# Higher-order Process Matrix Tomography of a passively-stable Quantum SWITCH

Michael Antesberger1,† Marco Túlio Quintino2 Philip Walther1,4 Lee A. Rozema1
1University of Vienna, Faculty of Physics, Vienna Center for Quantum Science
and Technology (VCQ) & Research Network Quantum Aspects of Space Time (TURIS)
2 Sorbonne Université, CNRS, LIP6, F-75005 Paris, France
3 Institute for Quantum Optics and Quantum Information (IQOQI), Austrian
Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
4 Christian Doppler Laboratory for Photonic Quantum Computer, Faculty of
Physics, University of Vienna, 1090 Vienna, Austria
†Correspondence to<EMAIL_ADDRESS>

(August 28, 2024)

###### Abstract

The field of indefinite causal order (ICO) has seen a recent surge in
interest. Much of this research has focused on the quantum SWITCH, wherein
multiple parties act in a superposition of different orders in a manner
transcending the quantum circuit model. This results in a new resource for
quantum protocols, and is exciting for its relation to issues in foundational
physics. The quantum SWITCH is also an example of a higher-order quantum
operation, in that it not only transforms quantum states, but also other
quantum operations. To date, no higher-order quantum operation has been
completely experimentally characterized. Indeed, past work on the quantum
SWITCH has confirmed its ICO by measuring causal witnesses or demonstrating
resource advantages, but the complete process matrix has only been described
theoretically. Here, we perform higher-order quantum process tomography.
However, doing so requires exponentially many measurements with a scaling
worse than standard process tomography. We overcome this challenge by creating
a new _passively-stable_ fiber-based quantum SWITCH using active optical
elements to deterministically generate and manipulate time-bin encoded qubits.
Moreover, our new architecture for the quantum SWITCH can be readily scaled to
multiple parties. By reconstructing the process matrix, we estimate its
fidelity and tailor different causal witnesses directly for our experiment. To
achieve this, we measure a set of tomographically complete settings, that also
spans the input operation space. Our tomography protocol allows for the
characterization and debugging of higher-order quantum operations with and
without an ICO, while our experimental time-bin techniques could enable the
creation of a new realm of higher-order quantum operations with an ICO.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

The formalism of higher-order quantum operations (HOQO) provides a framework
to view quantum operations as objects that can be subjected to transformations
[1, 2, 3, 4, 5, 6]. This framework is particularly useful for analysing
causality in quantum mechanics. Since it was first realized that quantum
mechanics allows for processes with an indefinite causal order (ICO) [7, 8,
6], the field of quantum causality has seen an increasing level of interest
[9]. These processes are interesting for a variety of foundational topics [10,
11, 12, 13, 14, 15], and also because it has been recognized that they can
lead to ICO-based enhancements that go even beyond “normal quantum
technology.” Examples of these advantages include applications in quantum
computing [16, 17, 18, 19], quantum communication [20, 21, 22, 23, 24, 25, 26,
27, 28], channel discrimination[29, 30], metrology [31], reversing quantum
dynamics [32, 33], and even thermodynamics [34]. Experimentally, work has
focused on either implementing various protocols [35, 36, 37, 38, 39, 40, 41,
42] or verifying the ICO of a given experimental implementation [43, 44, 45,
46].

In spite of this large body of work, there has not yet been a complete
experimental characterization of process with an ICO. Instead, previous work
on ICO has mainly focused on designing and measuring witnesses to essentially
provide a yes or no answer to the question “does this process have an
indefinite causal order?” On the one hand, this is because no concrete
protocol for a complete characterization has yet been presented. On the other
hand, this is because the number of experimental settings required for a
complete characterization has been prohibitive in past experiments. Here, we
overcome both of these hurdles, first presenting a protocol to perform
“higher-order process matrix tomography,” and then implementing a new
experimental method to realize the quantum SWITCH based on time qubits, based
on the proposal of [47]. Our new passively-stable implementation is based on
active optical elements, and it allows us to acquire sufficient data
(estimating almost 10,000 distinct probabilities) to fully reconstruct a
process matrix demonstrating an ICO for the first time.

In the two-party quantum SWITCH, Alice and Bob each act on a target system
(typically taken to be a qubit) in their local laboratories. This target qubit
is sent first to one party and then to the other. The order in which the
target qubit is shared between the two is coherently controlled via a second
_control_ qubit. If the control qubit is prepared in a superposition state,
then the two parties act on the target qubit in a superposition of orders
(Fig. 1a-c). The quantum SWITCH and the so-called quantum time flip [48, 49,
50], are the only processes which do not respect our standard notions of
causality that have been experimentally implemented to date. The quantum
SWITCH is an example of a HOQO, in the sense that its inputs are not only the
control and target qubits, but also Alice and Bob’s operations.

All experimental realizations of the quantum SWITCH have been accomplished by
encoding both the control and the target systems in a single photon.
Typically, the control system is encoded in a path degree-of-freedom, which
then determines the order that the photon is routed between the two parties.
In practice, this means that a photon is placed in a superposition of the two
paths using a beamsplitter, and these paths are then looped between two
parties in a manner mimicking the paths of Fig. 1c. The parties then act on a
different degree of freedom, such as polarization [35], time bins [37], or
orbital angular momentum [44]. The result of all of these approaches is
essentially a Mach-Zehnder interferometer, which must be phase
stabilized.111Polarization has also been used [44, 36] as a control system,
but also in this case the polarization is used to route the photon into two
paths in superposition, which also requires a stable phase. Stabilizing the
phase for long enough to acquire the required data for full higher-order
quantum process tomography presents a daunting experimental challenge.

We overcome this challenge by implementing a new passively-stable quantum
SWITCH. In our experiment, the control system is encoded in a time-bin qubit,
the target qubit is encoded in the polarization of the same photon, and active
optical switches are used to route the photon between the two parties in
superposition of both orders, as in the theoretical proposal of [47]. While
two other experiments have achieved an intrinsically stable phase, using a
Sagnac-like approach [37, 51], it is not clear how to scale these methods to
multiple-parties. Our approach, however, has straight-forward generalization
to multiple parties [47], making it a promising new experimental method to
create an ICO.

Recently, there has been some discussion in the community whether such
photonic implementations of the quantum switch are simulations of an ICO [52,
53], with some concluding that they may only have an ICO in a “weak sense”
[54], while others conclude that the experiments do have an ICO [55, 56], or
that they at least have a quantifiable resource advantage [57]. Here we do not
address this debate, but we make use of the mathematical formalism of
processes matrices and HOQO to describe our physical experiment.

Figure 1: The quantum SWITCH. Panel a) (b)) shows the causally ordered process
with the control in state $\ket{0}_{C}$ ($\ket{1}_{C}$), where Alice (Bob)
acts before Bob (Alice) on the target system. c) A superposition of orders
with the control qubit in the state
$\frac{1}{\sqrt{2}}\ket{0}_{C}+\ket{1}_{C}$. d) The principle of the process
tomography on the quantum SWITCH. Alice and Bob perform projective
measurements in three different basis and then prepare four different linearly
independent states in their output. The same input states are prepared at the
past-target. The past-control is fixed to the superposition state for
generating the indefinite order. Finally, the future-control is measured in
different basis, while the future-target is traced out (the two measurements
shown are those that are implemented experimentally; to span the space, a
third measurement is needed).

One method to certify ICO is made via the violation of a so-called causal
inequality [6]. This is a device-independent technique, similar to the use of
a Bell violation to verify entanglement [58]. Unfortunately, it is not yet
known if one can implement a quantum process that deterministically violates a
causal inequality; moreover, it has been shown that the quantum SWITCH cannot
violate causal inequalities [17]. Instead, in the first implementation of the
quantum SWITCH [35], the ICO was indirectly proven by performing a game, where
a player has to decide if two unitary gates either commute or anti-commute
(see App. B.1). By winning that game more than one could with a definite
causal order, it was concluded the experiment did not have a definite causal
order. This method can, in fact, be reframed, in terms of a causal witness
[13]. A causal witness is a measurement which can be used to verify if a
process is causally non-separable (i.e. if it has an ICO), and it has been
experimentally implemented for the quantum SWITCH [45, 44]. Unlike a causal
inequality, a causal witness is not device independent, requiring the
assumption that the experimenter knows the correct quantum description of the
experiment. Recently, progress has been made by relaxing the complete device
independent approach, allowing the certification of causal non-separability
under semi-device independent assumptions [59, 46, 60], Bell locality-like
assumptions [15, 45], and additional device independent no-signalling
assumptions [61, 62].

In this work, we implement full experimental higher-order process tomography
of the quantum SWITCH. For this goal, we generalise the ideas from quantum
state and quantum process tomography [2, 63, 4, 64, 65, 66, 67, 68, 69] to
tackle tomography of arbitrary higher-order processes, including those without
a definite causal order. In particular, we show that it is possible to
construct tomographically complete measurement settings on arbitrary quantum
processes by using a tomographically complete set of input states, spanning
the input state space; a tomographically complete set of measurement-
repreparation channels, spanning the input operation spaces; and a
tomographically complete set of quantum measurements spanning the output state
space (see Fig. 1d). We then employ these ideas to experimentally perform
higher-order quantum process tomography on the quantum SWITCH.

