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# Experiments on quantum causality

K. Goswami<EMAIL_ADDRESS>J. Romero<EMAIL_ADDRESS>Australian
Reserch Council Centre of Excellence for Engineered Quantum Systems, School of
Mathematics and Physics,
University of Queensland, QLD 4072 Australia

###### Abstract

Quantum causality extends the conventional notion of fixed causal structure by
allowing channels and operations to act in an indefinite causal order. The
importance of such an indefinite causal order ranges from the
foundational—e.g. towards a theory of quantum gravity—to the applied—e.g. for
advantages in communication and computation. In this review, we will walk
through the basic theory of indefinite causal order and focus on experiments
that rely on a physically realisable indefinite causal ordered process—the
quantum switch.

††preprint: AIP/123-QED

##  I Introduction

Causality is central to the way that we model the world. There is a definite
relationship between cause and effect—effects depend on their causes. This
relationship between cause and effect is called causal order, and this is
established by observing changes in the effect given some freely chosen
intervention on the cause. The effects and interventions happen in the
physical stage that is space-time Milburn and Shrapnel (2018). In conventional
quantum physics, every pair of events has a fixed causal order. Quantum
causality relaxes this constraint, it allows for scenarios “event A is in the
causal past of event B” and “event B is in the causal past of event A” to be
in superposition. Although counter-intuitive, this is plausible in the light
of general relativity—where the causal order is dynamic rather than fixed
Hardy (2007). We already know quantum theory is incompatible with local-
realistic theories that assign definite, pre-existing values to physical
observables Bell (1964), we can extend this idea to causal order and imagine
processes that have _indefinite_ causal order. Implicit in such a process is
the superposition of the order of events. It is this superposition of the
order of events—rather than indefinite cause-effect relationships— that has
been demonstrated in experiments so far.

Indefinite causal order is of interest to the foundational quantum physics
community because a fixed causal order might not be a feature of a theory that
successfully combines general relativity and quantum physics. For a more
pedagogical approach focusing on the foundational implications of quantum
causality, curious readers are invited to read the review by Brukner (2014).

From a pragmatic standpoint, indefinite causal order offers interesting
extensions of conventional quantum Shannon theory Ebler, Salek, and Chiribella
(2018); Chiribella  _et al._ (2018); Salek, Ebler, and Chiribella (2018) as
well as computationalChiribella  _et al._ (2013); Araújo, Costa, and Brukner
(2014); Araújo, Guérin, and Baumeler (2017) and communication complexity
advantages Guérin  _et al._ (2016). Our objective in this review is to provide
a summary of this type of experiments without discounting other experiments
that have broader foundational significance. Section II starts with a
mathematical description of a process with indefinite causal order. We give
the example of a _quantum switch_ Chiribella  _et al._ (2013), a device that
exhibits indefinite causal order. Section III focuses on how we can
experimentally detect indefinite causal order with the use of a _causal
witness_ Araújo  _et al._ (2015). Section IV discusses experiments that have
shown communication advantages by exploiting indefinite causal order. A common
requirement for all these experiments is to have two different carriers of
information, one as a control of the order and one as a target to operate on.
For Sections III and IV, we will broadly divide these implementations into two
categories based on the degree of freedom that is used to control the order:
those that use the path of a photon and those that use polarisation. There is
still an active discussion in the quantum causality community about the origin
of the communication advantages Ebler, Salek, and Chiribella (2018); Salek,
Ebler, and Chiribella (2018); Chiribella  _et al._ (2018) that have been
claimed in recent experiments Goswami  _et al._ (2020); Guo  _et al._ (2020).
We will summarise recent theoretical debates and comment on the relevance for
current experiments. Section V discusses computational advantages for three
tasks: distinguishing commuting and anti-commuting operations, evaluation
exchange game, and Hadamard promise problem. Finally, Section VI gives a
summary and outlook for the future. Note that in this review, we will often
refer to “causal order” simply as “order” for brevity.

Figure 1: Indefinite causal order. The order that two individual operations,
$\pazocal{N}_{p}$ and $\pazocal{N}_{q}$, act on a target qubit is controlled
by a control qubit $\left|\psi\right\rangle_{c}$. (a) When the control qubit
$\left|\psi\right\rangle_{c}{=}\left|0\right\rangle$, the operations have a
definite order $\pazocal{N}_{q}{\circ}\pazocal{N}_{p}$. (b) When the control
qubit $\left|\psi\right\rangle_{c}{=}\left|1\right\rangle$, the order is
$\pazocal{N}_{p}{\circ}\pazocal{N}_{q}$. (c) When the control is in a
superposition,
$\left|\psi\right\rangle_{c}{=}(\left|0\right\rangle{+}\left|1\right\rangle)/\sqrt{2}$,
the operations have an indefinite order. Reprinted with permission from K.
Goswami _et al_., “increasing communication capacity via superposition of
order," Phys. Rev. Res. 2, 033292 (2020) (Ref. Goswami  _et al._ , 2020).
Copyright 2020 by APS.

##  II Indefinite causal order

In conventional quantum mechanics, events take place in a definite order. We
can relax this assumption and consider a scenario where the order of the
events is not fixed or well-defined. The uncertainty in the order of events
can be attributed to classical randomness Lamport (1978)—when the order is
simply a mixture of the different possible orders—or quantum coherence—when
the order of events is a superposition of the different possible orders
Oreshkov, Costa, and Brukner (2012); Chiribella  _et al._ (2013). A scenario
for which the latter is true is said to have an _indefinite causal order_.

To represent the events, one performs _quantum operations_ incorporating any
combination of state preparation, transformation or measurement. The most
general operation, performed by a party $X$, is given by a completely positive
(CP) map $\pazocal{N}:X_{I}{\rightarrow}X_{O}$ — a transformation from an
input space $X_{I}{\equiv}\pazocal{L}\left(\pazocal{H}^{X_{I}}\right)$ to an
output space $X_{O}{\equiv}\pazocal{L}\left(\pazocal{H}^{X_{O}}\right)$. Here,
$\pazocal{L}\left(\pazocal{H}\right)$ denotes the space of bounded operators
over a Hilbert space $\pazocal{H}$. It is convenient to represent a CP map
$\pazocal{N}$ as a positive semidefinite matrix
$N^{X_{I}X_{O}}{=}[\mathbb{1}{\otimes}\pazocal{N}(|\mathbb{1}{\rangle}\\!{\rangle}\\!{\langle}\\!{\langle}{\mathbb{1}})]^{\pazocal{T}}{\in}X_{I}{\otimes}X_{O}$,
as given by the _Choi-Jamiołkowski (CJ) isomorphism_ Choi (1975); Jamiołkowski
(1972). Here $\pazocal{T}$ is the transpose, and
${|{\mathbb{1}}\rangle\\!\rangle}=\sum_{i}\left|i\right\rangle\\!\left|i\right\rangle$
is the unnormalised maximally entangled state where
$\\{\left|i\right\rangle\\}$ is an orthonormal basis in $X_{I}$. For a
deterministic quantum operation, e.g. a quantum channel, the CP map
$\pazocal{N}$ is trace preserving (TP), and the corresponding CJ isomorphism
satisfies $\operatorname{Tr}_{X_{O}}N^{X_{I}X_{O}}{=}\mathbb{1}^{X_{I}}$, with
$\mathbb{1}^{X_{I}}$ being the identity matrix in the input Hilbert space
$X_{I}$.

To encode the underlying causal structure between different quantum
operations, the framework of _process_ , represented by the _process matrix_ ,
was developed Oreshkov, Costa, and Brukner (2012). Based on the process
framework, the probability to realise the maps
$\left\\{\pazocal{N}_{1},\pazocal{N}_{2},\dots\right\\}$ in an experiment,
performed by the parties $\left\\{A,B,\dots\right\\}$, is given by the
_generalised Born rule_ Gutoski and Watrous (2006); Chiribella, D’Ariano, and
Perinotti (2009); Oreshkov, Costa, and Brukner (2012); Shrapnel, Costa, and
Milburn (2018)

$P(\pazocal{N}_{1},\pazocal{N}_{2},\dots)=\operatorname{Tr}\left[\left(N_{1}^{A_{I}A_{O}}\otimes
N_{2}^{B_{I}B_{O}}\otimes\dots\right)W\right],$ (1)

where $W{\in}A_{I}{\otimes}A_{O}{\otimes}B_{I}{\otimes}B_{O}{\otimes}\dots$ is
the _process matrix_. It is a positive semidefinite matrix which provides a
full specification of the possible correlations that can be observed between
the input and output systems. We call a process, $W_{\mathrm{sep}}$, _causally
separable_ when it can be represented as a convex combination of different
permutations of
events—$W_{\mathrm{sep}}{=}\sum_{\upsilon}p_{\upsilon}W^{\upsilon}$, where
$W^{\upsilon}$ is the corresponding process for the permutation $\upsilon$ and
$p_{\upsilon}$ is the probability of occurrence of the permutation $\upsilon$.
Every other process that does not decompose in this way is said to be
_causally nonseparable_ , i.e. they exhibit indefinite causal order.

