimental data. Having previously justified assumptions 1. and 2.
inside and outside the quantum theory, this implies that assumption 3. does
not hold, i.e., that the order of the operations in our experiment is
genuinely indefinite.

As a control, we perform two tests. First, we decrease the entanglement of the
joint control system by increasing the delay of the interferometer inside the
source (see the Methods - Section I.). In this case, each individual SWITCH
could still produce a causally non-separable processes if the control systems
were in a sufficiently coherent state. However, since we insert the control
systems in a mixed separable state, we cannot violate a Bell inequality with
their target systems. The Bell parameter versus the “source visibility” (the
two-photon visibility in its anti-correlated basis) is plotted in Fig. 5a).
The dashed line is a calculation of the expected Bell parameter, including the
imperfect visibility of the two interferometers. All the data points agree
with the expected trend within error. The small step at an entanglement
visibility of around $0.5$ was caused by a lower fringe visibility which
increased the systematic error in setting the phases $\phi_{1}$ and
$\phi_{1}+\pi/4$ (see Fig. 3).

Next, we decrease the degree of causal non-separability of the two processes.
To do this, we introduce distinguishing information between the paths
corresponding to the orders ${U}_{i_{\text{A}}}\preceq{U}_{i_{\text{B}}}$ and
${U}_{i_{\text{B}}}\preceq{U}_{i_{\text{A}}}$ (in only one SWITCH, squares in
Fig. 5b), and in both simultaneously, circles in Fig. 5b)) by lengthening one
of the paths with respect to the other, effectively reducing the visibility of
the interferometers comprising the SWITCHes. As this occurs, we transition
from a superposition of temporal orders to a mixture of them (in other words,
to a causally-separable process, which satisfies assumption 3.). If all three
assumptions are met, one cannot violate a Bell inequality between the two
systems. Indeed, we experimentally observe that as the visibility is
decreased, the Bell parameter also decreases (Fig. 5b). In this plot the
dashed lines are linear fits to the experimental data.

Figure 3: Input state characterization a)-b) State tomography of the target
qubits— The real (Panel a) ) and imaginary (Panel b) ) parts of the two-photon
polarization state measured before the two photons enter the SWITCHes. This
state has a fidelity $0.935\pm 0.004$ with the ideal state $\ket{HH}$, and a
concurrence of $0.001\pm 0.010$. c) Bell measurement on the order qubits— Each
curve is a measurement of a Bell correlation term
$C\bigl{(}\mathcal{O}_{1}(\phi_{1}),\mathcal{O}_{2}(\phi_{2})\bigr{)}$ on the
control qubits, wherein the phase of $\phi_{1}$ is fixed, and the phase
$\phi_{2}$ is scanned. As described in Eq. (8) of the Methods - Section I., we
test the Clauser-Horne-Shimony-Holt (CHSH) inequality PhysRevLett.23.880
achieving a violation of $2.59\pm 0.09$. For the data in the green curve, the
phase $\phi_{1}$ was nominally shifted by $\pi/4$ rad with respect to the blue
curve. The red shaded areas represent the regions where values of $\phi_{1}$
and $\phi_{2}$ correspond with those used to construct our CHSH parameter (Eq.
(8) of the Methods - Section I.). In particular,
$\mathcal{O}=(\mathcal{O}_{1},\mathcal{O}_{2})$ where
$\mathcal{O}_{i}(\phi_{1},\phi_{2})=cos(\phi_{i})\,\sigma_{x}+cos(\phi_{i})\,\sigma_{z}$.
These data confirm that the two photons start in a path-entangled state, and
the polarization state is initially separable.

Figure 4: Output state characterization. Panels a) and b) show the real and
imaginary parts, respectively, of the two-photon polarization state measured
after the photons leave the SWITCHes. For the data shown here, the two control
qubits were found to be in the same state (either
$\ket{+}_{1}^{O}\ket{+}_{2}^{O}$ or $\ket{-}_{1}^{O}\ket{-}_{2}^{O}$). This
state has a fidelity of $0.922\pm 0.005$ with the target state
$(\ket{HV}+\ket{VH})/\sqrt{2}$, and a concurrence of $0.95\pm 0.01$.
Performing a Bell measurement directly using this state results in a CHSH
parameter of $2.55\pm 0.08$.

Figure 5: Bell parameter in the presence of various decoherence sources. a)
For these data the initial entanglement of control qubits is decreased,
passing from the entangled state
$\frac{1}{\sqrt{2}}\bigl{(}\ket{0,0}-\ket{1,1}\bigr{)}$ to a mixture of
$\ket{0,0}$ and $\ket{1,1}$. We measure the Bell parameter both on the input
path qubits (squares) and output polarization qubits (circles) as the source
is decohered. Here, the Bell parameter is plotted versus the visibility of the
entangled state in its anti-correlated basis. The dashed line is a simulation
of the experimental results. b) For these data the coherence of the
superposition of the orders of operations inside the SWITCHes is decreased,
leading to a classical mixture of orders. To control this transition, we
decrease the visibility of either only one of the two interferometers
(circles) or of both interferometers at the same time (squares). Each graph
shows the Bell parameter plotted versus the visibility of one interferometer.
The dashed lines are linear fits to the data. The horizontal dashed blue line,
in both plots, is the classical limit for a Bell violation. When the state of
the control qubit is too decohered, we can no longer violate a Bell
inequality.

In our work we engineered a situation wherein the only way entanglement can be
transferred from one system to another is by means of causally non-separable
processes. In our experiment, this transfer takes place between different
internal DOFs of photon pairs. Although it is often easy to transfer the
entanglement from one DOF to another, this is typically done with a device
that directly couples the two DOFs; e.g., in the case of path-polarization
transfer, a PBS could be used, violating assumption 2.. In our experiment, we
used an entangled quantum SWITCH to accomplish this interchange. Our SWITCHes
do not contain any device which directly couples these DOFs (only waveplates,
which act solely on the polarization state, and 50/50 BSs, which act solely on
the path state). Rather, here the interchange occurs because the control qubit
(the path) governs the order of the application of gates on the target qubit
(the polarization). Then, since we begin with an entangled state of the
control qubits, this state is transferred to the target qubits via an
indefinite order of the application of the gates. In other words, by choosing
a specific set of operations, the superposition of these operations is mapped
onto a superposition of orthogonal states. As a result, this transfer of
entanglement is the signature of an indefinite temporal order.

## Conclusion

We entangled the temporal orders between two parties and experimentally showed
that resulting temporal orders is indefinite. By violating a Bell inequality
using the joint target system after the SWITCHes, we verified that the
temporal orders in a quantum SWITCH cannot be described by any underlying
(generalized probabilistic) theory in which the initial joint target state
does not violate Bell’s inequalities, the operations on the target states are
local, and they have a predefined order. This did not require the assumption
that the systems and the operations are described by quantum formalism.
Clearly, for our demonstration to be loophole-free (as proposed in mag ), the
standard Bell loopholes (fair-sampling and locality) would need to be closed.
Further loopholes can arise related to implementation of the quantum SWITCH
(for example, quantifying the number of gate uses via theory-independent
techniques is an open question). This is a relevant line of research, that has
ties to several foundational issues in quantum mechanics.