The rest of the paper is organized as follows. In Section II we introduce the
theory of quantum process matrices, using the quantum SWITCH as a paradigmatic
example, and we present our causal tomography protocol. In Section III, we
discuss our new passively-stable architecture for the quantum SWITCH. Section
IV presents our experimental results, and we conclude in Section V.

##  II Theory

###  II.1 Process matrices and the quantum switch

The expression quantum process is a general term used to refer to the dynamics
of quantum systems, and its precise meaning may depend on the context. For
instance, when analysing transformations between quantum states, quantum
process refers to a quantum channel, which may be unitary (associated with
closed quantum systems and mathematically described by unitary operators) or
non-unitary (associated with open quantum systems and mathematically described
by Completely Positive Trace Preserving (CPTP) linear maps). In this scenario
of transformations between states, one can experimentally determine the
dynamics by means of what is known as quantum process tomography [64, 65, 66].
To do so, a complete set of known quantum states is fed into an unknown
quantum process $\mathcal{E}$ and a complete set of measurements is performed
on the output of the underlying process for each input state [69]. When
performing standard process tomography, one reconstructs, for example, the chi
matrix $\chi$, which takes quantum sates as inputs and returns quantum states
as outputs [70]. The chi matrix is often called a process matrix; however, we
stress that this chi or process matrix is different from the process matrices
discussed in the field of HOQO and ICO, the case which we address here.

In this work, we analyse transformations between quantum channels, hence we
use the word process to describe the dynamics between quantum operations. An
operation that transforms a quantum operation is sometimes referred to as a
higher-order quantum operation. Such operations have found applications in
several branches of quantum information processing [3, 2, 4, 71, 72, 32, 73].
The formalism of higher-order transformations, is also particularly useful for
describing quantum processes which may not respect a definite causal order,
such as the notorious quantum SWITCH [8, 29], the main object analysed in this
work. In its most general form, the quantum SWITCH is a process that,
transforms a pair of unitary operators $(U_{A},U_{B})$ into another operator
which is a coherent superposition of the composition $U_{A}U_{B}$ and
$U_{B}U_{A}$. In mathematical terms, the quantum SWITCH is the transformation

$\displaystyle(U_{A},U_{B})\mapsto\outerproduct{0}{0}\otimes
U_{A}U_{B}+\outerproduct{1}{1}\otimes U_{B}U_{A},$ (1)

where the first system of the RHS of Eq. 1 is referred to as the control
system, since the order in which $U_{A}$ and $U_{B}$ will be performed may be
controlled by setting the control qubit state. The second system is referred
to the target system, since it is the system on which the unitary operators
act.

###  II.2 Choi-Jamiołkowski isomorphism and quantum operations

Higher-order transformations such as the quantum SWITCH may be conveniently
described by means of a process matrix [6], a formalism which is heavily based
on the Choi-Jamiołkowski (CJ) isomorphism [74, 75], a method to represent
linear maps as linear operators, and linear operators as vectors. Let
$\mathcal{H}_{\text{in}}$ and $\mathcal{H}_{\text{out}}$ be finite linear
(Hilbert) spaces associated with the input and output. Let
$U:\mathcal{H}_{\text{in}}\to\mathcal{H}_{\text{out}}$ be a linear operator,
its process vector
$|U\rangle\rangle\in\mathcal{H}_{\text{in}}\otimes\mathcal{H}_{\text{out}}$ is
the defined as222Upper-indices on vectors and operators will be used to
indicate where these objects act. For instance, in
$|U\rangle\rangle^{\mathcal{H}_{\text{in}}\mathcal{H}_{\text{out}}}$, the
upper indices indicate that
$|U\rangle\rangle\in\mathcal{H}_{\text{in}}\otimes\mathcal{H}_{\text{out}}$.
Whenever clear from context, we might drop the upper indices to avoid an
overly heavy notation.

$|U\rangle\rangle^{\mathcal{H}_{\text{in}}\mathcal{H}_{\text{out}}}:=\sum_{i}\ket{i}\otimes
U\ket{i},$ (2)

where $\\{\ket{i}\\}_{i}$ is the computational basis. Notice that, the Choi
vector of the identity operator is given by

$\displaystyle|\mathds{1}\rangle\rangle=\sum_{i}\ket{i}\otimes\ket{i},$ (3)

which is equivalent to a maximally entangled state up to normalisation.

Let $\mathcal{L}(\mathcal{H})$ be the set of all linear operators acting on
$\mathcal{H}$. Let
$\mathcal{C}:\mathcal{L}(\mathcal{H}_{\text{in}})\to\mathcal{L}(\mathcal{H}_{\text{out}})$
be a linear map, its Choi operator
$C\in\mathcal{L}(\mathcal{H}_{\text{in}}\otimes\mathcal{H}_{\text{out}})$ is
defined as

$C^{\mathcal{H}_{\text{in}}\mathcal{H}_{\text{out}}}:=\sum_{ij}\outerproduct{i}{j}\otimes\mathcal{C}(\outerproduct{i}{j}).$
(4)

Then, the action of any linear map $\mathcal{C}$ on a state $\rho$ can be
written in terms of the Choi operator $C$ as

$\displaystyle\mathcal{C}(\rho)=\Tr_{\text{in}}\left({\rho^{T}}^{\mathcal{H}_{\text{in}}}\otimes\mathds{1}^{\mathcal{H}_{\text{out}}}\,C^{\mathcal{H}_{\text{in}}\mathcal{H}_{\text{out}}}\right),$
(5)

where $\rho^{T}$ is the transpose of $\rho$ in the computational basis and
$\rho$ is an arbitrary density operator acting on $\mathcal{H}_{\text{in}}$.

The Choi-Jamiołkowski isomorphism is very useful to represent quantum
operations, due to the fact that a linear map
$\mathcal{C}:\mathcal{L}(\mathcal{H}_{\text{in}})\to\mathcal{L}(\mathcal{H}_{\text{out}})$
is completely positive if and only if its Choi operator
$C\in\mathcal{L}(\mathcal{H}_{\text{in}}\otimes\mathcal{H}_{\text{out}})$
respects $C\geq 0$, and the map $\mathcal{C}$ is trace-preserving if and only
if $\Tr_{\text{out}}(C)=\mathds{1}^{\mathcal{H}_{\text{in}}}$. Since quantum
channels are completely positive trace-preserving maps, all quantum channels
have a simple and direct characterisation in terms of their Choi operators.
Before finishing this subsection we also remark that if $\mathcal{C}$ is a
unitary channel, that is $\mathcal{C}(\rho)=U\rho U^{\dagger}$ for some
unitary operator $U$, direct calculation shows that its Choi operator may be
written as $C=|U\rangle\rangle\langle\langle U|$, where $|U\rangle\rangle$ is
defined in Eq. 2.

A quantum instrument is a quantum operation which has a classical and a
quantum output, and it formalises the concept of a quantum measurement which
has a post-measurement quantum state. Mathematically, a set of linear maps
$\\{\mathcal{C}_{i}\\}_{i}$,
$\mathcal{C}_{i}:\mathcal{L}(\mathcal{H}_{\text{in}})\to\mathcal{L}(\mathcal{H}_{\text{out}})$
is a quantum instrument if all $\mathcal{C}_{i}$ are completely positive and
$\mathcal{C}:=\leavevmode\nobreak\ \sum_{i}\mathcal{C}_{i}$ is trace
preserving. In the Choi operator picture, this is equivalent to having
$C_{i}\geq 0$ and
$\Tr_{\text{out}}\left(\sum_{i}C_{i}\right)=\mathds{1}^{\mathcal{H}_{\text{in}}}$.
A simple and useful class of quantum instrument is the class of measure-and-
reprepare instruments. In its most basic form, a measure and reprepare
instrument simply performs a measurement described by the operators333A set of
operators $\\{M_{i}\\}$ represents a quantum measurement (that is, it is a
POVM) if $M_{i}\geq 0$ and $\sum_{i}M_{i}=\mathds{1}$. $\\{M_{i}\\}$, and
reprepares some fixed state $\sigma$. Its linear map is described by
$\mathcal{R}_{i}(\rho):=\Tr(\rho\,M_{i})\sigma$, and its Choi operators are
given by
$R_{i}\in\mathcal{L}\left(\mathcal{H}_{\text{in}}\otimes\mathcal{H}_{\text{out}}\right)$,
with $R_{i}=M_{i}^{T}\otimes\sigma$.