One can physically realise such a coherent-control protocol, for example in a
_quantum switch_ Chiribella  _et al._ (2013); Chiribella (2012); Colnaghi  _et
al._ (2012). A quantum switch is comprised of two quantum systems—target,
$\rho_{t}$ and control,
$\rho_{c}{=}\left|\psi\middle\rangle\middle\langle\psi\right|_{c}$. When the
control is two-dimensional—a qubit—we consider two complete positive (CP)
operations $\pazocal{N}_{p}$ and $\pazocal{N}_{q}$ acting on the target
system. The control determines the order of the operations on the target: e.g.
when $\left|\psi\right\rangle_{c}{=}\left|0\right\rangle$, the resulting
operation is $\pazocal{N}_{q}{\circ}\pazocal{N}_{p}$, and for
$\left|\psi\right\rangle_{c}{=}\left|1\right\rangle$ the resulting operation
is $\pazocal{N}_{p}{\circ}\pazocal{N}_{q}$ (see Fig. 1). The action of the
quantum switch is to transform two individual operations $\pazocal{N}_{p}$ and
$\pazocal{N}_{q}$ to a new operation
$T(\pazocal{N}_{p},\pazocal{N}_{q}){=}\left|0\middle\rangle\middle\langle
0\right|_{c}{\otimes}\pazocal{N}_{q}{\circ}\pazocal{N}_{p}{+}\left|1\middle\rangle\middle\langle
1\right|_{c}{\otimes}\pazocal{N}_{p}{\circ}\pazocal{N}_{q}$. When the
operations $\pazocal{N}_{p}$ and $\pazocal{N}_{q}$ are quantum channels, we
can give the equivalent Kraus representation Kraus  _et al._ (1983) as

$\displaystyle
T({\pazocal{N}_{p}},{\pazocal{N}_{q}})_{ij}{=}\left|0\middle\rangle\middle\langle
0\right|_{c}{\otimes}{K^{(q)}_{i}}{K^{(p)}_{j}}{+}\left|1\middle\rangle\middle\langle
1\right|_{c}{\otimes}{K^{(p)}_{j}}{K^{(q)}_{i}}.$ (2)

Here, $\\{K^{(p)}_{i}\\}$ are the Kraus operators of
$\pazocal{N}_{p}$—$\pazocal{N}_{p}(\rho_{t}){=\sum_{i}K^{(p)}_{i}\rho_{t}K^{(p){\dagger}}_{i}}$.
Similarly, $\\{K^{(q)}_{i}\\}$ are the Kraus operators of $\pazocal{N}_{q}$.
With this, the joint input state $\rho_{c}{\otimes}\rho_{t}$ transforms to the
output state
$T(\pazocal{N}_{p},\pazocal{N}_{q})(\rho_{c}\otimes\rho_{t}){=}\sum_{i,j}T({\pazocal{N}_{p}},{\pazocal{N}_{q}})_{ij}(\rho_{c}{\otimes}\rho_{t})T({\pazocal{N}_{p}},{\pazocal{N}_{q}})_{ij}^{\dagger}$.
Thus the coherence in the control system translates to coherence in the order
of operations. For example, a control qubit in superposition,
$\left|\psi\right\rangle_{c}{=}(\left|0\right\rangle_{c}{+}\left|1\right\rangle_{c})/\sqrt{2}$
results in superposition of two different orders.

It is interesting to see the relationship between the switch operation
$T(\pazocal{N}_{p},\pazocal{N}_{q})$ and the corresponding process formalism.
In terms of the process formalism, a quantum switch consists of three
parties—$A$ corresponding to the operation $\pazocal{N}_{p}$, $B$
corresponding to $\pazocal{N}_{q}$, and $C$ is the party at the output of the
quantum switch who performs, in general, a joint operation occurring after
both $A$ and $B$. There are two possible situations—$A$ being in the causal
past of $B$ ($A\prec B\prec C$), and $B$ is in the causal past of $A$ ($B\prec
A\prec C$). The process matrix of the quantum switch is
$W={\left|w\middle\rangle\middle\langle w\right|}$, where

$\displaystyle\left|w\right\rangle{=}\frac{1}{\sqrt{2}}\left(\left|0\right\rangle_{c}^{C_{I}^{c}}\left|w^{A\prec
B\prec C}\right\rangle{+}\left|1\right\rangle_{c}^{C_{I}^{c}}\left|w^{B\prec
A\prec C}\right\rangle\right),$ (3) $\displaystyle\left|w^{A\prec B\prec
C}\right\rangle{=}{|{\mathbb{1}}\rangle\\!\rangle}^{\mathscr{T}A_{I}}{|{\mathbb{1}}\rangle\\!\rangle}^{A_{O}B_{I}}{|{\mathbb{1}}\rangle\\!\rangle}^{B_{O}C_{I}^{t}},$
(4) $\displaystyle\left|w^{B\prec A\prec
C}\right\rangle{=}{|{\mathbb{1}}\rangle\\!\rangle}^{{\mathscr{T}}B_{I}}{|{\mathbb{1}}\rangle\\!\rangle}^{B_{O}A_{I}}{|{\mathbb{1}}\rangle\\!\rangle}^{A_{O}C_{I}^{t}}.$
(5)

Here, ${\mathscr{T}}$ is the Hilbert space associated with the target
$\rho_{t}$, $C_{I}^{c}$ and $C_{I}^{t}$ denote party $C$’s input Hilbert space
of the control and target systems respectively. Here, the dimension of all the
systems in Eqs. (4) and (5) are the same, effectively the dimension of the
target system. With this we can represent the output of the quantum switch,
$T(\pazocal{N}_{p},\pazocal{N}_{q})(\rho_{c}{\otimes}\rho_{t})$ as

$\displaystyle T(\pazocal{N}_{p},\pazocal{N}_{q})(\rho_{c}{\otimes}\rho_{t})$
$\displaystyle{=}\operatorname{Tr}_{\mathscr{T}A_{I}A_{O}B_{I}B_{O}}\bigg{[}\bigg{\\{}\left(\rho_{c}^{C_{I}^{c}}{\otimes}\rho_{t}^{\mathscr{T}}\right){T}{\otimes}N_{p}^{A_{I}A_{O}}{\otimes}N_{q}^{B_{I}B_{O}}\bigg{\\}}.W\bigg{]}.$
(6)

Detecting whether a process has indefinite causal order is similar to
detecting entanglement in nonseparable quantum states. We discuss this in the
next section.

Figure 2: Causal witness $S$. A causal witness separates the processes that
exhibit indefinite causal order from those that do not. For a process $W$,
with indefinite causal order, there exists a causal witness, namely an
operator $S$ which results Tr$[S\cdot W]<0$. For any non-separable process
$W_{\mathrm{sep}}$ results Tr$[S\cdot W_{\mathrm{sep}}]{\geq}0$.

##  III Causal witness

In the same way that some nonseparable states can violate a Bell inequality
Bell (1964) in a device-independent manner, some causally nonseparable
processes can violate a _causal inequality_ Oreshkov, Costa, and Brukner
(2012). Violation of a causal inequality is impossible with space-like and
time-like correlations Oreshkov, Costa, and Brukner (2012), and there is
currently no known implementation of a violation that is experimentally
feasible in our laboratories.

The quantum switch—although causally nonseparable—does not violate a causal
inequality. In the same way that an entanglement witness Gühne and Tóth (2009)
can be used to detect entanglement in nonseparable quantum states that do not
violate a Bell inequality, a device-dependent quantity called _causal witness_
can be used to confirm whether a process is causally nonseparable Araújo  _et
al._ (2015). The formulation of the causal witness is governed by the
separating hyperplane theorem Rockafellar (1970), which states that it is
always possible to find a hyperplane that separates two disjoint, closed-
convex sets (see Fig. 2). Owing to the fact that causally-separable process
matrices form a closed-convex set, for a particular causally non-separable
process $W$ it is possible to construct a witness $S$ such that

$\displaystyle\operatorname{Tr}[S\cdot W]<0.$ (7)

The same operator $S$, for causally separable processes $W_{\mathrm{sep}}$,
results $\operatorname{Tr}[S\cdot W_{\mathrm{sep}}]\geq 0$.

The value of $\operatorname{Tr}[S\cdot W]$ can be obtained experimentally
after decomposing $S$ into different operators representing different CP maps
(which define the events $A,B,C$), and using Eq. (1) to write
$\operatorname{Tr}[S\cdot W]$ as a combination of the joint probabilities of
these maps to be realised. There have been two experiments demonstrating that
the causal witness can indeed detect indefinite causal order, differing in
their implementations of the quantum switch Rubino  _et al._ (2017a); Goswami
_et al._ (2018).

A quantum switch requires two systems, one to act as control and one as
target. Although the description of a quantum switch is agnostic to the
physical platform, experiments so far have been photonic in nature. One degree
of freedom (DOF) is used as a control and another DOF in the same particle is
used as a target. Mathematically, this is no different from the case of the
control and target systems residing in separate particles. In order to
describe the experiments measuring the causal witness, we classify the quantum
switch implementations based on the DOF used for control: either path of the
photon Rubino  _et al._ (2017a), or polarisationGoswami  _et al._ (2018). In
the formerRubino  _et al._ (2017a), event $A$ is represented by a set of
_measure-prepare operations_ on the polarisation ($\pazocal{N}_{p}$)—four
different measurement operations on the incoming polarisation and re-
preparation of the polarisation in one of the three pre-assigned states. The
events associated with $B$ are ten different unitary operations on the
polarisation ($\pazocal{N}_{q}$). In the second experiment Goswami  _et al._
(2018), both of the events $A$ (operations $\pazocal{N}_{p}$) and $B$
(operations $\pazocal{N}_{q}$) are chosen from six unitary operations acting
on the transverse spatial mode of the photon. In both cases, the event $C$
corresponds to a measurement operation on the control system ($\pazocal{C}$).
With this, one can obtain the value of $\operatorname{Tr}[S\cdot W]$
experimentally after decomposing $S$ into CP maps associated with the events
$A,B,C$,

$\displaystyle
S{=}\sum_{a,b,c,d}\alpha_{a,b,c,d}\rho_{t}^{(a)}{\otimes}{\pazocal{N}_{p}}^{(b)}{\otimes}{\pazocal{N}_{q}}^{(c)}{\otimes}\pazocal{C}^{d}.$
(8)

The superscipts $a,b,c,$ denote different choices for the operations on the
target system and $d$ labels the final measurement operation on the control.
Using the decomposition in Eq. 8 and Eq. 1, $\operatorname{Tr}[S\cdot W]$ is
evaluated as a combination of the joint probabilities of the maps. If the
causal witness $\langle S\rangle:=\operatorname{Tr}[S\cdot W]$ is negative we
can conclude the quantum switch indeed exhibits indefinite causal order, i.e.
the order of events $A$ and $B$ is in a superposition.