All previous work involving quantum processes with indefinite temporal orders
achieved their goal by superimposing the order of operations, rather than
entangling them. The first proposal to entangle the temporal orders was made
very recently mag . Here we show that the basis of this theoretical concept is
in fact experimentally accessible. Moreover, we exploit this resource as a new
means to validate indefinite causal structures. Techniques to characterize
these structures are becoming increasingly relevant, as it is known that these
processes can lead to linear advantages in query complexity and exponential
advantages in quantum communication tasks PhysRevA.88.022318 ;
PhysRevLett.113.250402 ; PhysRevA.92.052326 ; PhysRevLett.117.100502 .

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Acknowledgments: We thank C. Branciard, F. Costa, B. Dakić, A. Moqanaki and T.
Strömberg for useful discussions. Funding: G.R. acknowledges support from the
uni:docs fellowship programme at the University of Vienna. L.A.R. acknowledges
support from the Templeton World Charity Foundation (fellowship no. TWCF0194).
M.A. acknowledges support from the Excellence Initiative of the German Federal
and State Governments (Grant ZUK 81). M.Z. acknowledges support through an ARC
DECRA grant DE180101443, and ARC Centre EQuS (CE170100009). Č.B. acknowledges
support from the John Templeton Foundation, Innovationsfond of Austrian
Academy of Sciences (ÖAW), Austrian Science Fund (FWF) through the projects
no. I-2562-N27 and I-2906, as well as support from the European Commission via
Testing the Large-Scale Limit of Quantum Mechanics (TEQ) (No. 766900) project.
Č.B. and P.W. acknowledge support from the Doctoral Programme CoQuS (no.
W1210-4) and the research platform TURIS. P.W. also acknowledges support from
the European Commission, Photonic Integrated Compound Quantum Encoding
(PICQUE) (no. 608062), Quantum Simulation on a Photonic Chip (QUCHIP) (no.
641039) and the Austrian Science Fund (FWF) through the START Program
(Y585-N20) and NaMuG (P30067-N36), United States Air Force Office of
Scientific Research via QAT4SECOMP (FA2386-17-1-4011) and Red Bull GmbH.
Author contributions: G.R., L.A.R., Č.B. and P.W. designed the experiment.
G.R. built the set-up and carried out data collection. G.R. and L.A.R.
performed data analysis. F.M. built the single-photon source. M.Z., M.A. and
Č.B. developed the theoretical idea. L.A.R., P.W. and Č.B. supervised the
project. All authors contributed to writing the paper. Competing interests:
The authors declare that they have no competing interests. Data and materials
availability: All data needed to evaluate the conclusions in the paper are
present in the paper and/or the Supplementary Materials. Additional data
related to this paper may be requested from the authors.

## Materials and Methods

### Section I. Entangled Photon Source

A periodically-poled potassium titanyl phosphate (PPKTP) crystal, phase-
matched for collinear type-II spontaneous parametric down-conversion (SPDC),
converts one photon at $426$ nm into two photons at $852$ nm. The photonic
state after the crystal can be approximated to a Fock state of two photons in
two orthogonal polarization modes $|H,a\rangle|V,a\rangle$, where $a$
indicates the common spatial mode of the two photons defined in Fig. 1. Two
PBSs are used to separate and then recombine the two photons. Each photon
passes through a HWP set at $\pm 45^{\circ}$. The state after the second PBS
is therefore:
$\bigl{(}\ket{H,b}\ket{H,c}-\ket{H,b}\ket{V,b}+\ket{H,c}\ket{V,c}-\ket{V,b}\ket{V,c}\bigr{)}/\
2$, where $b$ and $c$ indicate the two output spatial modes of the second PBS.
By post-selecting on coincidences, only the part of the state with the photons
in two different spatial modes is kept, resulting in the polarization-
entangled state
$\bigl{(}|H,b\rangle|H,c\rangle-|V,b\rangle|V,c\rangle\bigr{)}/\ \sqrt{2}$. We
then use two PBSs and two HWPs (Fig. 1) to convert this state into a path-
entangled state:
$\bigl{(}\ket{0}_{1}\ket{0}_{2}-\ket{1}_{1}\ket{1}_{2}\bigr{)}/\ \sqrt{2}$,
where the notation is the same as specified in Fig. 1. A trombone delay line
in between the two PBSs is used to compensate temporal delay between the two
photons, and a multi-order QWP in one mode is tilted to compensate for
undesired phases between the two components of the final quantum state. The
delay line and the QWP can be also used to modify the final output state in a
controllable way. In particular, by unbalancing the two paths by the coherence
length of the down-converted photons, the entangled state can be converted
into a statistical mixture of the states $|0\rangle_{1}|0\rangle_{2}$ and
$|1\rangle_{1}|1\rangle_{2}$.

List of suppfigures 1 Entangled photon-pair source. a) The source — The beam
from a Toptica DL Pro HP 426 laser is focused on a 30-mm-long PPKTP crystal,
phase-matched for degenerate collinear type-II SPDC from $426$ nm to $852$ nm.
The phase-matching is finely tuned by controlling the temperature of the
crystal with a precision greater than $0.01K$. The emitted photons have a
bandwidth of approximately $0.2$ nm. After the crystal, the residual pump beam
is filtered, the photons are then collimated and sent to a set-up to create
entanglement by post-selection (as explained in the main text). The
entanglement is first produced in polarization and then converted into path
using polarizing beam splitters. The source produces $\approx 30.000$ path-
entangled photon pairs per second with a pump power of $8$ mW. b) Set-up used
to measure a Bell Inequality on the path qubits — The two paths composing each
qubit are interfered on a beam splitter (BS) projecting each qubit onto a
basis on the equator of the Bloch sphere (see main text for more details).

For our experiment, both the path and the polarization states of the photon
pairs are important. To characterize the polarization state, we can perform
two-qubit polarization state tomography using a QWP, a HWP and a PBS for each
photon (Fig. 1, Panel a) ). To characterize the path entanglement, we perform
a Bell measurement on the path qubits using the apparatus shown in Fig. 1,
Panel b), which is essentially composed of one Mach-Zehnder interferometer for
each photon. The phase of the interferometers sets the measurement bases
$\\{\frac{1}{\sqrt{2}}(\ket{0}+e^{-i\phi_{i}}\ket{1}),\frac{1}{\sqrt{2}}(\ket{0}-e^{-i\phi_{i}}\ket{1})\\}$.
Using these two interferometers we can measure all what is required for a CHSH
parameter:

$S=\left\lvert
C(o_{1},o_{2})+C(o_{1}^{\prime},o_{2})+C(o_{1},o_{2}^{\prime})-C(o_{1}^{\prime},o_{2}^{\prime})\right\rvert,$
(8)

where

$C(o_{1},o_{2})=\dfrac{N_{++}-N_{+-}-N_{-+}+N_{--}}{N_{++}+N_{+-}+N_{-+}+N_{--}}.$
(9)

Here, $N_{++}$ is the number of coincidence events between detectors labelled
$+$ for each photon in Fig. 1b., $N_{+-}$ the number of coincidence events
between detectors $+$ and $-$ for each photon, and so on.