###  II.3 The quantum switch as a process matrix

We are now in position to present process matrices which describe
transformations between the quantum channels of different parties. We start by
presenting the process vector describing Fig 1a, which is simply a process
where a system flows freely from a common past target space
$\mathcal{H}_{P_{\text{t}}}$ to Alice’s input space
$\mathcal{H}_{A_{\text{in}}}$. Alice may perform an arbitrary operation as the
state goes from $\mathcal{H}_{A_{\text{in}}}$ to
$\mathcal{H}_{A_{\text{out}}}$. Later, the state goes freely from Alice’s
output space $\mathcal{H}_{A_{\text{out}}}$ to Bob’s input space
$\mathcal{H}_{B_{\text{in}}}$. Bob may then perform an arbitrary operation as
the state goes from $\mathcal{H}_{B_{\text{in}}}$ to
$\mathcal{H}_{B_{\text{out}}}$. Finally, the state goes freely from Bob’s
output space $\mathcal{H}_{B_{\text{out}}}$ to a common future target space
$\mathcal{H}_{F_{\text{t}}}$. The process vector of this quantum process is

$\ket{A\to
B}:=|\mathds{1}\rangle\rangle^{P_{\text{t}},{A}_{\text{in}}}\otimes|\mathds{1}\rangle\rangle^{{A}_{\text{out}},{B}_{\text{in}}}\otimes|\mathds{1}\rangle\rangle^{{B}_{\text{out}},F_{\text{t}}}\,,$
(6)

and its process matrix is given by

$W_{A\to B}:=\outerproduct{A\to B}{A\to B},$ (7)

where $A\to B$ indicates that Alice acts before Bob. (Note that we have not
included Alice or Bob’s operations in this description; this will be
introduced in Sec. II.5.) Analogously, we may define the process where Bob
acts before Alice, which will lead to a process vector

$\ket{B\to
A}:=|\mathds{1}\rangle\rangle^{\mathcal{H}_{\text{P}},\mathcal{B}_{\text{in}}}\otimes|\mathds{1}\rangle\rangle^{\mathcal{B}_{\text{out}},\mathcal{A}_{\text{in}}}\otimes|\mathds{1}\rangle\rangle^{\mathcal{A}_{\text{out}},F_{\text{t}}}\,,$
(8)

and its process matrix is given by

$W_{B\to A}:=\outerproduct{B\to A}{B\to A}.$ (9)

The quantum SWITCH is a process which allows one to coherently alternate
between $\ket{A\to B}$ and $\ket{B\to A}$. For that, we allow the common past
and common future to have another system, denoted as a control system, which
will be able to coherently alternate between the ordered process. More
formally, the common past and common future space are now described by
$\mathcal{H}_{P}=\mathcal{H}_{P_{\text{c}}}\otimes\leavevmode\nobreak\
\mathcal{H}_{P_{\text{t}}}$ and
$\mathcal{H}_{F}=\mathcal{H}_{F_{\text{c}}}\otimes\mathcal{H}_{F_{\text{t}}}$
respectively, and the Choi vector of the quantum switch is given by

$\displaystyle\ket{w_{\text{switch}}}:=\ket{0}^{{P}_{c}}$
$\displaystyle\otimes\ket{A\to B}\otimes\ket{0}^{{F}_{c}}$
$\displaystyle+\ket{1}^{{P}_{c}}\otimes\ket{B\to A}\otimes\ket{1}^{{F}_{c}},$
(10)

which corresponds to the process matrix

$W_{\text{switch}}:=\outerproduct{w_{\text{switch}}}{w_{\text{switch}}}$ (11)

Almost all known applications of the quantum SWITCH, e.g., computational
advantages [17], channel discrimination [8], reducing communication complexity
[21], semi-device-independent [59] and device-independent certification of
indefinite causality [61, 76], do not require the general form of the quantum
SWITCH is presented in Eq. (11). Rather, in such applications, one starts with
the control qubit in the $\ket{+}:=\frac{\ket{0}+\ket{1}}{\sqrt{2}}$, so that
the process state corresponds to a coherent superposition of processes
described by:

$\displaystyle\ket{w_{\text{switch}}^{+}}:=\frac{\ket{A\to
B}\otimes\ket{0}^{{F}_{c}}+\ket{B\to A}\otimes\ket{1}^{{F}_{c}}}{\sqrt{2}}.$
(12)

Additionally, for all such applications, one does not make use of the future
target system, hence this qubit is often discarded. Mathematically, discarding
a system corresponds to the partial trace. Hence, we construct the simplified
version of the SWITCH as

$W_{\text{s}}^{+}:=\Tr_{F_{\text{t}}}\left(\outerproduct{w_{\text{switch}}^{+}}{w_{\text{switch}}^{+}}\right)$
(13)

In this work, we focus on the simplified quantum SWITCH, and as is usual in
the literature, we use “quantum SWITCH” to refer to the process described in
Eq. (13).

###  II.4 The general process matrix formalism

The process matrix formalism allows one to assign a matrix which perfectly
describes transformations between arbitrary quantum objects, in particular, to
transform quantum channels into quantum channels. The normalisation
constraints from quantum channels (or more general quantum objects) and a
generalised notion of completely positive inputs lead to constraints on valid
process matrices. In a nutshell, when focused on quantum channels, a matrix
$W$ is a process matrix if it is positive semidefinite and respect a set of
affine constraints arising from the channel normalisation conditions. These
affine constraints are described in several Refs. such as [6, 17] and may be
viewed as causal constraints (for instance, they prevent local loops or the
possibility of obtaining negative probabilities via Born’s rule).

###  II.5 Measuring a process matrix

One of the main applications of the process matrix formalism is to provide
mathematical methods to analyse the dynamics of a quantum process and to
predict the outcomes of measurements performed on a quantum process. Below, we
describe the scenario considered in this work:

Figure 2: Probing a quantum process $W$. Pictorial illustration on how to
probe a quantum process bipartite quantum process $W$ with a common past and
common future. Here $\rho_{w}$ are quantum states, $\\{A_{a|x}\\}$ and
$\\{B_{b|y}\\}$ are quantum instruments, and $\\{M_{c|z}\\}$ are quantum
measurements.

  1. 1.

$W\in\mathcal{L}\left(\mathcal{H}_{P}\otimes\mathcal{H}_{A_{\text{in}}}\otimes\mathcal{H}_{A_{\text{out}}}\otimes\mathcal{H}_{B_{\text{in}}}\otimes\mathcal{H}_{B_{\text{out}}}\otimes\mathcal{H}_{F}\right)$
is the process matrix which describes a bipartite scenario with a common past
(target) and common future (control).

  2. 2.

$\rho\in\mathcal{L}(\mathcal{H}_{P})$ is a quantum state on the common past
(target) space.

  3. 3.

$A_{a}\in\mathcal{L}(\mathcal{H}_{A_{\text{in}}}\otimes\mathcal{H}_{A_{\text{out}}})$
are the Choi operators of an instrument on Alice’s space.

  4. 4.

$B_{b}\in\mathcal{L}(\mathcal{H}_{B_{\text{in}}}\otimes\mathcal{H}_{B_{\text{out}}})$
are the Choi operators of an instrument on Bob’s space.

  5. 5.

$M_{c}\in\mathcal{L}(\mathcal{H}_{F})$ are the measurement operators on the
common future (control) space.

In the scenario described above, if $W$ is the process matrix, one inputs the
state $\rho$ into the common past, Alice performs the instrument
$\\{A_{a}\\}$, Bob performs the instrument $\\{B_{b}\\}$, the measurement
$\\{M_{c}\\}$ is performed in the future, the probability that Alice obtains
the outcome $a$, Bob obtains the outcome $b$, and the future obtains the
outcome $c$ is given by

$\displaystyle p(a,b,c)=\Tr\left(W\,\left(\rho^{P}\otimes
A_{a}^{A_{\text{in}}A_{\text{out}}}\otimes
B_{b}^{B_{\text{in}}B_{\text{out}}}\right)^{T}\otimes M_{c}^{F}\right).$ (14)

In practice, it is often convenient to have indices to label states,
instruments, and measurements. In this work, we will then use $\\{\rho_{w}\\}$
to denote a set of states acting in the common past, $\\{A_{a|x}\\}$ for a set
of instruments in Alice’s space ($a$ labels the classical outcome of the
instrument and $x$ the choice of instrument), $\\{B_{b|y}\\}$ for a set of
instruments in Bob’s space ($b$ labels the classical outcome of the instrument
and $y$ the choice of instrument), and $\\{M_{c|z}\\}$ for a set of
measurements in the future space ($c$ labels the classical outcome and $z$ the
choice of measurement). We can then define the _setting operators_ 444We use
the phrase “setting operator”, since each of these operators will correspond
to a single experimental setting. as

$\displaystyle S_{abc|xyzw}:=\left(\rho^{P}_{w}\otimes
A_{a|x}^{A_{\text{in}}A_{\text{out}}}\otimes
B_{b|y}^{B_{\text{in}}B_{\text{out}}}\right)^{T}\otimes M_{c|z}^{F}$ (15)

which leads us to the so-called “generalised Born’s rule”:

$\displaystyle p\left(abc|xyzw\right)=\Tr\left(W\,S_{abc|xyzw}\right).$ (16)

###  II.6 Process matrix tomography

The goal of quantum tomography is to completely characterise a quantum object
by performing known measurements on it. Before discussing process matrix
tomography, we revisit the standard case of quantum state tomography, where
one aims to characterise an unknown state by analysing the outcomes obtained
after performing known measurements on it. If
$M_{a|x}\in\mathcal{L}(\mathbb{C}_{d})$ are known measurement operators, one
can make use of the probabilities $p(a|x)=\Tr(\rho\,M_{a|x})$ to uniquely
reconstruct the unknown state $\rho$. When the set of operators
$\\{M_{a|x}\\}$ spans the linear space of $\mathcal{L}(\mathbb{C}_{d})$, the
operator $\rho$ may be obtained via $p(a|x)$ by standard linear inversion
methods.