###  III.1 Experiment with path as control

One way to implement a quantum switch is to use the path of photons as control
for the order of operations. In Fig.3, the authors used type II spontaneous
parametric down-conversion (SPDC) to generate 790 nm single photons Rubino
_et al._ (2017a). One photon was used as a herald and the other photon was
used as input to the interferometer shown in Fig.3. The target system
(polarisation) is prepared by a set of waveplates, and then sent to a 50/50
beamsplitter (BS). The beamsplitter sets the state of the control system
(path) to be in the superposition of two paths—one first going to $M^{A}$
(corresponding to operations $\pazocal{N}_{p}$ in Eq. 8) and one first going
to $M^{B}$ (corresponding to operations $\pazocal{N}_{q}$ in Eq. 8). The
operations $M^{A}$ in Fig. 3 are _measure-prepare_ operations on polarisation,
consisting of four possible measurements—implemented by a set of waveplates
followed by a polarising beamsplitter—and three possible state re-
preparations—implemented by a set of waveplates. The operations $M^{B}$ in
Fig. 3 are chosen from a set of ten unitary operations on polarisation,
realised by a train of quarter waveplate (QWP), half waveplate (HWP) and a
second QWP. After the two operations, the photon was detected in any of four
output ports labelled as in Fig. 3 where the first bit denotes the result of
the measurement in $M^{A}$ and the second bit indicates the output port of the
control qubit. The experiment was repeated with three different input target
states each of multiple copies to build statistics. The theoreticaly predicted
value of the witness is $-0.2842$, the experimentally obtained value is
$-0.202{\pm}0.029$. Because of the separate paths for each of the orders, the
dominant source of error in this experiment is the dephasing of the control
qubit due to path length difference and other systematic experimental
imperfections. The two paths should remain interferometrically stable: as the
imbalance in the path length grows, the value of the causal witness increases
until it becomes greater than zero for visibilities below $\sim 66\%$.

Figure 3: Quantum switch with path as control. A pair of single photons is
produced via SPDC (not shown). One photon serves as a herald while the other
is sent to the quantum switch. The target system is encoded in polarisation
and is prepared using waveplates. After state preparation, the photon is sent
to a 50/50 beamsplitter which prepares the control in a superposition of two
paths: one where $M^{A}$ comes first and one where $M^{B}$ comes first. Note
that there are two interferometers—the photon impinges on a different spot on
the waveplate depending on the order of the operations. The yellow (purple)
path corresponds to the case when the photon entering $M^{A}$ is
horizontally-(vertically-) polarised, denoted by the first digit in the
detectors being 0(1). The second digit in the detectors correspond to the
final measurement outcome where 0 (1) indicates that a photon exited from a
horizontally-drawn (vertically-drawn) port. Reprinted with permission from G.
Rubino _et al._ , “Experimental verification of an indefinite causal order,"
Sci. Adv. 3, e1602589 (2017) (Ref. Rubino  _et al._ , 2017a). Copyright 2017
by AAAS.

###  III.2 Experiment with polarisation as control

In another implementation of the quantum switchGoswami  _et al._ (2018),
photon polarisation was used as the control qubit for the order of operations
that act on the shape— transverse spatial mode—of the photon. Fig. 4 shows the
experiment schematic, with $A$ and $B$ corresponding to the operations
$\pazocal{N}_{p}$ and $\pazocal{N}_{q}$ in Eq. 2. A narrowband (with a
coherence length much longer than the length of the interferometer) 795 nm
weak coherent laser serves as the input to the first polarising beamsplitter
(PBS1) in Fig. 4. A vertically(horizontally) polarised photon undergoes
operations $A$($B$) first followed by operations $B$($A$). When the control
qubit is set to diagonal polarisation, the order of the operations are in a
superposition. The target qubit in the transverse spatial mode was set to a
first-order Hermite-Gaussian mode. For the causal witness, the operations in
$A$ and $B$ were chosen from a set of six Pauli unitaries
$\\{\mathbb{1},\sigma_{x},\sigma_{y},\sigma_{z},P{=}(\sigma_{y}{+}\sigma_{z})/\sqrt{2},Q{=}(\sigma_{x}{+}\sigma_{z})/\sqrt{2}\\}$.
They were implemented using a train of rotating prisms and cylindrical lenses,
as shown in Fig. 5. For each of the two operations, the polarisation was
measured in the diagonal/anti-diagonal basis. The results were combined as in
Eq. 8 and a causal witness value of $-0.171{\pm}0.009$ was obtained. This
value is different from the ideal value of the causal witness which is
$-0.248$. Unlike in Fig. 3, the two possible orders of $A$ and $B$ in Fig. 4
share a common path so path length difference is not an issue in this kind of
quantum switch implementation. However, the shape of light is highly sensitive
to any imperfections in the transverse plane, e.g. imperfect surface flatness
of the optical elements and non-ideal mode matching. The degradation of the
visibility that results from these imperfections lead to an increase in the
value of the causal witness. Despite the deviations from the ideal case, the
causal witness is still negative indicating indefinite causal order in the
quantum switch.

Figure 4: Quantum switch using polarisation as control. The polarising
beamsplitter PBS1 routes the photon into either events A or B, which realise
unitary operations $\pazocal{N}_{p}$ and $\pazocal{N}_{q}$ respectively. Event
C is a polarisation measurement which determines the Stokes parameter of the
photon in the diagonal/anti-diagonal basis. Lenses L1 and L2 are used as a
telescope to ensure mode-matching. Reprinted with permission from K. Goswami
_et al._ , “Indefinite causal order in a quantum switch," Phys. Rev. Lett.
121, (2018)(Ref. Goswami  _et al._ , 2018). Copyright 2018 by APS.  Figure 5:
Realising unitary operations in the transverse spatial mode. The set of six
Pauli operations
$\\{\mathbb{1},\sigma_{x},\sigma_{y},\sigma_{z},P{=}(\sigma_{y}{+}\sigma_{z})/\sqrt{2},Q{=}(\sigma_{x}{+}\sigma_{z})/\sqrt{2}\\}$
were implemented using inverting prisms (R), cylindrical lenses (C), and
spherical lenses (L). The prisms rotate the incoming transverse mode, the
cylindrical lenses give a $\pi/2$ relative phase shift, and the spherical
lenses or mode-matching. The half-waveplates (H), quarter-waveplates (Q), and
phase-plate ($\varphi$) are used to correct polarisation changes caused by
reflections in the prisms and $\phi$ represents a phase plate. Reprinted with
permission from K. Goswami _et al._ , “Indefinite causal order in a quantum
switch," Phys. Rev. Lett. 121, (2018) (Ref. Goswami  _et al._ , 2018).
Copyright 2018 by APS.

##  IV Bell’s theorem for temporal order

The framework of causal witness presented in the previous section is theory-
dependent in that it assumes that quantum mechanics is true. However, a large
class of _generalised probabilistic theories_ which respects local-realism—and
therefore satisfies Bell inequality—could potentially explain the experimental
outcomes without assuming an indefinite causal structure. A Bell inequality to
distinguish between these generalised probabilistic theories and those that
exhibit indefinite causal order was introduced by Zych et al. Zych  _et al._
(2019) An experiment that violates such a Bell inequality was implemented by
Rubino et al. Rubino  _et al._ (2017b) It consists of two _entangled quantum
switches_ with separable target systems in polarisation and control systems
entangled in path.

Figure 6: Experimental violation of Bell’s theorem for temporal order, as
described in Ref. Rubino  _et al._ , 2017b. (a) Two quantum switches with
polarisation as target and path has control were used to show a violation.
(PBS: polarising beamsplitter, QWP:quarter- waveplate, HWP:half-waveplateHC,
HC:half-waveplate compensation, BS: beamsplitter. All output ports of the
beamsplitters are monitored by single-photon detectors.)

Rubino  _et al._ (2017b) use the setup in Fig. 6 to violate the following no-
go theorem: No states, set of transformation, or measurements violate Bell’s
inequality given (1) the initial joint state of the target system does not
violate Bell’s inequality; (2) the laboratory operations are local
transformation, i.e. they do not increase the amount of violation of Bell’s
inequality; (3) the order of operations on the two target systems are pre-
defined.

When assumptions 1 and 2 are both satisfied, the violation of Bell’s
inequality can only be associated with indefinite causal order.

To ensure the validity of assumption (1) a state tomography of the input joint
target systems was performed. They obtained the polarisation state
$\left|HH\right\rangle(\left|00\right\rangle)$ with a fidelity of
$0.935{\pm}0.004$ with and a concurrence of $0.001{\pm}0.010$. Morevoer, a
probability measurement on the joint target systems showed that the joint
probability distribution was the product of the marginals, suggesting that the
initial joint target systems are reasonably separable.

Assumption (2) is enforced by keeping the two operations spatially separated
in the optical setup, thus eliminating the possibility of entangling
operations. Moreover, the local operations cannot increase the amount of
violation of Bell’s inequality. This was supported by disentangling both
initial target-control systems and performing corresponding probability
measurement at the output. The probability distribution turned out to be the
product of the marginals.

With the first two assumptions verified, Rubino et al. Rubino  _et al._
(2017b) violated the Bell inequality using two quantum switches, $T_{1}$ and
$T_{2}$, in a two-loop Mach-Zehnder interferometer as shown in Fig. 6 . They
start with separable target systems ,
$\left|0\right\rangle_{t}^{T_{1}}\left|0\right\rangle_{t}^{T_{2}}$, and
control systems entangled in path,
$(\left|0\right\rangle_{c}^{T_{1}}\left|0\right\rangle_{c}^{T_{2}}-\left|1\right\rangle_{c}^{T_{1}}\left|1\right\rangle_{c}^{T_{2}})/\sqrt{2}$.
The entanglement of the initial joint control system was verified with a Bell
test resulting in a Bell parameter of $2.58{\pm}0.09$. The local operations
applied were $U_{A}^{T_{1}}{=}U_{A}^{T_{2}}{=}\sigma_{z}$ and
$U_{A}^{T_{1}}{=}U_{A}^{T_{2}}{=}(\mathbb{1}{+}i\sigma_{x})/\sqrt{2}$,
implemented via a set of waveplates. At the output, the authors measured the
control systems in the diagonal/anti-diagonal basis via a 50-50 beamsplitter.
This made the resulting target systems maximally entangled,
$(\left|l\right\rangle_{t}^{T_{1}}\left|l\right\rangle_{t}^{T_{2}}-\left|r\right\rangle_{t}^{T_{1}}\left|r\right\rangle_{t}^{T_{2}})/\sqrt{2}$
with
$\left|l\right\rangle{=}(\left|0\right\rangle{+}i\left|1\right\rangle)/\sqrt{2}$
and
$\left|r\right\rangle{=}(\left|0\right\rangle{-}i\left|1\right\rangle)/\sqrt{2}$.
A state tomography of the output joint target system gave a fidelity of
$0.922{\pm}0.005$ and a concurrence of $0.95{\pm}0.01$. The Bell test on the
joint output target systems resulted in a Bell parameter of $2.55{\pm}0.08$.
This violation of the Bell inequality means that assumption (3) is invalid,
thus demonstrating indefinite causal order in a theory-independent manner.