### Section II. Data Analysis

In order to convert the coincidence counts into probabilities, we weight each
measured count rate by the net detection efficiency of the corresponding
detector pair. We estimate these efficiencies in two parts. First, we measure
the relative coupling efficiencies between the output ports $M_{1}$ and
$M_{2}$ of SWITCH S1, and $M_{3}$ and $M_{4}$ of SWITCH S2. Then, within each
output port, we measure the relative efficiency of the detector in the
transmitted port and the reflected port. We find relative efficiencies between
$\approx 0.85$ and $1$. For more details, see the Methods section of our
previous work Rubinoe1602589 .

The main source of error in our experiment was phase fluctuations. In the Bell
measurement, this dephasing is mainly due to two contributions. 1) Undesired
phase-shifts in the interferometer (which we estimated to be about
$0.97^{\circ}$). 2) Fluctuations of the source, which produces time varying
phase between the $\ket{HH}$ and $\ket{VV}$ terms. In our source, we estimate
this to be approximately $1.9^{\circ}$, which is caused by a combination of
fluctuations in the pump laser wavelength, and the phase-matching temperature.
We convert these errors into an error in the Bell parameter using Gaussian
error propagation. To calculate the error for the Bell measurements on the
polarization qubits after the SWITCHes, we consider the same error sources as
above (where now the phase shifts in the measurement interferometer are
replaced by phase shifts in the SWITCHes). However, we also consider errors
arising from setting the polarization measurements. Finally, to estimate the
errors in the results extracted from tomography (i.e., fidelity and
concurrence), we performed a Monte Carlo simulation considering the phase
fluctuations discussed above.

### Section III. Proof of No-Go Theorem for Temporal Orders

All previous experimental studies of causally non-separable processes
NatCommun.6 ; Rubinoe1602589  were dependent on the validity of the quantum
theory (i.e., they were theory-dependent) and all physically realizable
processes are known to satisfy all causal inequalities (see Suppl. Material)
NewJournPhys.17.102001 ; 1367-2630-18-9-093020 . The latter means, in the
spirit of a local hidden variable approach, that experimental data taken from
a given causally non-separable quantum process could be understood as arising
in causal manner in an underlying generalized probabilistic theory (GPT).

In our current work, we relate a violation of a Bell’s inequality to the
violation of a no-go theorem for temporal orders, as proposed in Ref. mag .
This results in a theory-independent proof of causal indefiniteness. In this
section, we provide a rigorous introduction to such no-go theorem for temporal
orders.

We will begin by giving a brief introduction to the basic elements of GPTs
which are necessary for our no-go theorem. A more detailed discussion of the
GPT framework can be found in Ref. GPT2 ; 1367-2630-13-6-063001 ;
PhysRevA.75.032304 .

In a GPT, a system is described by a state $\omega$ that specifies outcome
probabilities for all measurements that can be performed on it. A complete
representation of the state is given by specifying the outcome probabilities
of a so-called ‘fiducial set’. The smallest such set defines the number $d$ of
degrees of freedom of the system. We restrict our consideration here to binary
systems that have two perfectly distinguishable states and no more. For
example, the fiducial set for a two-level system in quantum theory consists of
the (three) probability outcomes of spin projections along $x$, $y$ and $z$.
The state space is a compact and convex set $\Omega$ embedded in a vector
space. The extremal states of $\Omega$ that cannot be decomposed as a convex
mixture of other states are called ‘pure states’. An effect $e$ is defined as
a linear functional on $\Omega$ that maps each state onto a probability, i.e.,
$e:\Omega\rightarrow[0,1]$, where $e(\omega)$ is the probability to obtain an
outcome on the state $\omega$. The linearity is required to preserve the
convex structure of the state space.

A transformation $U$ is a linear map from a state to a state, i.e.,
$U:\Omega\rightarrow\Omega$. The transformation is linear for the same reason
that probabilities have to be linear maps of states. The sequence of
transformations $U_{1}$, … , $U_{n}$, in which transformation $U_{1}$
‘precedes’ transformation $U_{2}$, which ‘precedes’ $U_{3}$, etc., is
represented by a composition of maps: $U_{n}\circ...\circ U_{1}$. This defines
a definite order of transformations, which we denote as
$U_{1}\preceq...\preceq U_{n}$.

We will now introduce the no-go theorem for temporal orders, which was
originally proposed in Ref. mag .

In the framework of a GPT, the state of a composite system shared between two
parties S1 and S2 is given by $\omega_{1,2}\in\Omega_{1,2}$, where
$\Omega_{1,2}$ is the state space of a composite system. The state of a
composite system is given by a multiplet consisting of the local states
$\omega_{1}\in\Omega_{1}$ and $\omega_{2}\in\Omega_{2}$ of individual systems,
the correlation tensor $\hat{T}$ and a potential global parameter $\xi$ GPT1 ;
GPT2 ; GPT3 ; 1367-2630-13-6-063001 :

$\omega_{1,2}=\omega_{1,2}(\omega_{1},\omega_{2},\hat{T},\xi)$ (10)

The fact that subsystems are themselves systems implies that each has a well-
defined reduced state $\omega_{1}$, $\omega_{2}$ which does not depend on
which transformations and measurements are performed on the other subsystem;
this is often referred to as ‘no-signaling’. We also assume that
transformations and measurements performed on subsystems commute with each
other, so that one correlation tensor is enough to describe correlations
between them. If this was not the case, we would need to introduce two
correlation tensors, one when S1 applies operations before S2, and the other
when S2 performs operations before S1. Finally, the states in GPT need not to
satisfy the local tomography condition (stating that reduced states and
correlation tensor completely describe the systems’ state) but may include a
global parameter $\xi$.

For the present case of binary systems, the components of state (10) are given
as

$\displaystyle\omega^{(i)}_{1}$
$\displaystyle=p^{(i)}(o_{1}=1)-p^{(i)}(o_{1}=-1),$ (11a)
$\displaystyle\omega^{(j)}_{2}$
$\displaystyle=p^{(j)}(o_{2}=1)-p^{(j)}(o_{2}=-1),$ (11b) $\displaystyle
T^{(i,j)}$
$\displaystyle=p^{(i,j)}(o_{1}\,o_{2}=1)-p^{(i,j)}(o_{1}\,o_{2}=-1).$ (11c)

Here, for example, $p^{(i)}(o_{1}=1)$ is the probability to obtain outcome
$o_{1}=1$ when the $i$-th measurement is performed on the first subsystem and
$p^{(i,j)}(o_{1}\,o_{2}=1)$ is the joint probability to obtain correlated
results (i.e., either $o_{1}=o_{2}=+1$ or $o_{1}=o_{2}=-1$) when the $i$-th
measurement is performed on the first subsystem and the $j$-th measurement on
the second one.

An effect $e_{1,2}$ that maps a state onto a probability for a pair of local
measurements is given by $e_{1,2}=e_{1,2}(r_{1},r_{2},r_{1}r^{T}_{2})$, where
$r^{T}$ denotes transposition of $r$ (Note that the global parameter does not
contribute to the probability for a pair of local measurements). The
probability to obtain effect $e_{12}$ in state prepared in state $\omega_{12}$
is given by

$p(e_{12}|\omega_{12})=\frac{1}{4}\left(1+(\omega_{1}\cdot
e_{1})+(\omega_{2}\cdot e_{2})+(e_{2}\cdot\hat{T}e_{1})\right),$ (12)

where $(x\cdot y)$ is the Euclidean scalar product between two $d$-dimensional
real vectors $x$ and $y$.