For qubit states, a standard set of tomographically complete measurements is
formed by the three Pauli observables $X$, $Y$, and $Z$, which are associated
with the measurement operators via their eigenprojectors:
$\big{\\{}\outerproduct{+}{+},\outerproduct{-}{-}\big{\\}}$,
$\big{\\{}\outerproduct{y_{+}}{y_{+}},\outerproduct{y_{-}}{y_{-}}\big{\\}}$,
$\big{\\{}\outerproduct{0}{0},\outerproduct{1}{1}\big{\\}}$, respectively,
where $\ket{\pm}=\frac{\ket{0}\pm\ket{1}}{\sqrt{2}}$ and
$\ket{y_{\pm}}=\frac{\ket{0}\pm i\ket{1}}{\sqrt{2}}$. In particular, the
standard measurement operators from the set

$\displaystyle\mathcal{S}:=\\{\outerproduct{\psi_{i}}{\psi_{i}}\\}_{i=1}^{4},$
(17)

where

$\displaystyle\outerproduct{\psi_{1}}{\psi_{1}}:=\outerproduct{0}{0},\hskip
5.12149pt\outerproduct{\psi_{2}}{\psi_{2}}:=\outerproduct{1}{1},$ (18)
$\displaystyle\outerproduct{\psi_{3}}{\psi_{3}}:=\outerproduct{+}{+},\outerproduct{\psi_{4}}{\psi_{4}}:=\outerproduct{y_{+}}{y_{+}}.$
(19)

These measurements are linearly independent, forming a (non-orthonormal) basis
for $\mathcal{L}(\mathbb{C}_{2})$.

We now consider the task of performing tomography of a qubit channel. As
discussed in section II.2, every quantum channel
$\mathcal{C}:\mathcal{L}(\mathcal{H}_{\text{in}})\to\mathcal{L}(\mathcal{H}_{\text{out}})$,
can be represented by its Choi operator
$C\in\mathcal{L}(\mathcal{H}_{\text{in}}\otimes\mathcal{H}_{\text{out}})$. In
this case, tomography can be carried out by preparing a set of states
$\\{\rho_{w}\\}_{w}$, $\rho_{w}\in\mathcal{L}(\mathcal{H}_{\text{in}})$ and
performing a complete set of measurements on each state. For qubits, the
standard measurements are

$\displaystyle\mathcal{M}:=\\{M_{i|j}\\}_{i=1,j=1}^{i=2,j=3},$ (20)

where $M_{i|j}$ is a POVM element, the label $i$ stands for the outcomes, and
$j$ for the choice of measurements. Hence, $\mathcal{M}$ is a set with three
dichotomic measurements with POVM elements given by

$\displaystyle M_{1|1}:=\outerproduct{0}{0},\hskip
11.38109ptM_{2|1}:=\outerproduct{1}{1}$ (21) $\displaystyle
M_{1|2}:=\outerproduct{+}{+},\hskip 6.54413ptM_{2|2}:=\outerproduct{-}{-}$
(22) $\displaystyle
M_{1|3}:=\outerproduct{y_{+}}{y_{+}},M_{2|3}:=\outerproduct{y_{-}}{y_{-}}.$
(23)

Note that, due to the normalisation of probabilities, the measurements of some
measurement operators are unnecessary. However, in practice, the orthogonal
measurements $M_{2|j}$ are often measured to aid in the data normalization. We
will include them here, with a view to our experiment.

From these input states and measurements, one estimates the probabilities
$p(a|x,w)=\Tr\Big(\mathcal{C}(\rho_{w})\,M_{a|x}\Big{missing})\,$, which can
also be written as

$\displaystyle p(a|x,w)=$ $\displaystyle\Tr\Big(C\,\left(\rho_{w}^{T}\otimes
M_{a|x}\right)\Big{missing}),$ (24) $\displaystyle=$
$\displaystyle\Tr\left(C\,S_{a|xw}\right)$ (25)

in the Choi formalism, where $S_{a|xw}:=\rho_{w}^{T}\otimes M_{a|x}$, where
$S_{a|xw}$ is a setting operator for standard quantum process tomography. Now,
one way to perform complete tomography is by ensuring that the setting
operators $S_{a|xw}$ span the space
$\mathcal{L}(\mathcal{H}_{\text{in}}\otimes\mathcal{H}_{\text{out}})$. Also,
thanks to a property usually referred to as “local tomography” [77] if the set
of operators $\\{\rho_{w}^{T}\\}_{w}$ spans
$\mathcal{L}(\mathcal{H}_{\text{in}})$ and the set $\\{M_{a|x}\\}_{a,x}$ spans
$\mathcal{L}(\mathcal{H}_{\text{out}})$, then the set of setting operators
$\left\\{\rho_{w}^{T}\otimes M_{a|x}\right\\}_{w,a,x}$ spans
$\mathcal{L}(\mathcal{H}_{\text{in}}\otimes\leavevmode\nobreak\
\mathcal{H}_{\text{out}})$. In other words, full quantum channel tomography is
always possible if one measures a set of characterised setting operators
$\\{S_{a|xw}\\}_{a,x,w}$ that span the space
$\mathcal{L}(\mathcal{H}_{\text{in}}\otimes\leavevmode\nobreak\
\mathcal{H}_{\text{out}})$.

In principle, measuring a set of setting operators $\\{S_{a|xw}\\}_{a,x,w}$
which span
$\mathcal{L}(\mathcal{H}_{\text{in}}\otimes\mathcal{H}_{\text{out}})$, is
actually “overkill”. More specifically, due to the normalisation condition
$\Tr_{\text{out}}(C)=\mathds{1}_{\text{in}}$, respected by quantum channels,
there are linear operators in
$\mathcal{L}(\mathcal{H}_{\text{in}}\otimes\mathcal{H}_{\text{out}})$ which
cannot be written as linear combinations of quantum channels,e.g.,
$\outerproduct{0}{0}_{\text{in}}\otimes\mathds{1}_{\text{out}}$. One can then
consider a set of operators $\\{S_{a|xw}\\}_{a,x,w}$ which spans the set of
quantum channels, a subspace with dimension strictly smaller than the
dimension of
$\mathcal{L}(\mathcal{H}_{\text{in}}\otimes\mathcal{H}_{\text{out}})$. In
particular, the linear space
$\mathcal{L}(\mathcal{H}_{\text{in}}\otimes\mathcal{H}_{\text{out}})$ has
dimension of $d_{\text{in}}^{2}\,d_{\text{out}}^{2}$, and the linear span of
quantum channels in
$\mathcal{L}(\mathcal{H}_{\text{in}}\otimes\mathcal{H}_{\text{out}})$ has
dimension of $d_{\text{in}}^{2}(d_{\text{out}}^{2}-1)$. We emphasize, however,
this does not represent a problem; in fact, in practice, more using an over-
complete measurement set is known to minimise the experimental errors in
standard quantum tomography [78].

Figure 3: Active generation and measurement of time-bin qubits. a) In this
row, an incident single photon is deterministically placed in a superposition
of the “short” and “long” time bins, using an ultra-fast optical switch
(UFOS). In order to achieve passive phase stability, the measurements, shown
in rows b) and c), are actually implemented using the same device as in row
a). However, for clarity, we have mirrored the device horizontally. b) This
row indicates our measurements in the Z-basis. Here, the UFOS remains in the
“bar state”. After traversing the device, the photon remains in a
superposition of the two incident time bins (but spread over two paths). In
this situation, simply resolving the arrival time of the photon projects into
the Z-basis. c) A schematic of our deterministic measurement in the Y-basis.
Here the UFOS alternate between the “cross” and bar states so that the short
(long) time bin now takes the long (short) path. In this manner, the two time
bins interfere on the beamsplitter, so that finding the photon in the upper
(lower) path corresponds to projecting the time bin onto $\ket{y+}$
($\ket{y-}$).