##  V Application in quantum Shannon theory

Aside from foundational implications, indefinite causal order brings about
some practical advantages. We first focus on an advantage for communication:
communicating through noisy channels. Optimising information transmission
through noisy channels is the primary goal of information theory. Quantum
physics played a major role in this endeavour by opening up new
possibilities—such as unprecedented security—when the information carriers are
themselves quantumWilde (2013); Yin  _et al._ (2020). A quantum information
theory that allows for indefinite causal order opens up further possibilities
Ebler, Salek, and Chiribella (2018); Salek, Ebler, and Chiribella (2018);
Chiribella  _et al._ (2018), one example is being able to communicate through
completely noisy channels. Several worksEbler, Salek, and Chiribella (2018);
Salek, Ebler, and Chiribella (2018); Chiribella  _et al._ (2018) have shown
that when two noisy channels, say $\pazocal{N}_{p}$ and $\pazocal{N}_{q}$, are
connected in an indefinite order, there is a significant improvement in both
classical and quantum information capacity. We first discuss the theoretical
aspects of this communication advantage and then proceed to discuss two
experiments that demonstrate these ideas.

A standard way of modelling a noisy quantum channel is to exploit generalised
Pauli channels. A Pauli channel is a probabilistic mixture of Pauli operations
associated with three common errors: $\sigma_{1}{\equiv}\sigma_{x}$ (bit
flip), $\sigma_{3}{\equiv}\sigma_{z}$ (phase flip), and
$\sigma_{2}{\equiv}\sigma_{y}$ (combination of bit flip and phase flip). We
denote as $\sigma_{0}{\equiv}I$, i.e. identity operation when there is no
error. The action of such a channel $\pazocal{N}_{p}$ on a quantum state
$\rho$ would be,

$\displaystyle\pazocal{N}_{p}(\rho){=}\sum_{i=0}^{3}p_{i}\sigma_{i}\rho_{t}\sigma_{i}^{\dagger}$
(9)

where $\sum_{i}p_{i}{=}1$. Let us consider two such channels $\pazocal{N}_{p}$
and $\pazocal{N}_{q}$ acting on the target qubit $\rho_{t}$. Note that we keep
the same notation—$\pazocal{N}_{p}$ and $\pazocal{N}_{q}$—as in the previous
section, only that we focus on the case that these are noisy channels rather
than an arbitrary operation on the target system. Let us also consider a
control qubit initially set to the state
$\rho_{c}{=}|\psi\rangle\langle\psi|_{c}$, where
$\left|\psi\right\rangle{=}\sqrt{\gamma}|0\rangle{+}\sqrt{1{-}\gamma}|1\rangle$.
The total output state of the switch becomesGoswami  _et al._ (2020)

$T[\pazocal{N}_{p},\pazocal{N}_{q}](\rho_{c}{\otimes}\rho_{t}){=}\left(\begin{array}[]{cc}A&B\\\
B&\tilde{A}\\\ \end{array}\right),$ (10)

with,

$\begin{array}[]{l}A{=}\gamma\pazocal{N}_{q}\circ\pazocal{N}_{p}(\rho_{t}),\\\
\tilde{A}{=}(1-\gamma)\pazocal{N}_{p}\circ\pazocal{N}_{q}(\rho_{t}),\\\
B{=}\sqrt{\gamma(1-\gamma)}(\epsilon_{+}(\rho_{t})-\epsilon_{-}(\rho_{t})),\\\
\end{array}$ (11)

where ($\epsilon_{-}$) $\epsilon_{+}$ represents an auxiliary trace non-
preserving map

$\displaystyle\epsilon_{+}(\rho_{t})$
$\displaystyle{=}\sum_{i=0}^{3}p_{i}q_{i}\,\rho_{t}+\sum_{i=0}^{3}r_{0i}\,\sigma_{i}\rho_{t}\sigma_{i}^{\dagger}$
(12) $\displaystyle\epsilon_{-}(\rho_{t})$
$\displaystyle{=}r_{12}\,{\sigma_{3}}\rho_{t}\sigma_{3}^{\dagger}{+}r_{23}\,\sigma_{1}\rho_{t}\sigma_{1}^{\dagger}{+}r_{31}\,\sigma_{2}\rho_{t}\sigma_{2}^{\dagger},$
(13)

with $r_{ij}{=}p_{i}q_{j}{+}p_{j}q_{i}$. Note that
$\pazocal{N}_{p}{\circ}\pazocal{N}_{q}(\rho_{t}){=}\epsilon_{+}(\rho_{t})+\epsilon_{-}(\rho_{t}){=}\pazocal{N}_{q}{\circ}\pazocal{N}_{p}(\rho_{t})$,
which means any definite order of $\pazocal{N}_{p}$ and $\pazocal{N}_{q}$ will
have the same effect on the target qubit. The presence of the nontrivial off-
diagonal block matrix $B$ in the joint output state (Eq. 10) arises from the
indefinite causal order and can lead to a communication advantage; $B$
vanishes in the case of definite order.

Before proceeding, it is important to introduce the standard measure of
classical and quantum information transmission in a channel, quantified via
the classical capacity $C$ and the quantum capacity $Q$ respectively. The
classical capacity is defined by the maximum rate of classical bits that can
be transmitted over asymptotic uses of the channel (i.e. when the number of
times a channel is used approaches infinity). A convenient measure to quantify
the classical capacity is given by the Holevo capacity $\chi$, which for a
channel $\pazocal{N}$ is defined by:

$\displaystyle\chi(\pazocal{N}){=}\underset{p(m),\rho_{m}}{{\mathrm{max}}}E\left(\sum_{m}p(m)\pazocal{N}(\rho_{m})\right){-}\sum_{m}p(m)E\left(\pazocal{N}(\rho_{m})\right).$
(14)

Here, $E(.)$ is the von Neumann entropy, and the maximum is over the input
ensembles $\\{p(m),\rho_{m}\\}$. Similarly, the quantum capacity $Q$ is the
maximum rate that qubits can be transmitted over asymptotic uses of the
channel. The one shot version of the quantum capacity is given by coherent
information $Q_{1}$:

$\displaystyle
Q_{1}=\underset{\rho}{{\mathrm{max}}}\,E\left(\pazocal{N}(\rho)\right){-}E\left[(\mathbb{1}{\otimes}\pazocal{N})\left(\left|\psi_{\rho}\middle\rangle\middle\langle\psi_{\rho}\right|\right)\right].$
(15)

Here, $\left|\psi_{\rho}\right\rangle$ is the purification of the input state
$\rho$. The purification can be obtained by adding an ancillary system
associated with an orthonormal basis $\\{\left|m\right\rangle\\}$ to the
system of $\rho$. Considering $\rho$ in a diagonalised form
$\sum_{m}\lambda_{m}\left|\lambda_{m}\middle\rangle\middle\langle\lambda_{m}\right|$,
the purification becomes
$\left|\psi_{\rho}\right\rangle{=}\sum_{i}\sqrt{\lambda_{m}}\left|\lambda_{m}\right\rangle\left|m\right\rangle$
Nielsen and Chuang (2000), where $\\{\lambda_{m}\\}$ and
$\\{\left|\lambda_{m}\right\rangle\\}$ are the eigenvalues and the
eigenvectors of $\rho$ respectively. The optimisation of the coherent
information is done over all possible input states $\rho$Wilde (2013).

###  V.1 Experiment with polarisation as the control

The schematic in Fig. 4 can be used to verify the communication advantage
proposed by Ebler et al.Ebler, Salek, and Chiribella (2018). The polarisation
was used as the control for the order of Pauli operators. The Pauli operators
act on the transverse spatial mode, which was initially set to a first-order
Hermite-Gaussian mode. The Pauli operators were implemented using pairs of
rotating prisms which were rotated depending on the operationsGoswami  _et
al._ (2020). All possible pairs the Pauli operations were implemented to be
able to reconstruct the channels $\pazocal{N}_{p}$ and $\pazocal{N}_{q}$ with
the relevant probability distributions Goswami  _et al._ (2020).

Figure 7: Logarithm of Holevo capacity $\chi$ when (a) both channels are
depolarising channels and (b) one channel is $\sigma_{z}$ channel and the
other is a depolarising channel, for varying depolarising strength $q$
(x-axis). The solid blue line is the capacity for definite causal order. The
solid black line is the ideal capacity for the case of indefinite causal
order. The red dots are experimentally-derived capacities with the orange
shade showing the uncertainty due to non-ideal visibility. Reprinted with
permission from K. Goswami _et al_., “increasing communication capacity via
superposition of order," Phys. Rev. Res. 2, 033292 (2020) (Ref. Goswami  _et
al._ , 2020). Copyright 2020 by APS.

_Classical communication with two depolarising channels_ Ebler et al.Ebler,
Salek, and Chiribella (2018) proposed that two completely depolarising
channels $\pazocal{N}(\rho){=}(\sum_{i}\sigma_{i}\rho\sigma_{i}$)/4 connected
in an indefinite causal order is able to transmit a finite amount of
information. This is particulaly surprising given that any definite ordered
combination of these channels will scramble all the classical information.
Fig. 7 shows the logarithm of the Holevo capacity as a function of
depolarising strength $q$. The theoretical prediction for the Holevo capacity
of two fully-depolarising ($q{=}1$) channels arranged in a superposition of
order is $0.049$ bits. As shown in Fig. 7.a, the experiment showed a
transmission of $(3.4{\pm}0.2){\times}10^{-2}$ bitsGoswami  _et al._ (2020).