The product state is represented by
$\omega_{p}=\omega_{p}(\eta_{1},\eta_{2},\eta_{1}\eta^{T}_{2},\xi_{p})$, where
the correlation tensor is of a product form. If we perform a pair of local
measurements on the arbitrary product state, the outcome probability
factorizes into the product of the local outcome probabilities.

We next introduce a pair of local (reversible) transformations
$(U_{1},U_{2}):\Omega_{12}\rightarrow\Omega_{12}$ as a linear map from the
space of states of a composite system to itself:

$(U_{1},U_{2})(\omega_{12})=(U_{1}\omega_{1},U_{2}\omega_{2},U_{1}\hat{T}U^{T}_{2},\xi^{\prime}),$
(13)

where the global parameter $\xi^{\prime}$ is changed under the transformations
in general. Since testing our Bell inequality involves only local
transformations and measurements, it is sufficient to specify effects for
those measurements.

In our experiment, $\omega_{1}$ and $\omega_{2}$ themselves are states of
composite systems each consisting of a ‘control’ and a ‘target’ subsystem.
Hence, the entire system under investigation consists of four subsystems, a
control and a target subsystem of S$1$ and a control and a target system of
S$2$. The overall state is

$\omega_{1,2,3,4}=\omega_{1,2,3,4}(\omega^{T}_{1},\omega^{C}_{1},\omega^{T}_{2},\omega^{C}_{2},...,\hat{T}^{ij},...,\hat{T}^{ijk},...,\hat{T}^{1234},\Xi)$
(14)

where $C$ and $T$ refer to the terms ‘control’ and ‘target’ subsystem,
$\hat{T}^{ij}$, $\hat{T}^{ijk}$ and $\hat{T}^{1234}$ are correlation
(sub)tensors describing correlations between pairs $\\{i,j\\}$, triple
$\\{i,j,k\\}$ and quadruple $\\{1,2,3,4\\}$ of subsystems, respectively, and
$\Xi$ is the set of all global parameters.

The no-go theorem concerns the reduced state of the two target systems as
given by

$\omega^{T}_{1,2}=\omega^{T}_{1,2}(\omega^{T}_{1},\omega^{T}_{2},\hat{T}^{TT},\xi^{T})$
(15)

where $\omega^{T}_{1}$ and $\omega^{T}_{2}$ are states of the target systems
of S1 and S2, $\hat{T}^{TT}$ is their correlation tensor and $\xi^{T}$ is the
corresponding global parameter.

Leveraging these definitions, we now present three assumptions, which are the
fulcrum of our no-go theorem for a definite local causal order.

1\. The initial joint state of the target system $\omega_{1,2}^{T}$ does not
violate a Bell inequality.

Suppose that the two observers can each perform a measurement
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$, respectively. We label $m_{1}$ and
$m_{2}$ as the measurement choices of S1 and S2 and $o_{1}$ and $o_{2}$ as the
corresponding outcomes. Under these conditions, we suppose that our input
state $\omega^{T}_{1,2}$ can be described through a local hidden variables
theory (i.e., in Bell’s terms, a theory that satisfy ‘local causality’), and
therefore it is associated to the probability distribution

$p(o_{1},o_{2}|m_{1},m_{2},\omega^{T}_{1,2})=\int\rho(\lambda)\;p(o_{1}|m_{1},\lambda,\omega^{T}_{1,2})\;p(o_{2}|m_{2},\lambda,\omega^{T}_{1,2})\;d\lambda,$
(16)

where $\lambda$ is often referred to as a ‘hidden variable’. We implicitly
assume the ‘freedom of choice’ condition — the assumption that the choices of
the measurement settings are independent of $\lambda$ — is fulfilled.

2\. The laboratory operations are represented by local transformations
$U_{i}^{T}$ on the target systems. They do not increase the ‘amount’ of
violation of Bell inequalities on the target systems.

This is satisfied in the considered class of GPTs by definition because the
“amount of violation of Bell inequalities” is obtained by maximization over
all local transformations of type (13) (or convex mixtures therefrom) and our
‘laboratory operations’ are assumed to be of such type. For concreteness, let
us consider the Clauser-Horne-Shimony-Holt (CHSH) inequality
PhysRevLett.23.880  — a similar reasoning applies to different forms of Bell
inequalities. Following the Peres-Horodecki criterion PhysRevLett.84.2726 ;
PhysRevLett.84.2722 , the maximal value of the CHSH inequality in quantum
mechanics is given in terms of two largest absolute values of the correlation
tensor singular values, say $t_{1}$ and $t_{2}$, as
$\sqrt{t_{1}^{2}+t_{2}^{2}}$. The singular-value elements cannot increase
under local transformations (they are invariant under reversible local
operations).

3\. The order of S1 and S2’s operations on the target system is well defined.
Suppose first that the orders of application of the local operations performed
inside SWITCH S1 ${U}_{1_{A}}^{T}\preceq{U}_{1_{B}}^{T}\preceq\dots$ and those
performed inside SWITCH S2
$\,{U}_{2_{A}}^{T}\preceq{U}_{2_{B}}^{T}\preceq\dots$ are fixed. Since an
ordered sequence of local transformations is still a local transformation, if
a state undergoes such transformation on S1’s and S2’s sides, the amount of
violation of Bell’s inequalities cannot be increased (and in particular, the
singular values of the correlation tensor cannot increase and with them also
not violation of the CHSH inequality). The amount cannot be increased even if
the order of operations is chosen with a given probability distribution due to
convexity. The mutual order between S1’s and S2’s operations is irrelevant,
since we have assumed the two classes of operations to commute.

######  Theorem.

No states, set of transformations and measurements which obey assumptions
1.-3. can result in violation of a Bell inequality.

###### Proof.

Following 2., suppose that the initial target state $\omega_{1,2}^{T}$ does
not violate a Bell inequality. This means that Eq. (16) is fulfilled. Because
of 3., operations in S1’s and in S2’s laboratories are applied in a definite
order, say ${U}_{1_{A}}^{T}\preceq{U}_{1_{B}}^{T}\preceq\dots$ in S1’s and
${U}_{2_{A}}^{T}\preceq{U}_{2_{B}}^{T}\preceq\dots$ in S2’s side. The state
evolves, therefore, under a composition of the local operations as

$\dots\;({U}_{1_{\text{B}}}^{T},{U}_{2_{\text{B}}}^{T})\circ({U}_{1_{\text{A}}}^{T},{U}_{2_{\text{A}}}^{T})(\omega^{T}_{1,2}).$

Let us restrict ourselves to the case of only two transformations per SWITCH
(${U}_{\text{A}}^{T}$ and ${U}_{\text{B}}^{T}$). After the pairs of operations
are applied in order ${U}_{1_{\text{A}}}^{T}\preceq{U}_{1_{\text{B}}}^{T}$ and
${U}_{2_{\text{A}}}^{T}\preceq{U}_{2_{\text{B}}}^{T}$ on the two sides, the
state becomes