Finally, we now consider tomography of process matrices
$W\in\leavevmode\nobreak\
\mathcal{L}\left(\mathcal{H}_{P}\otimes\mathcal{H}_{A_{\text{in}}}\otimes\mathcal{H}_{A_{\text{out}}}\otimes\mathcal{H}_{B_{\text{in}}}\otimes\mathcal{H}_{B_{\text{out}}}\otimes\mathcal{H}_{F}\right)$,
such as the quantum switch illustrated in Fig. 1d. As discussed before, one
way to perform tomography is to measure setting operators $S_{abc|xyzw}$ which
span the linear space
$\mathcal{L}\left(\mathcal{H}_{P}\otimes\mathcal{H}_{A_{\text{in}}}\otimes\mathcal{H}_{A_{\text{out}}}\otimes\mathcal{H}_{B_{\text{in}}}\otimes\mathcal{H}_{B_{\text{out}}}\otimes\mathcal{H}_{F}\right)$.
Also, thanks to local tomography, we may consider sets of states and
measurements which span the local space individually. We then consider the set
of states given by Eq. 17 and the set of measurements is given by Eq. 20. For
tomography of a higher-order process matrix, we then consider the set of
measure-and-reprepare instruments (to be used as inputs for Alice and Bob’s
channels) given by all combinations of the two sets above, that is

$\displaystyle\mathcal{R}:=\\{R_{i|(j,k)}\\}_{i=1,j=1,k=1}^{i=2,j=3,k=4},$
(26)

where

$\displaystyle
R_{i|(j,k)}:=M_{i|j}\otimes\outerproduct{\psi_{k}}{\psi_{k}}^{T}.$ (27)

The interpretation of Eq. (27) is the following, first, the measurement $j$
with POVM elements $\\{M_{i|j}\\}_{i}$ is performed, then, the state
$\ket{\psi_{k}}$ is prepared. Notice that, in our measure-and-reprepare
instruments, the prepared state $\ket{\psi_{k}}$ is independent of the
measurement choice $j$ and the obtained outcome $i$. One can then perform full
tomography with the setting operators

$\displaystyle S_{abc|xyzw}$
$\displaystyle:={\outerproduct{\psi_{w}}{\psi_{w}}^{T}}^{P_{\text{t}}}\otimes
A_{a|x}^{A_{I}A_{O}}\otimes B_{b|y}^{B_{I}B_{O}}\otimes
M_{c|z}^{F_{\text{c}}},$ (28)

where $\outerproduct{\psi_{w}}{\psi_{w}}$ are the $4$ different quantum states
in $\mathcal{S}$, $A_{a|x}$ and $B_{b|y}$ are each the $2\times 3\times 4=24$
instrument elements555Note that here we identify $x=(j,k)$, that is, $x$ can
assume $12=3\times 4$ different values. of set $\mathcal{R}$ defined in Eq.
(26), and $2\times 3=6$ measurement elements for the future control space are
the measurements $M_{c|z}$ of $\mathcal{M}$ (Eq. (20)). In total, we then
measure $4\times 24\times 24\times 6=13824$ different settings. We remark that
in this tomography approach, we do not make use of any constraints on the
process matrices. One could also to reduce the number of required settings by
imposing the assumption that valid process matrices necessarily belong to a
particular linear subspace, as discussed in subsection II.4.

In an ideal theoretical scenario, if we obtain the probabilities
$p\left(abc|xyzw\right)=\Tr\left(W\,S_{abc|xyzw}\right)$ for a tomographically
complete set of setting operators, standard linear inversion will uniquely
identify the process $W$. However, due to finite statistics, we never obtain
the exact probability $p\left(abc|xyzw\right)$, but an approximation from
measured frequencies. Also, due to measurement precision and other possible
sources of errors, we cannot expect to obtain an exact reconstruction of the
process matrix. Indeed, performing direct linear inversion often results in
unphysical quantum states or processes. Instead, we aim to estimate a physical
process matrix that agrees best with the experimental data.

In order to estimate our experimental process matrix $W_{\text{exp}}$, we
perform a fitting routine to find the process matrix that best describes our
measured data. We find that minimizing the least absolute residuals works
quite well. To do this, we numerically search for a process matrix
$W_{\text{exp}}$ that minimizes the following expression:

$r=\frac{1}{N_{\text{Settings}}}{\sum_{abcxyzw}\Big{|}p_{\text{exp}}(abc|xyzw)-\Tr(W_{\text{exp}}\,S_{abc|xyzw})}\Big{|},$
(29)

where $N_{\text{Settings}}$ is the number of setting operators, and the
minimisation is further subject to the constraint that $W_{\text{exp}}$ is a
valid process matrix. This minimisation can be performed by means of
semidefinite programming (SDP), and may be implemented with the help of
numerical libraries such as MOSEK. Our MATLAB code implementing this, is
available at [79]. The first term under the root are the experimentally
measured probabilities, while the second term corresponds to what is predicted
by quantum theory for the characterised settings $S_{abc|xyzw}$. Since
$W_{\text{exp}}$ is the only unknown quantity, the minimization of Eq. 29
delivers a process matrix that fits best to our experimental data, making no
assumptions about the specific form of the process matrix.

##  III Experiment

###  III.1 Time-Bin Quantum SWITCH

Figure 4: Experimental setup. a) The complete experiment. The individual
sections are indicated with colors. The red section shows the Sagnac SPDC
source that generates heralded single photons. In the yellow section, we
illustrate the asymmetric MZI to generate and measure the time-bin control
qubit. The orange section shows the target qubit preparation stage, which
consists of a PBS and two waveplates. The green area hosts the fiber-based
quantum SWITCH. The heralded and heralding photons are detected using SNSPDs,
shown in the blue area. By triggering off of the detection event of a
heralding photon we use a pulse generator to control the optical switches in
the setup. The sub-panels i) - vi) in panel b) show the functionality of the
quantum SWITCH. By controlling the state of the optical switches, we route the
two time-bins in different orders through Alice and Bob’s quantum channels.
After the SWITCH operation, the target qubit has experienced the action of the
quantum channels in a different order depending on the state of the time-bin
qubit.

To date, most previous implementations of the quantum SWITCH were based on
bulk optics. Since photonic quantum SWITCHes are essentially interferometers,
inevitable phase drifts limit the measurement time or require active
stabilization [45, 35]. Furthermore, since adding more parties means that the
dimension of the control system must be increased, scaling up previous
architectures requires more and more spatial modes to be transmitted through
the same optic, making it difficult to create a SWITCH with more than two
parties. Here, we present a passively-stable, fiber-based architecture for the
quantum SWITCH where the control system is encoded in a time degree of freedom
of the photon. Thus, in our architecture, although the dimension of the
control system must still be increased at the same rate, this can be done
using additional time bins, but only one spatial mode must traverse each
optical element. Furthermore, by using the same interferometer to prepare and
measure the control system, all phase fluctuations cancel out, making our
setup passively stable. This is important for process tomography, as we must
perform many measurements, and the experiment must remain stable during this
time.

To create the time-bin qubit that we will use to control the order, we start
by generating a photon pair, $\lambda=1550$ nm, using spontaneous parametric
down conversion (SPDC). One photon of the pair is directly detected to herald
the other photon, setting a time reference for the experiment. The second
photon is sent to a 50/50 beamsplitter—a fiber directional coupler (FDC)—which
splits the incoming mode into two fibers of different lengths. We then
deterministically recombine these two fiber paths using an ultra-fast fiber
optical switch (UFOS), see Fig. 3a (and also the yellow section of Fig. 4a)
[80]. To do so, we generate an electronic pulse, triggered off of the timing
reference generated by detecting the first photon. This pulse is sent to the
UFOS which change its state to first route the “photon component” from the
short path, followed by the “photon component” in long path, into the upper
output mode of the UFOS-MZI (Fig. 3a ii and iii). The result is that the
second photon is left in an equal superposition of two time bins in a single
fiber (Fig. 3a iv). Note that because the short time bin is transmitted
through the FDC, while the long time bin is reflected, one mode picks up a
reflection phase, while the other does not. Hence, in our experiment we
prepare the control qubit in the state
$\ket{y-}_{\text{C}}=(\ket{S}_{\text{C}}-i\ket{L}_{\text{C}})/\sqrt{2}$, where
we have labelled the modes as the “short” (“long”) state $\ket{S}_{\text{C}}$
($\ket{L}_{\text{C}})$, when it has taken the short (long) fiber path of the
interferometer. The spacing between these two time bins is $150$ ns, which is
set by the response time of our UFOS.

The UFOS we use to route the photon are BATi 2x2 Nanona fiber switches. In
addition to creating the time bin qubit, they allow us to route the photon in
a controlled way through the quantum SWITCH. Our UFOSs have a response time of
60 ns, with a maximal duty-cycle of 1 MHz, and a cross-channel isolation
greater than 20 dB for any polarisation (see [81, 82] for more details).