_Classical communication with one unitary and one depolarising channel._ A
more dramatic example of information-theoretic advantage is possible when one
channel is a Pauli operation and the other is a completely depolarising
channel. A definite-ordered combination of these two channels result to a
zero-capacity channel. However, when these channels are arranged in an
indefinite order, it is possible to achieve deterministic classical
information transmission (solid black line in Fig. 7.b) through the system
even in the case of full depolarisation strength ($q{=}1$)Goswami  _et al._
(2020). As shown in Fig. 7.b, for all values of $q$ the experimental Holevo
capacity of the indefinite-ordered case (orange curve) is significantly higher
than the definite-ordered case (blue curve); for $q{=}1$, Holevo capacity is
$0.64{\pm}0.02$ bits.

Figure 8: (a) Experimental setup Guo  _et al._ (2020). Pairs of photons were
produced by spontaneous parametric down-conversion. One of the photons acted
as a herald while the other is input to the interfereometer. The polarisation
of the photon was used as target and the photon path acted as the control
qubit. The noisy channels (here denoted by $\epsilon_{\vec{p}}$ and
$\pazocal{F}_{\vec{q}}$) each consist of waveplates. (HWP: half wave plate;
QWP: quarter wave plate; PBS: polarizing beam splitter; BS: 50/50
beamsplitter; RM: mirror; FC: fiber coupler; SPD: single photon detector; DL:
trombone-arm delay line.)(b) Experimental (cyan) classical capacity (denoted
here by $C_{1}$) for the case of two fully-depolarising channels combined in
indefinite order compared with ideal theoretical classical capacity (red). (c)
One-shot coherent information $Q_{1}$ for the channels($\pazocal{N}_{p}$ and
$\pazocal{N}_{q}$) as a function of q [0,1]. Blue dots show the experimental
results, the corresponding theoretical prediction is shown by the red line.
For comparison, the one-shot coherent information of the two dephasing
channels combined in a definite order is also shown by the black curve.
Reprinted with permission from Y. Guo _et al._ , “Experimental transmission of
quantum information using a superposition of causal orders," Phys. Rev.Lett.
124, (2020) (Ref. Guo  _et al._ , 2020). Copyright 2020 by APS.

###  V.2 Experiment with path as the control

To show communication advantage, Guo et al. implemented a quantum switch (see
Fig. 8.a) using path as the control for the order and polarisation as the
target qubitGuo  _et al._ (2020). The Pauli operators are realised using a set
of waveplates. The noisy channels are constructed according to the relevant
probability distributions (similar to Eq. 9) using the individual
instantiations of the different Pauli operations.

_Classical communication advantage:_ Similar to the classical communication
advantage that was shown by Goswami et al. Goswami  _et al._ (2020), Guo et
al. also experimentally demonstrated that two fully depolarising channels
acting in an indefinite order can lead to nonzero information capacity Guo
_et al._ (2020). They showed this for different input polarisation states
$\\{\left|H\right\rangle,\left|V\right\rangle\\}$,
$\\{\left|D\right\rangle,\left|A\right\rangle\\}$ and
$\\{\left|R\right\rangle,\left|L\right\rangle\\}$, achieving one-shot
classical capacities of of $0.0422{\pm}0.0010$ $0.0426{\pm}0.0011$
$0.0432{\pm}0.0009$ bits, respectively. Fig. 8.b shows these values compared
with the ideal value of 0.049 bits.

_Quantum communication through zero capacity channel:_ The authors used the
combination of a bit flip channel
$\pazocal{N}_{p}(\rho){=}(1-p)\rho{+}p\sigma_{1}\rho\sigma_{1}$ and a phase
flip channel $\pazocal{N}_{q}(\rho){=}(1-q)\rho{+}q\sigma_{3}\rho\sigma_{3}$
with $p{=}q{=}1/2$. A definite-ordered combination of these channels has zero
classical and quantum capacity. However, Salek et al. has proposed that it is
possible to transmit heralded, noiseless qubits through the channel
$\epsilon_{-}$ (Eq. 13 ) if the channels are arranged in an indefinite
orderSalek, Ebler, and Chiribella (2018). Guo et al. have shown experimentally
that indeed this is the case, they achieved a heralded one-shot coherent
information ($Q_{1}$) of $0.812{\pm}0.003$. Fig. 8.c is a plot of $Q_{1}$ as a
function of the parameter $q$. For comparison, $Q_{1}$ of a definite-ordered
combination of $\pazocal{N}_{p}$ and $\pazocal{N}_{q}$ is shown by the black
curve.

_Quantum communication with entanglement-breaking channels:_ Entanglement-
breaking channels cannot transmit quantum information Wilde (2013). Two copies
of the entanglement-breaking channels
$\pazocal{N}(\rho){=}1/2(\sigma_{1}\rho\sigma_{1}{+}\sigma_{2}\rho\sigma_{2})$
arranged in a a definite order yields a zero-capacity channel Chiribella  _et
al._ (2018). However, when these channels are in an indefinite order, the
resulting auxiliary channels $\epsilon_{\pm}$ (Eq. 12 and 13) are unitary
channels. This results in deterministic noiseless qubit transmission. In the
experiment, the authors achieved a one-shot coherent information of
$0.855{\pm}0.004$.

###  V.3 Communication advantage: further discussion

Although this is a review of experiments in causality, for the sake of
completeness, we are highlighting an active discussion. When the communication
enhancement based on superposition of causal order were introduced in Refs.
Ebler, Salek, and Chiribella (2018); Salek, Ebler, and Chiribella (2018);
Chiribella  _et al._ (2018); Goswami  _et al._ (2018), two alternative
resources—coherent superposition of path by Abbott et al.Abbott  _et al._
(2018) and coherent superposition of intermediate quantum operations by Guérin
et al. Guérin, Rubino, and Brukner (2019)—were proposed. Both these works
showed that the activation—increase from zero to nonzero—of both classical and
quantum capacities is possible even for a process that does not exhibit
indefinite causal order. Specifically, Abbott  _et al._ (2018) demonstrated
that the superposition of two paths, each containing a depolarising channel
can lead to a higher classical capacity than superposition of causal order
(with a caveat that the activated capacity is dependent on channel
implementation, unlike superposition of causal order). However, it turned out
that advantages associated with superposition of path is not universal. For
certain combinations of noisy channels—two entanglement breaking channels in
the case of quantum communication Chiribella  _et al._ (2018) and one
depolarising and one unitary channel in the case of classical communication
Goswami  _et al._ (2018)—superposition of causal order always outperforms
superposition of path. The other resource, introduced by Guérin et al.Guérin,
Rubino, and Brukner (2019), showed that a coherent superposition of
intermediate unitary operations (referred to as superposition of direct pure
processes, SDPP) can also lead to a communication advantage. As could be
expected, superposition of indefinite causal order does not present an
advantage over SDPP because of the presence of a clean channel in the latter
in the first place.These subsequent works which also show a communication
advantage motivates two interesting questions: (1) Can the communication
enhancement in the quantum switch be explained by an alternative resource? and
(2) How do we put different resources in a comparative framework?

In response to (1), Guérin et al.Guérin, Rubino, and Brukner (2019),argues
that as noisy operations were acting only on the target qubit, the control
qubit provides a clean ‘side channel’ to transmit information. As a counter to
this argument, Kristjánsson et al. Kristjánsson  _et al._ (2019) showed that
theoretically there is no side channel generated in the quantum switch and
thus indefinite causal order is a resource in itself. One subtle thing to
highlight is that in experimental implementations the vacuum plays a part
Oreshkov (2019). This is different from the ideal quantum switch which is a
mathematical framework dependent solely on the local channels, and the
background causal structure. The role of vacuum is prevalent in the protocols
of Abbott et al. Abbott  _et al._ (2018) and Guérin et al. Guérin, Rubino, and
Brukner (2019) (what Kristjánsson et al. refers to as _vacuum-extended_
channels).

Motivated by (2), Kristjánsson et al. Kristjánsson  _et al._ (2019) presented
a resource-theoretic framework of indefinite causal order. A recent experiment
by Rubino et al.Rubino  _et al._ (2020) compared all three
resources—superposition of order, superposition of path, SDPP—and confirmed
what has been already been shown in theory that coherent superposition of
intermediate unitary operations always yields higher information theoretic
advantage compared to the other resources. All these discussions provide a
clear motivation for a generalised resource theory to encompass different
types of coherent controls the advantages they bring about.

##  VI Application in computation and communication complexity

Figure 9: (a) Experimental setup Procopio  _et al._ (2015) to distinguish
whether a pair of unitary gates commute or anti-commute. The photons were
generated via spontaneous parametric down-conversions. One photon is used as a
herald, and the other is the input to the Mach-Zehnder interferometer. The
path, which is used as the control, is coherently superposed by the input
beamsplitter (BS). The gates which consist of the train of waveplates in each
arm, act on the polarisation (b) The probability that a photon is measured in
either Port 0 or Port 1 for different combinations of Pauli gates and
identity. Green (blue) bars correspond to the commuting (anti-commuting)
combinations. (c) Same as (b), only that the gates are chosen from a random
set of 50 commuting and 50 anti-commuting sets of gates. Reprinted with
permission from L. M. Procopio _et al._ ,“Experimental superposition of orders
of quantum gates," Nat. Commun. 6, (2015) (Ref. Procopio  _et al._ , 2015).
Copyright 2015 by Nat. Commun.

Conventional quantum information theory departed from classical information by
considering information carriers that exhibit quantum behaviour. The resulting
new paradigm of computation—quantum computation—offers considerable advantage
for some specific tasks, e.g. factoring. As we have seen in the previous
sections, indefinite causal order adds another aspect of quantum behaviour by
allowing for superpositions of the order that operations act on a target
systemChiribella  _et al._ (2013); Araújo, Costa, and Brukner (2014), it is
thus natural to ask whether this can result to further computational
advantage. Quantum states in the quantum circuit model evolve according to a
sequence of quantum gates that have a fixed order—what advantages arise with
coherent control of this order?

In this section, we will focus on experiments which demonstrate that
indefinite causal order presents advantages for some computational and
communication complexity tasks. The quantum switch is again central to all
these demonstrations.