$\omega^{T}_{1,2}\,{}^{\prime}=({U}_{1_{\text{B}}}^{T},{U}_{2_{\text{B}}}^{T})\circ({U}_{1_{\text{A}}}^{T},{U}_{2_{\text{A}}}^{T})(\omega^{T}_{1,2})=\bigl{(}{U}_{1_{\text{B}}}^{T}\circ{U}_{1_{\text{A}}}^{T},{U}_{2_{\text{B}}}^{T}\circ{U}_{2_{\text{A}}}^{T}\bigr{)}(\omega^{T}_{1,2})$
(17)

which is still local due to 1. \- 2.. Hence

$p(o_{1},o_{2}|m_{1},m_{2},\omega^{T}_{1,2}\,{}^{\prime})=\int\rho(\lambda)\;p(o_{1}|m_{1},\lambda,\omega^{T}_{1,2}\,{}^{\prime})\cdot
p(o_{2}|m_{2},\lambda,\omega^{T}_{1,2}\,{}^{\prime})\;d\lambda.$ (18)

In general, the order of operations does not need to be fixed, but can be
specified probabilistically by a further hidden variable $\nu$, whose
different values correspond to different permutations of the order of
operations. We obtain

$p(o_{1},o_{2}|m_{1},m_{2})=\iint\rho(\lambda,\nu)\;p(o_{1}|m_{1},\lambda,\omega^{T}_{1,2}\,{}^{\nu})\cdot
p(o_{2}|m_{2},\lambda,\omega^{T}_{1,2}\,{}^{\nu})\;d\lambda d\nu,$ (19)

where $\rho(\lambda,\nu)$ is the joint probability distribution over the two
types of variables and $\omega^{T}_{1,2}\,{}^{\nu}$ is the final state of the
target systems upon application of the transformations in order $\nu$.

Hence we conclude that a local target state subjected to the action of a set
of local operations applied in a predefined order can by no means lead to the
violation of Bell inequalities, even if the order is chosen probabilistically
in each run of the experiment. This concludes the proof. ∎

### Section IV. Relation between the present work and Ref mag

In Ref. mag , the position of a massive object serves as a ‘control’ quantum
system and a quantum system (e.g., a photon) that is exchanged between Alice’s
and Bob’s laboratory as a ‘target’ system. By putting the massive object in a
macroscopic superposition of two positions, one closer to Alice’s and the
other closer to Bob’s position, one induces a relative time dilation between
Alice’s and Bob’s laboratory. The superposition of massive objects can
effectively lead to ‘entanglement’ of the temporal order between local
operations, enabling the violation of a Bell-type inequality. In the
conceptual framework of general relativity, the resource for the violation is
a ‘non-classical space-time’ created by macroscopic superposition of large
masses. In the second-quantized picture, the superposition can be seen as
entanglement in the Fock basis, and the scheme enables one to ‘swap’ this
entanglement to the final entanglement of the target systems. Unfortunately,
the physical demands of the proposal make that experiment infeasible. However,
quantum control of the order of events can also be achieved without the use of
gravitational interaction. This can be done, for example, in an extended
quantum circuit model, wherein the order of applied quantum gates is
coherently controlled by an ancillary system (the quantum SWITCH). The
difference between the two scheme is that in the gravitational scheme, the
spatio-temporal distance of any pair of events in a space-time region is
influenced by a superposition state of the mass, whereas in the linear optical
implementation, only the order of the gates applied on the propagating system
(e.g., photons) is in an indefinite order.

### Section V. Experimental Theory-Independent Proof of Assumption 1.

Recall that Assumption 1. is that the initial target states does not violate a
Bell’s inequality. In the notation introduced above, the initial target state
is $\omega_{1,2}^{T}$. Our demonstration of assumption 1. presented here is
based solely on experimental data and the framework of GPTs, and hence is
theory independent. Our goal is to prove that the input state is a product
state, and is thus local.

Let us denote the probabilities for measurement outcomes as measured on
reduced states of the target system of S$1$ and S$2$ as
$p(o_{1}|m_{1},\omega^{T}_{1})$ and $p(o_{2}|m_{2},\omega^{T}_{2})$,
respectively. If the state is a local product state then the probability for
joint outcomes, as measured on the composite system of the two target
subsystems in the initial state $\omega^{T}_{1,2}$, is factorisable, i.e., it
can be expressed as

$p(o_{1},o_{2}|m_{1},m_{2},\omega^{T}_{1,2})=p(o_{1}|m_{1},\omega^{T}_{1})\cdot
p(o_{2}|m_{2},\omega^{T}_{2}).$ (20)

We experimentally performed a large set of measurements on the input target
states and checked for this property. The measurements we made are
tomographically complete, in quantum theory. In a GPT, this might not be the
case, because the GPT system may have more degrees of freedom than a quantum
system. Nevertheless, in principle by performing all possible local
transformations and local measurements, one can demonstrate that in the
subspace in which the transformations and measurements of the final test of
Bell’s inequalities are performed, the effective state of the targets is a
product state.

Tables 1 and 2 show the values of the probabilities
$p(o_{1},o_{2}|m_{1},m_{2},\omega^{T}_{1,2})$ (which, for brevity, is
indicated as $p_{1,2}$ in the Tables) in the first four columns and the
marginalized probability products $p(o_{1}|m_{1},\omega^{T}_{1})\cdot
p(o_{2}|m_{2},\omega^{T}_{2})$ (denoted as $p_{1}\cdot p_{2}$ in the Table) in
the last four columns. It can be seen that the two sets of probabilities agree
well. More quantitatively, let us define the root-mean-square (RMS) distance
between the two sets of probabilities as

$\text{d}=\sqrt{\frac{1}{N}\sum_{o_{1},o_{2}}\sum_{m_{1},m_{2}}\Bigl{(}p(o_{1},o_{2}|m_{1},m_{2},\omega^{T}_{1,2})-p(o_{1}|m_{1},\omega^{T}_{1})\cdot
p(o_{2}|m_{2},\omega^{T}_{2})\Bigr{)}^{2}},$ (21)

where $N$ is the number of data points. Evaluating this over our results, we
obtain a RMS distance of $0.6\cdot 10^{-2}\pm 2.7\cdot 10^{-2}$, indicating
that, within error, the two distributions are very close. Thus we can conclude
that assumption 2. holds.