Having created the time bin control qubit, we need to apply the quantum SWITCH
operation to the target system, which we encode in the polarization degree-of-
freedom of the same photon. To route the photon, we use two additional UFOSs
and follow the protocol illustrated in Fig. 4b.i-vi. In particular, we send a
voltage pulse train consisting of three low levels and two high levels to the
UFOS’s. During each low level, the fiber switches are in a “cross state”
(output modes are swapped with respect to the input), while during a high
level the switch state is set to the “bar state” (input modes transmitted to
output modes). As the time-bins approach the quantum SWITCH (Fig. 4b.i) the
UFOS’s are initially in the cross-state, which routes the short time bin
$\ket{S}_{\text{C}}$ through Bob’s quantum channel (Fig. 4b.ii). Then the
UFOSs change to the bar state (Fig. 4b.iii), which sends $\ket{L}_{\text{C}}$
through Alice’s channel, while $\ket{S}_{\text{C}}$ travels over a fiber from
the RHS-UFOS to the LHS-UFOS. Then the UFOSs see a low voltage level, and
their state is set to cross (Fig. 4b.iv). This sends $\ket{S}_{\text{C}}$
through Alice local laboratory, while $\ket{L}_{\text{C}}$ loops back to the
LHS switch. In Fig. 4b.v the UFOS’s are in bar state and hence,
$\ket{L}_{\text{C}}$ passes through Bob’s channel. At this point
$\ket{S}_{\text{C}}$ exits the quantum SWITCH. Finally, the fiber switches are
set to the cross state (Fig. 4b.vi) so that $\ket{L}_{\text{C}}$ leaves the
quantum SWITCH. At this point, depending on the control state, the target
system has experienced a different order of Alice and Bob’s actions, which, as
we will describe shortly, act on the polarization state of the photon. Note,
that all the lengths of the fibers in the quantum SWITCH are set to ensure the
correct routing of time-bins spaced by $150$ ns.

The time-bin quantum SWITCH from Fig. 4b is placed in the full fiber-based
setup (Fig. 4a), in which the time-bins are prepared and measured. The quantum
SWITCH itself is shown in the green section of panel a). The type-II SPDC
photon source[83] is shown in the red section666Our source is in a Sagnac
configuration, although for this experiment we only pump the Sagnac loop in
one direction so as not to generate polarization entanglement. Here, a PBS
reflects the heralding photon to a single photon detector (blue area), while
the other photon is transmitted to the Mach-Zehnder-like time-bin generation
interferometer explained above (yellow section). Following this, we have a
photon encoding a time-bin qubit in the state
$(\ket{S}_{\text{C}}-i\ket{L}_{\text{C}})/\sqrt{2}$ in the “upper” output of
the interferometer (clockwise direction). The counter-clockwise path of the
loop (lower UFOS-MZI output mode) hosts an optical circulator with an empty
port to filter out misguided photons, which can arise from the imperfect
extinction ratio of our UFOSs. Next, the target system is encoded in the
photon’s polarization. For this we use a polarizing beam splitter (PBS) and a
set of a quarter- and a half-waveplate, shown in the orange section of Fig.
4a. Then we apply the 2-SWITCH operation to the target system described above
(green section of Fig.4a and Fig.4b), where we implement Alice and Bob’s
instruments using short free-space sections containing waveplates and
polarizers.

After exiting the SWITCH, the photon follows the fiber loop in clockwise
direction and approaches the MZI used for time-bin generation (yellow
section); now, from the opposite direction in the lower path. At this point,
we can decide to measure the control qubit in the computational ($Z$) or a
superposition ($Y$) basis. These measurements are illustrated in Fig. 3b and
c, respectively. For measurements in the computational basis, both time-bins
are routed by the UFOS along the lower path of the MZI, after which the two
time bins split up at the FDC, and are then sent to detectors in the blue
region. By measuring the arrival time, with respect to the herald detection,
we can distinguish between the short and long time bins. To measure in a
superposition basis, we use UFOS-MZI to send the time bins through the
opposite paths of the interferometer ($\ket{S}_{C}$ takes the long path and
$\ket{L}_{C}$ takes the short path) so that they arrive at the FDC at the same
time. In this case, interference occurs at the FDC, and detecting a photon at
exiting the upper (lower) port corresponds to projecting the control qubit in
$\ket{y+}$ ($\ket{y-}$).

With this in place, we collect the measurement statistics from different
measurement settings by detecting coincidence events between the heralding
photon and the FDC output or the circulator output. For each experimental
configuration, we record $\approx 1600$ coincidence counts ($\approx 21,000$
total single photon counts) over $10$ s at the FDC and circulator output. The
photon source generates $\approx 1,480,000$ single photons ($\approx 116,000$
coincidence events) in $10$ s before the experiment. Thus, our entire quantum
SWITCH experiment has an overall insertion loss of $\approx 18$ dB. All of our
measurements are carried out with superconducting nanowire single photon
detectors (SNSPD) from PhotonSpot Inc. The result, for a representative set of
measurements, is shown in Fig. 5. Therein, the bars are the theory for an
ideal quantum SWITCH with the control qubit in $\ket{y-}$, described by the
process matrix $W^{y-}$, while the points are our experimentally measured
data. Already, one can observe good agreement between theory and experiment.
Using unitary operations (rather than measure and reprepare instruments), we
can also play the anti-commuting/commuting gate discrimination game, as in
Ref. [35]. We find a success probability of $0.974\pm 0.018$, indicating a
high-fidelity of our implementation (the full details of this measurement are
presented in the Appendix B.1).

Figure 5: Experimentally Estimated Probabilities. A small subset of the
experimentally estimated probabilities. The bars are the theory for the ideal
process matrix $W^{y-}$, and the points are the experimental estimates. The
upper (lower) panel shows measurements of the control qubit in the $Z$ ($Y$)
basis. The red bars are for outcomes $\ket{0}$ and $\ket{y-}$, and the blue
for outcomes $\ket{1}$ and $\ket{y+}$.

###  III.2 Experimental Process Matrix Tomography

We will now present our experimental reconstruction of the process matrix of
our time-bin quantum SWITCH. As discussed in Sec. II, to probe the underlying
process, Alice and Bob must each implement a complete set of instruments. In
our experiment, the target system is encoded in the polarization state of the
photon, so Alice and Bob must act on this degree of freedom. Rather than the
measurement-repreparation instruments defined in Eqs. 26 and 27, we use a
slightly modified form $\tilde{R}:=\tilde{R}_{i|(j,k)}$ presented in the
Appendix Eqs. 79 and 80. In particular, Alice and Bob each have access to
three different measurement bases $\tilde{M}_{i|j}$ where $j\in{1,2,3}$
defines the measurement, and $i$ defines the outcome. For each $j$,
$\tilde{M}_{j}:=\tilde{M}_{1|j}-\tilde{M}_{2|j}$ is the observable associated
with the POVM $\\{\tilde{M}_{1|j},\tilde{M}_{2|j}\\}$. The specific operators
we implement are defined in Eq. 61 and 70. For example, $j=1$ corresponds to
the $Z$ basis: $\tilde{M}_{1}:=\tilde{M}_{1|1}-\tilde{M}_{2|1}=Z$.

Experimentally, we implement these measurements using a polarizer fixed to
transmit horizontally polarized light $\ket{H}$. We set the measurement basis
using a quarter waveplate and a half waveplate before the polarizer to the
angles given in Eq. 70. To implement the second part of the instrument—the
repreparations—we must prepare one of four different states. We experimentally
accomplish this using another quarter- and half-waveplate to rotate the
photon’s polarization if it is transmitted through the polarizer. This allows
us to prepare one of the four $\ket{\tilde{\phi}_{k}}$ states listed in Eq.
78. Thus, overall, both parties can implement the 24 different measurement-
repreparation operators defined in Eqs. 79 and 80 (6 different measurement
operators times 4 different repreparations)

In addition to Alice and Bob’s channels, we must send in a complete set of
target states, and perform measurements on the control qubit after the SWITCH.
To this end, we first prepare the target qubit in the four different input
states $\ket{\tilde{\psi}_{w}}$ given in Eq. 60. We set these states using the
quarter-half waveplate pair mounted in the target preparation stage, shown in
the orange area of Fig. 4a (the exact waveplate angles that we use are listed
in Eq. 60).

Finally, at the output of the SWITCH we must measure the state of the control
qubit. This procedure is illustrated in Fig. 3 panels b) and c). As discussed
in Sec. III.1, we use the same beamsplitter to measure and prepare the time-
bin qubit, but from opposite directions. As a result, the phase of this
measurement basis is fixed to the $Y$ basis. In our notation in the Appendix
Eqs. 45 and 52, this corresponds to a measurement $\tilde{C}_{1|2}$ and
$\tilde{C}_{2|2}$. Experimentally, a $\tilde{C}_{1|2}$ versus
$\tilde{C}_{2|2}$ result depends on which port of the FDC the photon exits. As
described above, we can additionally measure in the $Z$ basis by fixing the
UFOS-MZI to the bar state on the return trip such that the short and long
time-bins do not interfere at the TDC and observing the arrival time of the
time-bins. If we find the photon arrives earlier, this is associated with an
$\tilde{C}_{1|1}$ detection event, while if it arrives later corresponds to
$\tilde{C}_{2|1}$.