###  VI.1 Distinguishing between commuting and anti-commuting gates

We first focus on the following task: Given that a pair of gates either
commute or anti-commute, how do we distinguish between these two cases? A
quantum circuit with a fixed order of these gates would require at least two
uses of one of the gates. In contrast, a quantum circuit with an indefinite
order of these two gates require only a single use of each gate Chiribella
(2012).

Consider two gates $U_{1}$ and $U_{2}$ acting on a target qubit
$\left|\psi\right\rangle_{t}$. A control qubit $\left|\psi\right\rangle_{c}$
controls the order that these two gates are applied:
$\left|0\right\rangle_{c}$ leads to the order $U_{1}U_{2}$ and
$\left|1\right\rangle_{c}$ leads to the order $U_{2}U_{1}$. Placing $U_{1}$
and $U_{2}$ in a quantum switch and setting the control qubit to a
superposition thus results to a superposition in the order that these gates
are applied to the target qubit. More specifically, for an input joint state
of
$(\left|0\right\rangle_{c}+\left|1\right\rangle_{c})/\sqrt{2}\otimes\left|\psi\right\rangle_{t}$,
and a Hadamard operation applied at the output of the control qubit, the
resulting joint state is,

$\displaystyle\frac{1}{2}\left(\left|0\right\rangle_{c}\otimes\\{U_{1},U_{2}\\}\left|\psi\right\rangle_{t}+\left|1\right\rangle_{c}\otimes[U_{1},U_{2}]\left|\psi\right\rangle_{t}\right).$
(16)

If the control qubit is measured to be in $\left|0\right\rangle_{c}$, the
gates commute, if the control qubit is measured to be in
$\left|1\right\rangle_{c}$, the gates anti-commute.

Procopio et al.Procopio  _et al._ (2015) showed this advantage using a quantum
switch that uses path as control and polarisation as the target qubit for the
gates $U_{1}$ and $U_{2}$. They used a quantum switch similar to the one used
by Rubino et al.Rubino  _et al._ (2017a) for testing the causal witness, in
this case replacing $M^{A}$ and $M^{B}$ by $U_{1}$ and $U_{2}$. Photons from
spontaneous parametric down-conversion were used, one as a herald and one as
input to the quantum switch. As shown in Fig. 9.a, the quantum switch consists
of a Mach-Zender interferometer with each arm containing a loop. The train of
quarter-waveplates and half-waveplates in each arm act as the gates. The input
50/50 beamsplitter creates the superposition of transmitted
($\left|0\right\rangle_{c}$) and reflected ($\left|1\right\rangle_{c}$) paths
for the control qubit. The two paths are again coherently combined at the
final 50/50 beamsplitter, which acts as the Hadamard operation for the control
qubit. The direction that the photon exits—measured by the single-photon
detectors at either port 0 or port 1—reveals whether the gates are commuting
or anti-commuting.

Fig. 9.b. shows the probability that a photon exits from a port, given that
the quantum switch has $U_{1}$ and $U_{2}$ as the gates. Note that $U_{1}$ and
$U_{2}$ were chosen from the three Pauli operations
$\\{\sigma_{x},\sigma_{y},\sigma_{z}\\}$ and the identity operation
$\mathbb{1}$. Fig. 9.b shows the results for all 16 combination of gates. As
expected, when the gates are commuting, the photon exits port 0 (green bars),
otherwise if the gates are anti-commuting, the photon exits port 1 (blue
bars). The success rate—defined as the probability to exit the correct port—is
$0.973\pm 0.016$. Fig. 9.c shows the results for 50 randomly generated
commuting unitaries and 50 randomly generated anti-commuting unitaries. The
authors report a success of $0.976\pm 0.015$ in this case. Note that these
success rates are both larger than the highest success rate possible with a
definite-order quantum circuit which is $0.9288$.

Figure 10: (a) Experimental setup for showing an advantage in communication
complexity Wei  _et al._ (2019). Photon pairs were produced from spontaneous
parametric down-conversion. One photon is used as a herald and the other is
fed to the interferometer via circulator (Cir) and beamsplitter (BS) which
prepares the control in a superposition of the two paths. In the Sagnac loop,
Alice and Bob implement the unitary operations by using variable delays and a
phase modulator (PM). The binary digits were represented by either the long
option or the short option. For the long option, the authors stored the pulse
train by using a 7 km long fiber spool (FS). The result of the computation was
obtained from the outputs of the superconducting nanowire single photon
detectors (SNSPD2 and SNSPD3). (b) The transmitted information with respect to
the system size($n$). Red points show the experimental results with the best
fit line using a quantum switch. For $n{=}12$ both classical (back) and
quantum causally definite (blue) protocols require more transmitted bits,
indicating a clear advantage for using a quantum switch. (c) The parameter
$\gamma$ represents the ratio of the transmitted information $Q$ over the
lower bound on the transmitted information $C$ for a given quantum or
classical causally separable protocol. For $n=12$, $\gamma<1$ indicating that
a quantum switch requires less transmitted bits for the EE game. Reprinted
with permission from K. Wei _et al._ , “Experimental quantum switching for
exponentially superior quantum communication complexity," Phys. Rev. Lett.
122, (2019) (Ref. Wei  _et al._ , 2019). Copyright 2019 by APS.

###  VI.2 Communication complexity

Communication complexity quantifies the cost of communication requirements in
solving problems when the input is distributed to two or more parties Nielsen
and Chuang (2000). Guérin  _et al._ (2016) suggested that indefinite causal
order can lead to a reduction in the communication complexity of the _exchange
evaluation (EE)_ game. Consider bit strings $\vec{x}$,
$\vec{y}\in\\{0,1\\}^{n}$ and Boolean functions
$f,g:\\{0,1\\}^{n}\rightarrow\\{0,1\\}$ with $f(\vec{0}){=}g(\vec{0}){=}0$,
where $\vec{0}$ is a string of zeros. Alice has $(\vec{x},f)$ and Bob has
$(\vec{y},g)$, and both of them can signal a third party, Charlie. Charlie’s
task is to evaluate the _exchange evaluation function_
$\mathrm{EE}(\vec{x},f,\vec{y},g){=}f(\vec{y}){\oplus}g(\vec{x})$, where
$\oplus$ denote addition modulo 2. The communication is strictly one-way i.e.
Charlie cannot signal to Alice and Bob.

Guérin  _et al._ (2016) proposed that superposition of causal order between
Alice and Bob—such that communication direction is also in a superposition—can
achieve deterministic, exponential communication advantage in terms of the
number of bits that Alice and Bob communicate. Charlie can compute the
exchange evaluation function by measuring the control qubit of a quantum
switch.

Wei  _et al._ (2019) implemented a variant of the protocol proposed by Guérin
_et al._ (2016) using a quantum switch that uses path as a control for the
order. For the target system, they used $d{=}2^{n+1}$-qudit obtained from the
arrival time of the same photon. The arrival time was divided into $2^{n+1}$
time bins which then make up an orthonormal basis $\left|z\right\rangle$ with
$z{\in}\\{0,1,\dots,2^{n+1}-1\\}$. The $EE$ game can be translated to a game
that distinguishes commuting unitaries from anti-commuting unitaries, similar
to the task in the previous section. Wei  _et al._ (2019) introduced two
unitaries $U_{A}\equiv X^{x}_{d}D_{d}(f)$ and $U_{B}\equiv X^{y}_{d}D_{d}(g)$,
where $D_{d}(f)=Z_{d}^{f(z)d/{2z}}$ and $X_{d}$ and $Z_{d}$ are high-
dimensional Pauli operators acting on the basis-set
$\\{\left|z\right\rangle\\}$ which satisfy $Z_{d}X_{d}=\exp(2\pi
i/d)X_{d}Z_{d}$. With these definitions,
$[U_{A},U_{B}]\left|0\right\rangle_{t}=0$ if $f(y)\oplus g(x)=0$,
$\\{U_{A},U_{B}\\}\left|0\right\rangle_{t}=0$ if $f(y)\oplus g(x)=1$, where
the subscript $t$ denotes the target system.

The problem can be solved by using a quantum switch with operations $U_{A}$
and $U_{B}$, and input control in
$\left|+\right\rangle_{c}=1/\sqrt{2}(\left|0\right\rangle_{c}+\left|1\right\rangle_{c})$,
and input target in $\left|0\right\rangle_{t}$. A Hadamard operation applied
to the control qubit results to an overall state

$\displaystyle\frac{1}{2}\left(\left|0\right\rangle_{c}\otimes\\{U_{A},U_{B}\\}\left|0\right\rangle_{t}{-}\left|1\right\rangle_{c}\otimes[U_{A},U_{B}]\left|0\right\rangle_{t}\right).$
(17)

Measuring the control qubit in the basis $\left|z\right\rangle$ then reveals
whether $f(\vec{y}){\oplus}g(\vec{x})=0$ or $f(\vec{y}){\oplus}g(\vec{x})=1$.

The experiment uses a fibre-based Sagnac interferometer, as shown in Fig.
10.aWei  _et al._ (2019). Photon pairs are produced from SPDC, one of the
photons is used as a herald and another is fed to the interferometer. The
preparation of the control qubit into a superposition of paths is done by a
combination a circulator and a fibre beamsplitter. Alice and Bob in the Sagnac
loop each have variable delays and a phase modulator to implement the unitary
operations $U_{A}$ and $U_{B}$. Alice and Bob individually uses the phase
modulator twice, once when the photon enters the Sagnac loop, and once when it
exits the loop. The photon interference in the loop ensures that it exits at
either of the output ports of the interferometer depending on the answer to
the EE game.

Fig. 10.b shows the transmitted bits as a function of system size $n$ (i.e.
for when there are $2^{n}$ choices of delay for Alice and Bob). The red dashed
line is the best fit for the experimental data. For comparison, the black and
blue curves show the required transmitted bits for the best classical protocol
and the best fixed-order quantum protocol, respectively. For a system size
$n=12$, the indefinite-ordered protocol using the quantum switch beats both
classical and best fixed-order quantum protocols. Fig. 10.c compares the
performance of the indefinite-ordered protocol to fixed-order processes by
taking the ratio of transmitted information through the quantum switch ($Q$)
and the minimum transmitted information through either causally separable
quantum or classical protocol $C$, i.e. $\gamma=Q/C$. For $n=12$, this ratio
is less than for both the classical and quantum-definite ordered processes
indicating a clear advantage in using a quantum switch.