Table 1: Comparison between the two-states probabilities $p(o_{1},o_{2}|m_{1},m_{2},\omega^{T}_{1,2})$ and the products of marginalized single-state probabilities $p(o_{1}|m_{1},\omega^{T}_{1})\cdot p(o_{2}|m_{2},\omega^{T}_{2})$ for the input target states. - Part I. The compatibility between the two sets of probabilities shows the separability of the input target state $\omega_{1,2}^{T}$. The choice of the measurement bases is inspired by the bases commonly adopted for polarization measurements in classical optics. We indicate with ‘H’ and ‘V’ the states of horizontal and vertical polarization, with ‘D’ and ‘A’ the diagonal and anti-diagonal states, with ‘R’ and ‘L’ the circular polarization states right- and left-handed. The experimental error associated to each of these probabilities is $0.01$. Measur. Basis | $p_{1,2}$ | $p_{1,2^{\perp}}$ | $p_{1^{\perp},2}$ | $p_{1^{\perp},2^{\perp}}$ | $p_{1}\cdot p_{2}$ | $p_{1}\cdot p_{2^{\perp}}$ | $p_{1^{\perp}}\cdot p_{2}$ | $p_{1^{\perp}}\cdot p_{2^{\perp}}$
---|---|---|---|---|---|---|---|---
H, H | 0.97 | 0.03 | 0.00 | 0.00 | 0.97 | 0.03 | 0.00 | 0.00
H, V | 0.01 | 0.99 | 0.00 | 0.00 | 0.01 | 0.99 | 0.00 | 0.00
H, A | 0.58 | 0.41 | 0.00 | 0.00 | 0.59 | 0.41 | 0.00 | 0.00
H, D | 0.42 | 0.58 | 0.00 | 0.00 | 0.42 | 0.58 | 0.00 | 0.00
H, R | 0.39 | 0.61 | 0.00 | 0.00 | 0.39 | 0.61 | 0.00 | 0.00
H, L | 0.61 | 0.38 | 0.00 | 0.00 | 0.62 | 0.38 | 0.00 | 0.00
V, H | 0.00 | 0.00 | 0.96 | 0.04 | 0.00 | 0.00 | 0.96 | 0.04
V, V | 0.00 | 0.00 | 0.03 | 0.97 | 0.00 | 0.00 | 0.03 | 0.97
V, A | 0.00 | 0.00 | 0.61 | 0.39 | 0.00 | 0.00 | 0.61 | 0.39
V, D | 0.00 | 0.00 | 0.38 | 0.61 | 0.00 | 0.00 | 0.38 | 0.61
V, R | 0.00 | 0.00 | 0.35 | 0.64 | 0.00 | 0.00 | 0.35 | 0.64
V, L | 0.00 | 0.00 | 0.64 | 0.36 | 0.00 | 0.00 | 0.64 | 0.36
A, H | 0.39 | 0.01 | 0.53 | 0.02 | 0.41 | 0.01 | 0.54 | 0.02
A, V | 0.01 | 0.37 | 0.02 | 0.54 | 0.02 | 0.39 | 0.02 | 0.54
A, A | 0.24 | 0.16 | 0.34 | 0.21 | 0.26 | 0.16 | 0.34 | 0.21
A, D | 0.18 | 0.22 | 0.21 | 0.33 | 0.18 | 0.24 | 0.23 | 0.31
A, R | 0.16 | 0.25 | 0.18 | 0.35 | 0.16 | 0.27 | 0.19 | 0.34
A, L | 0.26 | 0.14 | 0.36 | 0.19 | 0.27 | 0.14 | 0.36 | 0.19
D, H | 0.55 | 0.02 | 0.45 | 0.02 | 0.53 | 0.02 | 0.45 | 0.02
D, V | 0.01 | 0.57 | 0.02 | 0.45 | 0.01 | 0.54 | 0.01 | 0.46
D, A | 0.32 | 0.26 | 0.29 | 0.18 | 0.32 | 0.24 | 0.28 | 0.20
D, D | 0.23 | 0.35 | 0.18 | 0.29 | 0.22 | 0.34 | 0.18 | 0.29
D, R | 0.21 | 0.37 | 0.16 | 0.31 | 0.19 | 0.37 | 0.16 | 0.31
D, L | 0.35 | 0.22 | 0.32 | 0.17 | 0.34 | 0.20 | 0.31 | 0.18
Table 2: Comparison between the two-states probabilities $p(o_{1},o_{2}|m_{1},m_{2},\omega^{T}_{1,2})$ and the products of marginalized single-state probabilities $p(o_{1}|m_{1},\omega^{T}_{1})\cdot p(o_{2}|m_{2},\omega^{T}_{2})$ for the input target states. - Part II. The compatibility between the two sets of probabilities shows the separability of the input target state $\omega_{1,2}^{T}$. Measur. Basis | $p_{1,2}$ | $p_{1,2^{\perp}}$ | $p_{1^{\perp},2}$ | $p_{1^{\perp},2^{\perp}}$ | $p_{1}\cdot p_{2}$ | $p_{1}\cdot p_{2^{\perp}}$ | $p_{1^{\perp}}\cdot p_{2}$ | $p_{1^{\perp}}\cdot p_{2^{\perp}}$
---|---|---|---|---|---|---|---|---
R, H | 0.65 | 0.02 | 0.33 | 0.01 | 0.64 | 0.02 | 0.33 | 0.01
R, V | 0.01 | 0.66 | 0.01 | 0.30 | 0.02 | 0.66 | 0.01 | 0.31
R, A | 0.39 | 0.27 | 0.22 | 0.13 | 0.40 | 0.26 | 0.21 | 0.14
R, D | 0.29 | 0.39 | 0.12 | 0.19 | 0.28 | 0.40 | 0.13 | 0.19
R, R | 0.27 | 0.41 | 0.11 | 0.20 | 0.26 | 0.42 | 0.12 | 0.19
R, L | 0.41 | 0.25 | 0.22 | 0.12 | 0.41 | 0.24 | 0.22 | 0.13
L, H | 0.32 | 0.01 | 0.63 | 0.04 | 0.32 | 0.02 | 0.63 | 0.03
L, V | 0.01 | 0.32 | 0.03 | 0.64 | 0.01 | 0.32 | 0.02 | 0.65
L, A | 0.18 | 0.14 | 0.42 | 0.27 | 0.19 | 0.13 | 0.41 | 0.28
L, D | 0.14 | 0.21 | 0.23 | 0.41 | 0.13 | 0.22 | 0.24 | 0.40
L, R | 0.12 | 0.23 | 0.22 | 0.43 | 0.12 | 0.23 | 0.22 | 0.43
L, L | 0.21 | 0.12 | 0.44 | 0.24 | 0.21 | 0.12 | 0.43 | 0.25

### Section VI. Experimental Theory-Independent Proof of Assumption (2b)

In this Section we wish to experimentally prove Assumption (2b), which states
that the laboratory operations do not couple (i.e. generate non-classical
correlations between) the control and the target systems, within a given party
S$i$. To do so, we first prepare the control and target state in a product
state. We then set the state of the control system to one wherein the local
operations are applied in a definite order. Note that this is a device-
dependent step. Nevertheless, it can be described within GPTs, and hence is
still theory-independent. We then need to verify that the control and target
systems are still in a product state after the SWITCH. We do this using the
same technique we used to verify that the target qubits began in an input
state (Methods - Section IV.).