In order to be complete on the future control space, we would require an
additional measurement of the control qubit in the $X$ basis, i.e. we need the
measurements $M_{1|2}$ and $M_{2|2}$ from Eq. 22. In our experiment, this
could be achieved using a fast phase modulator to apply the appropriate phase
between the short and long path only on the reverse direction. However, we do
not implement this here. Instead, we impose an additional constraint on our
tomographic reconstruction. We require that
$\Tr\left(W_{\text{exp}}{X}^{F}\right)=0$; where
$X=\outerproduct{+}{+}-\outerproduct{-}{-}$. Given the passive phase stability
of our experiment, this is a very good assumption. We verify this assumption,
by comparing reconstructions with and without this constraint. In particular,
we find that the fidelity between the process matrices reconstructed with and
without this constrain is $0.999982$, well below our experimental error.

Overall, this results in $24\times 24\times 4\times 4=9216$ setting operators
of the form given in Eq. 81 (number of Alice’s settings $\times$ number of
Bob’s settings $\times$ number of target states $\times$ number of control
measurements). However, for the control measurements, we have access to both
ports of the FDC beamsplitter simultaneously (i.e there is a detector in each
output port of the beamsplitter), giving rise to $4608$ different experimental
configurations. From these data we can calculate the probabilities for each
given setting operator. Experimentally, we measure count rates associated with
each setting operator, which we must then normalize to convert into the
required probabilities. To do so, we make use of the normalization condition
over the outcomes of all three measurements

$\displaystyle\sum_{abc}p(abc|xyzw)=1.$ (30)

Thus, we define a normalization constant for every value of $x$, $y$, $z$, and
$w$

$\displaystyle N_{xyzw}=\sum_{abc}C(abc|xyzw),$ (31)

where $C(abc|xyzw)$ are the number of coincidences measured between the
heralding detector and the detectors after the FDC, corresponding to the
setting operator defined by $a$, $b$, $c$, $x$, $y$, $z$, and $w$. Then our
experimentally estimated probabilities are defined as

$\displaystyle p_{\text{exp}}(abc|xyzw)=\frac{C(abc|xyzw)}{N_{xyzw}}.$ (32)

A small subset of the resulting probabilities are plotted in Fig. 5. Then, by
minimizing Eq. 29 (with the setting operators $S_{abc|xyzw}$ replaced by the
experimental setting operators $\tilde{S}_{abc|xyzw}$ from Eq. 81) we can
reconstruct the process matrix $W_{\text{exp}}$. Our MATLAB code implementing
this minimization is available at [79].

##  IV Results

###  IV.1 Fidelity

The experimentally obtained $64\times 64$ process matrix and the ideal matrix
are plotted in Fig. 6 as a 3D-bar chart, where panel a) shows the real part,
and panel b) the imaginary part. The solid bars are the experimentally
reconstructed process matrix, while the transparent bars are the theoretical
process matrix $W^{y-}_{s}$. The x and y axis numerically label the basis
elements. The relatively close agreement between the target process matrix and
our experimental process matrix is already evident in this figure.

Figure 6: Process tomography data. This figure shows the experimentally recreated process matrix of the quantum SWITCH. a) represents the real part of $W_{\text{SWITCH}}$ and b) the imaginary part. The color gradient along the x-axis does not have a physical meaning; rather, it is color coded in order to identify the individual elements of this $64$x$64$ matrix better. Additionally, the ideal process matrix is represented via the layer of semi-transparent bars. Calculating the fidelity between these two process matrices results in $F(W_{\text{exp}},W_{s}^{y-})=0.920\pm 0.001$. | ${G}_{\text{y-,res}}^{\text{GR}}$ | ${G}_{\text{exp,res}}^{\text{GR}}$ | ${G}_{\text{y-,all}}^{\text{GR}}$ | ${G}_{\text{exp,all}}^{\text{GR}}$
---|---|---|---|---
$W^{y-}_{s}$ | $-0.5834$ | $-0.5525$ | $-0.5834$ | $-0.5512$
$W_{\text{exp}}$ | $-0.387\pm 0.003$ | $-0.431\pm 0.003$ | $-0.370\pm 0.003$ | $-0.431\pm 0.002$
Table 1: Generalized Robustness Witness Analysis. A summary of the different generalized robustness witnesses constructed. The witnesses $G_{i,j}$ are labelled by two subscripts. The first indicates if the witness was designed for the ideal process matrix $W^{y-}$ (subscript ”$y-$”) or the experimental process matrix $W_{\text{exp}}$ (subscript ”exp”). The second subscript indicates if the restricted measurement set (subscript ”res”) or the complete measurement set (subscript ”all”) was used for the witness. The first row shows the value of the witness for $W^{y-}$, and the second row shows the experimental values, which were evaluated as $\Tr[G_{i,j}W_{\text{exp}}]$.  | ${G}_{\text{y-,res}}^{\text{WN}}$ | ${G}_{\text{exp,res}}^{\text{WN}}$ | ${G}_{\text{y-,all}}^{\text{WN}}$ | ${G}_{\text{exp,all}}^{\text{WN}}$
---|---|---|---|---
$W^{y-}_{s}$ | $-2.296$ | $-2.174$ | $-2.767$ | $-2.624$
$W_{\text{exp}}$ | $-1.64\pm 0.02$ | $-1.76\pm 0.01$ | $-1.96\pm 0.02$ | $-2.112\pm 0.02$
Table 2: White Noise Witness Analysis. A summary of the different white noise
witnesses constructed. The witnesses $G_{i,j}$ and process matrix labels are
labelled by two subscripts described in the caption of Tab. 1.

To further assess the agreement between our experiment and theory, we estimate
the fidelity of the measured process matrix $W_{\text{exp}}$ to the ideal
matrix $W_{s}^{y-}$. Since every valid process matrix normalized by its trace
is a valid quantum state, we use the conventional expression for calculating
the fidelity
$F(\sigma,\rho)=\Tr\big(\sqrt{\sqrt{\sigma}\rho\sqrt{\sigma}}\big{missing})$
with $\sigma$ and $\rho$ being different density matrices [70]. This results
in a fidelity of

$F(W_{\text{exp}},W_{\text{SWITCH}})=0.920\pm 0.001\,,$ (33)

where the error arises is estimated using a Monte Carlo simulation of the
entire reconstruction procedure accounting for finite measurement statistics
and small waveplate errors of $1^{\circ}$. Especially given the high-dimension
of our process matrix, this fidelity indicates that our experiment is quite
close to theory.

To quantify the agreement between our experimental data and $W_{\text{exp}}$
we compare the residuals of our fit $r$ (defined in Eq. 29) to the average
statistical error of our data $\eta_{\text{stat}}$. The residuals $r$ can be
interpreted as the disagreement between the outcome predicted by
$W_{\text{exp}}$ and the measured experimental outcome, averaged over all
measurement settings. For our fit $r=0.0089$, indicating an excellent match to
our experimental data. We estimate our statistical errors as follows. First,
we treat the probability to obtain an outcome $abc$ as a binomial variable:
either we detect a photon or we do not. Then we estimate the variance of that
setting as $N_{xyzw}\leavevmode\nobreak\ p(abc|xyzw)\times(1-p(abc|xyzw))$,
where $N_{xyzw}$ is the number of photons detected in all outcomes associated
with $xyzw$ (defined in Eq. 31). Finally, we compute the average error per
setting as

$\displaystyle\eta_{\text{stat}}=\frac{1}{N_{\text{settings}}}\sum_{abc|xyzw}\frac{p(abc|xyzw)(1-p(abc|xyzw){\color[rgb]{1,0,1})}}{\sqrt{C}}.$
(34)

This is simply the standard error of each setting operator averaged over all
settings. Evaluating this for our data, we find $\eta_{\text{stat}}=0.0056$.
Given that $\eta_{\text{stat}}\approx r$, we conclude that our process matrix
fits our data well.

###  IV.2 Causal Non-Separability

A bipartite process matrix without a common past is causally non-separable
when it cannot be written as a classical mixture of causally ordered processes
[6, 13]. When considering bipartite processes with a common past, such as the
quantum switch considered in this work, there are different non-equivalent
definitions of indefinite causality. In Ref. [13, 43], a bipartite process
matrix $W\in\leavevmode\nobreak\
\mathcal{L}\left(\mathcal{H}_{P}\otimes\mathcal{H}_{A_{\text{in}}}\otimes\mathcal{H}_{A_{\text{out}}}\otimes\mathcal{H}_{B_{\text{in}}}\otimes\mathcal{H}_{B_{\text{out}}}\otimes\mathcal{H}_{F}\right)$
with common past and common future is said to be causally separable if it can
be written as a convex sum of causally-ordered process matrices. That is, if
we can write

$\displaystyle W=pW^{A>B}+(1-p)W^{B>A},$ (35)

where $p\in[0,1]$, and $W^{A>B}$ and $W^{B>A}$ are causally-ordered process
(objects also referred to as quantum combs [4, 8], see Appendix C).
Alternatively, Ref. [11] proposes the concept of extensible causally-separable
processes. This leads to a definition which differs from the one in Eq. (35),
but is equivalent to the definition of causal-separability presented in Ref.
[84], which considers not only convex mixtures of causally-ordered processes,
but also incoherent (hence, classical) control of causal order. The analysis
and the numbers presented in this section and in the main part of this paper
were obtained via the definition of [43], which is the one presented in Eq.
(35). However, we stress that the results of our work are not qualitatively
affected by the different before-mentioned definitions, in the sense that, in
all cases, the process we obtain after tomography is not causally separable,
and it is robust against different kinds of noise. In Appendix C, we present a
more detailed discussion of such definitions and how they make small
quantitative changes in the numbers presented here.