Figure 11: Experimental setup for the quantum 4-switch, as described in Ref.
Taddei  _et al._ , 2020. An input photon was coherently divided into four
spatial modes using a four-core fibre beamsplitter (4CF-BS), placed between
two multiplexer/demultiplexer (DMUX) units. The four output modes were then
sent to the quantum 4-switch gate. Each spatial mode was associated with a
unique permutation of the four unitary polarisation operations. The photons
enters through the ‘IN’ side and exit through the ‘OUT’ side as shown. The
notation, for example, “$\leftarrow$A" means “from A" and “A $\leftarrow$"
means “to A". After leaving the quantum-4 switch, the four spatial modes were
recombined using a second 4CF-BS. The outputs were connected directly to
single-photon detectors (APD).

###  VI.3 Hadamard promise problem

So far we have only considered a quantum switch where the control is a
qubit—controlling the order of only two operations. The quantum switch can be
generalised to a quantum $n$-switch, with a control that specifies the order
of $n$ operations. In this case, a $d$-dimensional control system (where
$d\leq n!$) creates a coherent superposition of $d$ permutations of all the
quantum operations.Araújo, Costa, and Brukner (2014) To formulate the
generalisation, consider a set of unitary operations $\\{U_{i}\\}$. We define
$\Pi_{x}{=}U_{\sigma_{x}(n)}.U_{\sigma_{x}(n-1)}{\dots}U_{\sigma_{x}(1)}$ with
$\sigma_{x(i)}$ being the $i$-th element of the $x$-th permutation of the
gates, where $x{\in}\\{1,2,\dots,d\\}$. The action of the quantum switch
$T_{n}$ on the input state
$\left|x\right\rangle_{c}\otimes\left|\psi\right\rangle_{t}$ becomes,

$\displaystyle
T_{n}(\left|x\right\rangle_{c}\otimes\left|\psi\right\rangle_{t}){=}\left|x\right\rangle_{c}\otimes\Pi_{x}\left|\psi\right\rangle_{t}.$
(18)

Taddei et al.Taddei  _et al._ (2020) used this generalisation of a quantum
switch to show a computational advantage for the Hadamard promise problem:
Given a $d$-dimensional Hadamard matrix $M_{d}$Hedayat and Wallis (1978) with
all $+1$ entries along its first row and first column, and a unitary-gate
oracle $\pazocal{O}$ fulfilling the promise $\Pi_{x}{=}m_{x,y}\Pi_{0}$—where
$\\{m_{x,y}\\}={\pm}1$ are the entries of $M_{d}$—for some column $y\in[P]$ of
$M_{d}$, compute $y$. Note that here, $[P]$ is defined as $[P]:={0,1,...P-1}$
,i.e. $P$ permutations of an $n$-letter alphabet.

After the series of steps: (1) initialise joint system to
$\left|0\right\rangle_{c}\otimes\left|\psi\right\rangle_{t}$ (2) apply $M_{d}$
on the control (3) apply $T_{n}$ on the joint system (4) apply $M_{d}^{-1}$ on
the control, the resulting state is,

$\displaystyle\frac{1}{d}M_{d}^{-1}T_{n}M_{d}(\left|0\right\rangle_{c}\otimes\left|\psi\right\rangle_{t}){=}\left|y\right\rangle_{c}\otimes\Pi_{0}\left|\psi\right\rangle_{t}.$
(19)

From this state, the value of $y$ can be read by measuring the control in the
computational basis.

Using the quantum $n$-switch, the total number of oracle queries—the query
complexity $R$—of the Hadamard promise problem is $R{=}n$ for all $P{\leq}n!$.
At least for the case of $P{=}n!$, this is quadratically lower in $n$ when
compared with the best definite-ordered algorithm.

Fig. 11 shows the experimental setup used to verify this computational
advantageTaddei  _et al._ (2020) for the case of $d{=}4$, i.e. using a quantum
4-switch. The path was used to control the different permutations of the order
of the operations. The implementation is based on multi-core optical fibres
where it is convenient to separate the photon into different paths.
Polarisation (of the same photon) was used as the target qubit. In Fig. 11
1546 nm pulses are sent through a combination of demultiplexers (DMUX) and
4-core fibre beamsplitter (4CF-BS) to realise the Hadamard operation.

Each single-mode output in this operation becomes an input to the quantum
4-switch, where each one corresponds to a different permutation of the
polarisation operations ${U_{i}}$. Each single-mode fibre input of the quantum
4-switch connects to a different 4-core fibre unit. The different permutations
of the polarisation operations $U_{i}$ were coherently applied and realized
using controllable liquid crystal retarders (LCR). After the LCR light is
coupled back into another 4-core fibre on the OUT-side, which are in turn
connected via a demultiplexer to single-mode fibres that connect to the next
4-core fibre back on the quantum 4-switch’s IN-side. After the quantum
4-switch, a second Hadamard operations is applied to the control using another
DMUX-4CF-BS-DMUX train. The output of this is directly connected to single-
photon detectors, where the detection of a photon in $y$-th detector gives the
value of $y$ in the Hadamard promise problem.

For two sets of unitary operationsTaddei  _et al._ (2020), one consisting of
$\\{\mathbb{1},Z,X\\}$ and one consisting of
$\\{\mathbb{1},Z,X,(Z{+}X)/\sqrt{2}\\}$ the average probability of success
were $0.948{\pm}0.005$ and $0.959{\pm}0.008$ respectively. This success rate
was achieved using only 4 queries, compared to 9 queries for the best known
fixed-order quantum circuit that can solve the Hadamard promise problem.

##  VII Summary and Concluding Remarks

We have presented the experiments in the nascent field of quantum causality.
To facilitate the logical flow of this review, the experiments were not
presented in chronological order. The first experiment was on the
computational advantage in distinguishing commuting and anti-commuting gates
Procopio  _et al._ (2015), followed by works on detecting indefinite causal
order Rubino  _et al._ (2017a, b); Goswami  _et al._ (2018), works that show
communication complexity and communication capacity advantages Guo  _et al._
(2020); Goswami  _et al._ (2020); Wei  _et al._ (2019), and most recently the
quantum 4-switch Taddei  _et al._ (2020). The field is enlivened by both
foundational and practical aspects. Indefinite causal order will most likely
likely be a feature of any theory that successfully combines general
relativity and quantum mechanics and we enthusiastically await experiments
that will have foundational importance in this area. In the interim, we have
provided a summary of the equally important first experiments in quantum
causality which are mostly pragmatic in nature.

As shown by the experiments of Rubino  _et al._ (2017a) and Goswami  _et al._
(2018), a causal witness enables us to detect indefinite causal order in a
process like the quantum switch. More strongly, Rubino  _et al._ (2017b) have
shown that indefinite causal order can be witnessed in a theory-independent
manner using a Bell’s theorem formulated for temporal order. The quantum
switch has been used to enable communication through depolarising
channels—which normally scrambles any information whether classical or
quantum—as demonstrated in the experiments of Goswami  _et al._ (2020) and Guo
_et al._ (2020). Computational advantage has also been demonstrated for tasks
that can be considered as variants of channel discrimination. Procopio  _et
al._ (2015) have shown that indefinite causal order in a quantum switch can be
used to distinguish beteween commuting and anti-commuting gates with only one
use of each gate. A reduction in communication complexity was shown by Wei
_et al._ (2019) by casting the exchange evaluation game as a task of
distinguishing between commuting and antic-commuting operations. Taddei  _et
al._ (2020) demonstrated that using a quantum switch to solve the Hadamard
promise problem leads to a quadratically smaller number of queries compared
with the best definite-ordered algorithm. These practical demonstrations
reinforce the idea that indefinite causal order is a resource which can be
exploited for various communication and computation tasks.

Although all the experiments we have reviewed are photonic in nature, there is
no fundamental reason that indefinite causal order cannot be achieved in other
platforms. Photonic implementations of the quantum switch abound because it is
convenient, a photon is an indivisible system that possesses several degrees
of freedom which can be easily controlled and measured. Any two degrees of
freedom can serve as control and target systems for the quantum switch.
Current implementations of the quantum switch can be grouped broadly in terms
of the degree of freedom used as control: path or polarisation. Using path as
the control means that the various input and output locations within the same
optical element—e.g. where the photons impinge on a waveplate as in Fig.
9.a—are not co-located. These implementations assume that regardless of the
path that a photon takes, the same set of operations act on the target system.
We emphasise that this is a reasonable assumption. We mention this only to
clarify any confusion that might arise when one compares the theoretical
proposal of a quantum switch—where the maps themselves have one input and
output port—with an experimental implementation. If one uses polarisation as a
control for the order, it is possible for the operations to be spatially co-
located regardless of the order, hence operations are not spatially
distinguishable. However, this leaves other degrees of freedom to be used as
the target system, e.g. transverse spatial mode or time-bins, which are
practically more challenging to manipulate compared to polarisation.

Most of the experiments so far (with the exception of Taddei  _et al._ (2020))
used a qubit for the control. This is limited to indefinite causal order
between only two operations. Whereas this has not stopped experiments and
theorists alike from showing the advantages to be had from having indefinite
causal order, it is of interest whether higher-dimensional extensions can lead
to better advantages, and whether these can also be shown experimentally.
Having qudits both for control—to enable more permutations of the order—and
target—to increase information capacity—can lead to fruitful extensions
especially towards scalability. Multi-core fibre optic technology has already
enabled the first quantum 4-switch, we look forward to further experiments
that involve qudits using maturing technologies that manipulate all the
photonic degrees of freedom Mounaix  _et al._ (2019).

As a final remark, we point out that although our experiments clearly
reproduce the output of a quantum switch, there remains the subtlety that all
our implementations are a function of the channels in the quantum switch and
the vacuum. For a proper accounting in a resource theory of indefinite causal
order, we specify that if the input to a channel is the vacuum, the channel
acts as an identity channelKristjánsson  _et al._ (2019). Although this does
not stop experiments from showing the various communication or computational
advantages that have been or will be discovered, this subtlety will certainly
play a role in evaluating various resource theories of coherent control.