Tables 3 and 4 report the values of the probabilities
$p(o_{C},o_{T}|m_{C},m_{T},\omega_{1})$ (which, for brevity, is indicated as
$p_{C,T}$ in the Tables) compared with the marginalized probability products
$p(o_{C}|m_{C},\omega^{C}_{1})\cdot p(o_{T}|m_{T},\omega^{T}_{1})$ (denoted as
$p_{C}\cdot p_{T}$ in the Tables). For these measurements, because of
challenges setting the basis to measure the path degree of freedom, we did not
perform a tomographically-complete set of measurements. Instead, we measured
the control system (encoded in the path) only in one basis
($\\{\ket{+},\ket{-}\\}$) and performed a complete set of measurements on the
target system. Nevertheless, using our knowledge of quantum mechanics and of
our device, measurements in $\\{\ket{+},\ket{-}\\}$ basis result in the
strongest correlations. For the measurements we performed, the displayed
output probabilities $p(o_{C},o_{T}|m_{C},m_{T},\omega_{1})$ are very close to
those correspondent to a separable state. This is indicated by the fact that
the RMS distance (Eq. 21) between these two sets of probabilities (the
measured joint probabilities $p(o_{C},o_{T}|m_{C},m_{T},\omega_{1})$ and that
given by the product $p(o_{C}|m_{C},\omega^{C}_{1})\cdot
p(o_{T}|m_{T},\omega^{T}_{1})$) is $3.1\cdot 10^{-2}\pm 1.7\cdot 10^{-2}\pm
1.0\cdot 10^{-2}$ (where the first error is a statistical error while the
second is a systematic error) when the control system is prepared in the state
correspondent to $\ket{0}$, and $4.9\cdot 10^{-2}\pm 1.6\cdot 10^{-2}\pm
1.0\cdot 10^{-2}$ when the control system is prepared in the state
correspondent to $\ket{1}$. This thus confirms the validity of assumption (2b)
in a theory-independent manner.

Table 3: Comparison between the two-states probabilities $p(o_{C},o_{T}|m_{C},m_{T},\omega_{1})$ and the products of marginalized single-state probabilities $p(o_{C}|m_{C},\omega^{C}_{1})\cdot p(o_{T}|m_{T},\omega^{T}_{1})$ for the control and the target states when applying the gates in the order $\mathcal{A}(\omega)\preceq\mathcal{B}(\omega)$. The two sets of probabilities associated to the control and the target states in output are compatible within experimental errors. Measur. Basis | $p_{C,T}$ | $p_{C,T^{\perp}}$ | $p_{C^{\perp},T}$ | $p_{C^{\perp},T^{\perp}}$ | $p_{C}\cdot p_{T}$ | $p_{C}\cdot p_{T^{\perp}}$ | $p_{C^{\perp}}\cdot p_{T}$ | $p_{C^{\perp}}\cdot p_{T^{\perp}}$
---|---|---|---|---|---|---|---|---
H | 0.19 | 0.27 | 0.29 | 0.25 | 0.22 | 0.24 | 0.26 | 0.28
V | 0.26 | 0.21 | 0.26 | 0.26 | 0.25 | 0.23 | 0.27 | 0.25
D | 0.19 | 0.29 | 0.34 | 0.19 | 0.25 | 0.22 | 0.28 | 0.25
R | 0.43 | 0.02 | 0.53 | 0.02 | 0.44 | 0.02 | 0.53 | 0.02
L | 0.01 | 0.48 | 0.01 | 0.50 | 0.01 | 0.48 | 0.01 | 0.50
Table 4: Comparison between the two-states probabilities $p(o_{C},o_{T}|m_{C},m_{T},\omega_{1})$ and the products of marginalized single-state probabilities $p(o_{C}|m_{C},\omega^{C}_{1})\cdot p(o_{T}|m_{T},\omega^{T}_{1})$ for the control and the target states when applying the gates in the order $\mathcal{B}(\omega)\preceq\mathcal{A}(\omega)$. The two sets of probabilities associated to the control and the target states in output are compatible within experimental errors. Measur. Basis | $p_{C,T}$ | $p_{C,T^{\perp}}$ | $p_{C^{\perp},T}$ | $p_{C^{\perp},T^{\perp}}$ | $p_{C}\cdot p_{T}$ | $p_{C}\cdot p_{T^{\perp}}$ | $p_{C^{\perp}}\cdot p_{T}$ | $p_{C^{\perp}}\cdot p_{T^{\perp}}$
---|---|---|---|---|---|---|---|---
H | 0.33 | 0.17 | 0.23 | 0.27 | 0.28 | 0.22 | 0.28 | 0.22
V | 0.15 | 0.35 | 0.34 | 0.16 | 0.25 | 0.26 | 0.24 | 0.25
R | 0.03 | 0.52 | 0.01 | 0.44 | 0.02 | 0.53 | 0.02 | 0.43
L | 0.45 | 0.02 | 0.51 | 0.01 | 0.45 | 0.02 | 0.51 | 0.02
D | 0.28 | 0.23 | 0.27 | 0.22 | 0.28 | 0.23 | 0.27 | 0.22

## Supplementary Material

### Quantum SWITCH and Causal Inequalities

The quantum SWITCH NatCommun.6 ; Rubinoe1602589  has been shown not to violate
causal inequalities, making it impossible to use such violation as a theory-
independent proof that causal order of operations in the SWITCH is indefinite.
Here, we briefly re-examine such reasoning following Refs.
NewJournPhys.17.102001 ; 1367-2630-18-9-093020 .

We introduce the $x$, $y$ and $z$ indices to refer, respectively, to the
measurements choices of Alice, Bob and Charlie. We call $a$, $b$ and $c$ their
respective measurement results. It is always possible to re-write
$p(a,b,c|x,y,z)$ as

$p(a,b,c|x,y,z)=p(c|a,b,x,y,z)p(a,b|x,y,z)$ (22)

It should be noticed that regardless of the causal order between operations in
Alice’s and Bob’s laboratory, the operation in Charlie’s laboratory always
occurs after them. In other words, his operation is in the future light cone
of both Alice’s and Bob’s operations. Thus, $a,b$ cannot depend on $z$, so

$p(a,b|x,y,z)=p(a,b|x,y)$ (23)

As we previously observed, after tracing out in Charlie’s laboratory in the
SWITCH, the process matrix of Alice and Bob is causally separable. Hence, one
can rewrite $p(a,b|x,y)$ in the form of a convex mixture, obtaining

$p(a,b,c|x,y,z)=p(c|a,b,x,y,z)\bigl{[}\zeta\cdot p^{A\preceq
B}(a,b|x,y)+(1-\zeta)\cdot p^{B\preceq A}(a,b|x,y)\bigr{]}$ (24)

We can combine the probabilities $p^{A\preceq B}(a,b|x,y)$ ($p^{B\preceq
A}(a,b|x,y)$) and $p(c|a,b,x,y,z)$ as a product of the probability respecting
the order $A\preceq B$ ($B\preceq A$) with the probability respecting the
order $\\{A,B\\}\preceq C$

$p(a,b,c|x,y,z)=\zeta\cdot p^{A\preceq B\preceq C}(a,b,c|x,y,z)+(1-\zeta)\cdot
p^{B\preceq A\preceq C}(a,b,c|x,y,z)$ (25)

Therefore, the SWITCH is a process whose probabilities have a ‘causal model’,
i.e., it can always be understood as arising from events that are causally
ordered or from a convex mixtures of causally ordered events. Hence, it
satisfies all causal inequalities.