###### Acknowledgements.

We thank the editors for the invitation to write this review. We thank Dr.
Fabio Costa, Dr. Cyril Branciard, Dr. Márcio Taddei, Prof. Giulio Chiribella,
and anonymous referees for useful discussions. This work has been supported
by: the Australian Research Council (ARC) by Centre of Excellence for
Engineered Quantum Systems (EQUS, CE170100009), the RTP scholarship from the
University of Queensland for KG, and Westpac Research Fellowship and L’Oreal-
UNESCO FWIS grant for JR. We thank Dr Markus Rambach and Michael Kewming for
reading early drafts of this review. We acknowledge the traditional owners of
the land on which the University of Queensland is situated, the Turrbal and
Jagera people.

## Data availability statement

The data that supports the findings of the articles reviewed in this review
paper are available within the articles and their supplementary material.

## References

  * Milburn and Shrapnel (2018) G. Milburn and S. Shrapnel, “Classical and quantum causal interventions,” Entropy 20, 687 (2018).
  * Hardy (2007) L. Hardy, “Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure,” J. Phys. A: Math. Gen. 40, 3081–3099 (2007).
  * Bell (1964) J. S. Bell, “On the Einstein-Poldolsky-Rosen paradox,” Physics 1, 195–200 (1964).
  * Brukner (2014) Č. Brukner, “Quantum causality,” Nat. Phys. 10, 259 (2014).
  * Ebler, Salek, and Chiribella (2018) D. Ebler, S. Salek, and G. Chiribella, “Enhanced communication with the assistance of indefinite causal order,” Phys. Rev. Lett. 120, 120502 (2018).
  * Chiribella  _et al._ (2018) G. Chiribella, M. Banik, S. S. Bhattacharya, T. Guha, M. Alimuddin, A. Roy, S. Saha, S. Agrawal, and G. Kar, “Indefinite causal order enables perfect quantum communication with zero capacity channel,” (2018), arXiv:1810.10457 [quant-ph] .
  * Salek, Ebler, and Chiribella (2018) S. Salek, D. Ebler, and G. Chiribella, “Quantum communication in a superposition of causal orders,” (2018), arXiv:1809.06655 [quant-ph] .
  * Chiribella  _et al._ (2013) G. Chiribella, G. M. D’Ariano, P. Perinotti, and B. Valiron, “Quantum computations without definite causal structure,” Phys. Rev. A 88, 022318 (2013).
  * Araújo, Costa, and Brukner (2014) M. Araújo, F. Costa, and Č. Brukner, “Computational Advantage from Quantum-Controlled Ordering of Gates,” Phys. Rev. Lett. 113, 250402 (2014).
  * Araújo, Guérin, and Baumeler (2017) M. Araújo, P. A. Guérin, and Ä. Baumeler, “Quantum computation with indefinite causal structures,” Phys. Rev. A 96 (2017).
  * Guérin  _et al._ (2016) P. A. Guérin, A. Feix, M. Araújo, and Č. Brukner, “Exponential communication complexity advantage from quantum superposition of the direction of communication,” Phys. Rev. Lett. 117, 100502 (2016).
  * Araújo  _et al._ (2015) M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, “Witnessing causal nonseparability,” New J. Phys. 17, 102001 (2015).
  * Goswami  _et al._ (2020) K. Goswami, Y. Cao, G. A. Paz-Silva, J. Romero, and A. G. White, “Increasing communication capacity via superposition of order,” Phys. Rev. Research 2, 033292 (2020).
  * Guo  _et al._ (2020) Y. Guo, X.-M. Hu, Z.-B. Hou, H. Cao, J.-M. Cui, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, and G. Chiribella, “Experimental transmission of quantum information using a superposition of causal orders,” Phys. Rev. Lett. 124 (2020).
  * Lamport (1978) L. Lamport, “Time, clocks, and the ordering of events in a distributed system,” Commun. ACM 21, 558–565 (1978).
  * Oreshkov, Costa, and Brukner (2012) O. Oreshkov, F. Costa, and Č. Brukner, “Quantum correlations with no causal order,” Nat. Commun. 3, 1092 (2012).
  * Choi (1975) M.-D. Choi, “Positive semidefinite biquadratic forms,” Linear Algebra Appl. 12, 95–100 (1975).
  * Jamiołkowski (1972) A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,” Rep. Math. Phys 3, 275–278 (1972).
  * Gutoski and Watrous (2006) G. Gutoski and J. Watrous, “Toward a general theory of quantum games,”  _Proceedings of 39th ACM STOC_ , , 565–574 (2006), quant-ph/0611234 .
  * Chiribella, D’Ariano, and Perinotti (2009) G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Theoretical framework for quantum networks,” Phys. Rev. A 80, 022339 (2009).
  * Shrapnel, Costa, and Milburn (2018) S. Shrapnel, F. Costa, and G. Milburn, “Updating the born rule,” New J. Phys. 20, 053010 (2018).
  * Chiribella (2012) G. Chiribella, “Perfect discrimination of no-signalling channels via quantum superposition of causal structures,” Phys. Rev. A 86, 040301 (2012).
  * Colnaghi  _et al._ (2012) T. Colnaghi, G. M. D’Ariano, S. Facchini, and P. Perinotti, “Quantum computation with programmable connections between gates,” Phys. Lett. A 376, 2940–2943 (2012).
  * Kraus  _et al._ (1983) K. Kraus, A. Böhm, J. D. Dollard, and W. H. Wootters,  _States, Effects, and Operations Fundamental Notions of Quantum Theory_, Vol. 190 (1983).
  * Gühne and Tóth (2009) O. Gühne and G. Tóth, “Entanglement detection,” Phys. Rep. 474, 1–75 (2009), arXiv:0811.2803 [quant-ph] .
  * Rockafellar (1970) R. T. Rockafellar,  _Convex Analysis_ (Princeton University Press, 1970).
  * Rubino  _et al._ (2017a) G. Rubino, L. A. Rozema, A. Feix, M. Araújo, J. M. Zeuner, L. M. Procopio, Č. Brukner, and P. Walther, “Experimental verification of an indefinite causal order,” Science Advances 3, e1602589 (2017a).
  * Goswami  _et al._ (2018) K. Goswami, C. Giarmatzi, M. Kewming, F. Costa, C. Branciard, J. Romero, and A. White, “Indefinite causal order in a quantum switch,” Phys. Rev. Lett. 121 (2018).
  * Zych  _et al._ (2019) M. Zych, F. Costa, I. Pikovski, and Č. Brukner, “Bell’s theorem for temporal order,” Nat. Commun. 10 (2019).
  * Rubino  _et al._ (2017b) G. Rubino, L. A. Rozema, F. Massa, M. Araújo, M. Zych, Č. Brukner, and P. Walther, “Experimental entanglement of temporal orders,” (2017b), arXiv:1712.06884 [quant-ph] .
  * Wilde (2013) M. M. Wilde, “Quantum information theory,” Cambridge University Press (2013).
  * Yin  _et al._ (2020) J. Yin, Y.-H. Li, S.-K. Liao, M. Yang, Y. Cao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, S.-L. Li, _et al._ , “Entanglement-based secure quantum cryptography over 1,120 kilometres,” Nature , 1–5 (2020).
  * Nielsen and Chuang (2000) M. Nielsen and I. Chuang, “Quantum computation and quantum information,” Cambridge University Press (2000).
  * Abbott  _et al._ (2018) A. A. Abbott, J. Wechs, D. Horsman, M. Mhalla, and C. Branciard, “Communication through coherent control of quantum channels,” (2018), arXiv:1810.09826 [quant-ph] .
  * Guérin, Rubino, and Brukner (2019) P. A. Guérin, G. Rubino, and Č. Brukner, “Communication through quantum-controlled noise,” Phys. Rev. A 99 (2019).
  * Kristjánsson  _et al._ (2019) H. Kristjánsson, G. Chiribella, S. Salek, D. Ebler, and M. Wilson, “Resource theories of communication with quantum superpositions of processes,” (2019), arXiv:1910.08197 [quant-ph] .
  * Oreshkov (2019) O. Oreshkov, “Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics,” Quantum 3, 206 (2019).
  * Rubino  _et al._ (2020) G. Rubino, L. A. Rozema, D. Ebler, H. Kristjánsson, S. Salek, P. A. Guérin, A. A. Abbott, C. Branciard, Č. Brukner, G. Chiribella, and P. Walther, “Experimental quantum communication enhancement by superposing trajectories,” (2020), arXiv:2007.05005 [quant-ph] .
  * Procopio  _et al._ (2015) L. M. Procopio, A. Moqanaki, M. Araújo, F. Costa, I. Alonso Calafell, E. G. Dowd, D. R. Hamel, L. A. Rozema, Č. Brukner, and P. Walther, “Experimental superposition of orders of quantum gates,” Nat. Commun. 6 (2015).
  * Wei  _et al._ (2019) K. Wei, N. Tischler, S.-R. Zhao, Y.-H. Li, J. M. Arrazola, Y. Liu, W. Zhang, H. Li, L. You, Z. Wang, and et al., “Experimental quantum switching for exponentially superior quantum communication complexity,” Phys. Rev. Lett. 122 (2019).
  * Taddei  _et al._ (2020) M. M. Taddei, J. Cariñe, D. Martínez, T. García, N. Guerrero, A. A. Abbott, M. Araújo, C. Branciard, E. S. Gómez, S. P. Walborn, L. Aolita, and G. Lima, “Experimental computational advantage from superposition of multiple temporal orders of quantum gates,” (2020), arXiv:2002.07817 [quant-ph] .
  * Hedayat and Wallis (1978) A. Hedayat and W. D. Wallis, “Hadamard matrices and their applications,” Ann. Statist. 6, 1184–1238 (1978).
  * Mounaix  _et al._ (2019) M. Mounaix, N. K. Fontaine, D. T. Neilson, R. Ryf, H. Chen, and J. Carpenter, “Time reversal of optical waves,” arXiv:1909.07003 (2019).