Causal witnesses, violation of Bell inequalities for temporal orders, and
violation of causal inequalities build a hierarchy of the notion of ‘the lack
of causality’. Let us clarify further what we mean by that. The weakest notion
of the lack of causality is that of causal non-separability, which is
formulated using quantum theory and hence is theory-dependent. A violation of
a causal inequality is the strongest notion as it is formulated solely in
terms of observable probabilities $p(a,b|x,y)$ without any assumption about
the underlying theory — it is therefore device-independent. Violation of the
newly-proposed Bell inequalities for temporal order should be considered, in
our view, a stronger proof of lack of causality than the measurement of a
causal witness but a weaker proof than a violation of a causal inequality. The
reason it is weaker than a causal inequality violation is that, although it
too is formulated in terms of the probabilities $p(a,b|x,y,\omega)$, it also
involves the notion of state $\omega$ (like in Eq. (15) of Methods - Section
III.) and the order of operations on it - this causes the proof to be device-
dependent. However, it can be defined for any GPT and hence is theory-
independent, and thus more general than the notion of a causal witness.
Although the quantum SWITCH violates a weaker notion of causality, shaped for
quantum theory, it cannot violate the stronger (device-independent) notion of
a causal inequalities. The open question addressed in our work is: “can we
still use the quantum SWITCH to perform a theory-independent proof of
indefinite causal orders?”. The answer is affirmative, and here we present an
experimental demonstration of this.


experimented with it to test whether LPath principles can be interpreted as a
heuristic to inspire methods that approximate sufficient statistics that can
work reasonably well, and we observed that categorical decoders work similarly
with Guassian decoders.

In addition, we also experimented with VAEs with slightly varied training
objectives as detailed in Appendix B.5.1 where we added though we did not
observe a significant difference in the final AUROC. In Table 1, we report the
best test AUROC in our experiments following the convention in prior works.

##  Appendix D Ablation Studies

###  D.1 COPOD on four cases

To verify that the dataset can be divided into four cases as depicted in
Figure 1, we separate the dataset into four cases and use our methods on each
case. We use the modes of the IID and OOD distributions on mse reconstruction
($u(\mathbf{x})$) and the norm of the latent code ($v(\mathbf{x})$) to decide
where the two distributions are considered to overlap, see Figure 8(f) for a
visualization.

The four cases correspond to:

Case 1: $\mathbf{z}_{\text{overlap}}$ \+ $\mathbf{x}_{\text{overlap}}$
Case 2: $\mathbf{z}_{\text{separable}}$ \+ $\mathbf{x}_{\text{overlap}}$
Case 3: $\mathbf{z}_{\text{overlap}}$ \+ $\mathbf{x}_{\text{separable}}$
Case 4: $\mathbf{z}_{\text{separable}}$ \+ $\mathbf{x}_{\text{separable}}$

Results are reported in Table 3. We can see that the order of the performances
respect their conjectured level of difficulty. Our method performs
considerably better than other statistics, primarily on Case 1. If we make the
overlapping region smaller, for example, by using more extreme quantiles, Case
1 will have fewer samples and the OOD detection would become more difficult.

(a) Histogram of $u(\mathbf{x})$ (MSE)

(b) Histogram of $v(\mathbf{x})$ (z_norm)

(c) $\mathbf{x}_{\text{separable}}$: Outside regions of $u(\mathbf{x})$ (MSE)

(d) $\mathbf{z}_{\text{separable}}$: Outside regions of $v(\mathbf{x})$
(z_norm)

(e) $\mathbf{x}_{\text{overlap}}$: Overlap region of $u(\mathbf{x})$ (MSE)

(f) $\mathbf{z}_{\text{overlap}}$: Overlap region of $v(\mathbf{x})$ (z_norm)

Figure 8: How overlap and outside regions are defined in Appendix D.1

.

| Case 1 | Case 2 | Case 3 | Case 4
---|---|---|---|---
$v(\mathbf{x})$ | 0.75 | 0.74 | 0.93 | 0.96
$u(\mathbf{x})$ | 0.93 | 0.98 | 1.00 | 0.98
ELBO | 0.83 | 0.87 | 1.00 | 0.96
Ours | 0.99 | 0.99 | 1.00 | 0.99
Table 3: COPOD results for four different cases using various statistics.

In this dataset, $u(\mathbf{x})$ alone outperforms $v(\mathbf{x})$ in Table 3.
We can see that ELBO’s performance is somewhere between $u(\mathbf{x})$ and
$v(\mathbf{x})$. This showcases the arithmetic cancellation discussed in
Section 2. Our LPath method, in contrast, does not suffer from it and can
combine their strengths to achieve stable and superior performances.

###  D.2 Individual statistics

To empirically validate how ($u(\mathbf{x}),v(\mathbf{x}),w(\mathbf{x})$)
complement each other suggested by Theorem 3.8, we use individual component
alone in first stage and fit the second stage COPOD as usual. We notice
signigicant drops in performances. We fit COPOD on individual statistics
$u(\mathbf{x})$, $v(\mathbf{x})$, $w(\mathbf{x})$ and show the results in
Table 4. We can see that our original combination in Table 1 is better
overall.

| OOD Dataset
---|---
Statistic | SVHN | CIFAR100 | Hflip | Vflip
$u(\mathbf{x})$ | 0.96 | 0.59 | 0.54 | 0.59
$v(\mathbf{x})$ | 0.94 | 0.56 | 0.54 | 0.59
$w(\mathbf{x})$ | 0.93 | 0.58 | 0.54 | 0.61
$v(\mathbf{x})$ & $w(\mathbf{x})$ | 0.94 | 0.58 | 0.54 | 0.60
$u(\mathbf{x})$ & $v(\mathbf{x})$ | 0.97 | 0.61 | 0.53 | 0.61
$u(\mathbf{x})$ & $w(\mathbf{x})$ | 0.98 | 0.61 | 0.54 | 0.61
Table 4: COPOD on individual statistics. IID dataset is CIFAR10.

###  D.3 MD

To test the efficacy of ($u(\mathbf{x}),v(\mathbf{x}),w(\mathbf{x})$) without
COPOD, we replace COPOD by a popular algorithm in OOD detection, the MD
algorithm Lee et al. (2018) and report such scores in Table 1. The scores are
comparable to COPOD, suggesting ($u(\mathbf{x}),v(\mathbf{x}),w(\mathbf{x})$)
is the primary contributor to our performances.

###  D.4 Latent dimensions

One hypothesis on the relationship between latent code dimension and OOD
detection performance is that lowering dimension incentivizes high level
semantics learning, and higher level feature learning can help discriminate
OOD v.s. IID. We conducted experiments on the below latent dimensions and
report their AUC based on $v(\mathbf{x})$ (norm of the latent code) in Table 5

Latent dimension | 1 | 2 | 5 | 10 | 100 | 1000 | 3096 | 5000
---|---|---|---|---|---|---|---|---
$v(\mathbf{x})$ AUC | 0.39 | 0.63 | 0.52 | 0.45 | 0.22 | 0.65 | 0.76 | 0.59
Table 5: Lower latent code dimension doesn’t help to discriminate in practice.

Clearly, lowering the dimension isn’t sufficient to increase OOD performances.