One method to quantify the degree to which our quantum process is causally
non-separable is by using a causal witness. A causal witness is a Hermitian
non-negative operator ${G}$ such that $\Tr\left({G}W_{\text{sep}}\right)\geq
0$ for all causally-separable processes. However, for all causally–non-
separable processes (such as the quantum SWITCH) one can always find a witness
${G}$ such that $\Tr\left({G}W^{y-}_{s}\right)<\leavevmode\nobreak\ 0$.
Without additional constraints, the quantity $\Tr\left({G}W\right)$ does not
have a physical meaning, and may be artificially inflated by multiplying the
witness $G$ by some constant. However, by setting additional normalisation
constraints on the witness $G$, one may identify the quantity
$\Tr\left({G}W\right)$ with how much noise the process $W$ can tolerate until
it becomes causality separable. More concretely, let
$\mathds{1}_{W}:=\frac{\mathds{1}}{d_{P}d_{A_{O}}d_{B_{O}}}$ be the “white
noise process”, which simply consists of discarding everything and outputting
white noise, and let $W_{\text{sep}}$ be an arbitrary causally-separable
process matrix. Ref. [13] shows that the problem

$\displaystyle\min\Tr(GW),$ (36) $\displaystyle\text{s.t.
}\Tr\left({G}W_{\text{sep}}\right)\geq 0,\quad\forall W_{\text{sep}}$ (37)
$\displaystyle\Tr(G)\leq\Tr(\mathds{1}_{W})$ (38)

is equivalent to its dual formulation

$\displaystyle\min-r,$ (39) $\displaystyle\text{s.t.
}\frac{W+r\mathds{1}_{W}}{1+r}\text{ is causally separable }.$ (40)

Hence, under the normalisation constraint $\Tr(G)\leq\leavevmode\nobreak\
\Tr(\mathds{1}_{W})$, we ensure the identity $\Tr(GW)=-r$, which allows us to
interpret $\Tr(GW)$ as how robust $W$ is against white noise.

Alternatively, one may also consider the normalisation $\Tr(G\Omega)\leq 1$,
where $\Omega$ is an arbitrary process matrix. In this case, the equivalent
problem reads as

$\displaystyle\min-r,$ (41) $\displaystyle\text{s.t.
}\frac{W+r\Omega}{1+r}\text{ is causally separable },$ (42)

where $\Omega$ is an arbitrary process matrix. In this case, the value
$\Tr(GW)=-r$ is typically called the “generalised robustness”; it may be
viewed as the amount of noise one needs to add to $W$ to make it causally
separable in the worst case scenario.

In order for the witness $G$ to fit the setting operators implemented in our
experiment, we impose an additional structure on the witness $G$ which is
given by

${G}=\sum_{abcxwyz}\alpha_{a,b,c,x,y,z,w}S_{abc|xyzw},$ (43)

where $\alpha_{a,b,c,x,y,z,w}$ are arbitrary real numbers and $S_{abc|xyzw}$
are the setting operators of our experiment (see Sec. II.5). Additionally, for
fixed setting operators, finding the maximal violation of a witness $G$ with
the normalization constraints related to white and generalised noise is a
Semidefinite Program (SDP) [13], and can be efficiently solved numerically
[85]. Our code doing so is also available at [79].

With these tools, we can construct a variety of witnesses. First, we can
construct witnesses using the complete measurement set (Eq. 28) or our
restricted measurement set (Eq. 81). We can further design witnesses for two
different process matrices $W^{y-}_{s}$ or $W_{\text{exp}}$. This results in
four witnesses: ${G}_{\text{y-,all}}$, ${G}_{\text{y-, res}}$,
${G}_{\text{exp,all}}$, ${G}_{\text{exp, res}}$. Where the subscript $y-$
(exp) indicates that the witness was optimized for the ideal (experimental)
process matrix, and the subscript all (res) indicates that the witness was
computed using the complete (restricted) measurement set. We can then further
construct witness for either the generalized or white noise robustness.

The results of the generalized robustness witnesses are summarized in Tab. 1.
The first row of Tab. 1 shows the value of the four witnesses evaluated using
$W^{y-}_{s}$, and the second row shows the experimental values, estimated
using $W_{\text{exp}}$. In this case, we see that
$\Tr(W^{y-}G_{\text{y-,res}}^{\text{GR}})\approx\leavevmode\nobreak\
\Tr(W^{y-}G_{\text{y-,all}}^{\text{GR}})$ and
$\Tr(W^{\text{exp}}G_{\text{exp,res}}^{\text{GR}})\approx\leavevmode\nobreak\
\Tr(W^{\text{exp}}G_{\text{exp,all}}^{\text{GR}})$. In other words, the
generalized robustness evaluated either with the complete or restricted
setting operators is equal within experimental error. Evidently, the
additional measurement on the future control system does not affect the
generalized robustness. More interesting for the generalized robustness
witnesses, is the performance of the witnesses optimized for our experimental
process matrix $W_{\text{exp}}$. Examining the performance of our
experimentally estimated witnesses (the bottom row of Tab. 1), we see that the
witnesses designed specifically for our experimental process matrix increase
the generalized robustness. In particular,
$\Tr(W_{\text{exp}}G_{\text{exp,all}}^{\text{GR}})>\leavevmode\nobreak\
\Tr(W_{\text{exp}}G_{\text{y-,all}}^{\text{GR}})$ and
$\Tr(W_{\text{exp}}G_{\text{exp,res}}^{\text{GR}})>\leavevmode\nobreak\
\Tr(W_{\text{exp}}G_{\text{y-,res}}^{\text{GR}})$. This would not be readily
possible without performing process matrix tomography.

In Tab. 2 we summarize the results of our white noise witness analysis. In
this case, we see a significant difference between the witnesses constructed
with the restricted and complete measurement sets, with the complete
measurements sets revealing a higher white noise robustness in all cases.
Furthermore, in the second row, we can see that each step progressively
improves the experimental white noise robustness. The first entry
$\Tr(G_{\text{y-,res}}^{\text{WN}}W_{\text{exp}})=-1.65\pm 0.02$ shows the
value that one would obtain for our setup without performing process matrix
tomography. i.e. the witness was designed for the ideal process and uses only
the experimentally implementable measurement settings. In the next column,
$\Tr(G_{\text{exp,res}}^{\text{WN}}W_{\text{exp}})=-1.76\pm 0.01$ is improved
by tailoring the witness for our experiment; however, still using only
experimentally implementable measurements. In the next two columns, we improve
both of these values further by computing the witness assuming the complete
measurement set. The final entry,
$\Tr(G_{\text{exp,all}}^{\text{WN}}W_{\text{exp}})=-2.11\pm 0.02$ is
significantly higher than the first entry, clearly showing the power of full
process matrix tomography. Process matrix tomography allows us to compute
properties of the experimental process, without having direct experimental
access to them and we can precisely tailor our analysis to our experimental
conditions. In the Appendix tables 3 and 4, we show the same analysis, for the
alternative definition of causal non-separability. The trends observed therein
are the same, although the absolute values of the robustnesses are smaller.

###  IV.3 Worst-Case Process Tomography

Figure 7: Worst-Case Process Tomography. Plots of the worst-case generalized
robustness (a) and white noise robustness (b) causal witnesses versus the
allowed deviation from the experimental data. For these plots, the tomography
routine attempted to find a process matrix which maximized the witness (i.e.
it searched for the “most causally separable” process matrix), while still
agreeing with our experimental data with an average error of $\epsilon$. For
all witnesses considered, the process which minimizes $\epsilon$ is also the
most causally non-separable. For $\epsilon<0.089$ no valid process matrix is
found.

For the process tomography results presented in Sec. IV.1, we found the
process matrix that fit to our data best, by minimizing Eq. 29. As discussed
there, this resulted in a process matrix that describes our data very well.
However, one could ask “Are there other causally-separable process matrices
that describe the data almost as well?” To answer this question, we perform a
“worst-case” version of our process matrix tomography. To do so, rather than
minimizing expression Eq. 29, we find the process matrix that maximizes the
generalized or white noise robustness,
$\Tr(W_{\text{worst}}{G}_{\text{j,all}}^{\text{GR}})$ or
$\Tr(W_{\text{worst}}{G}_{\text{j,all}}^{\text{WN}})$, respectively. We do
this using the witnesses designed for the ideal process matrix and the
originally reconstructed experimental process matrix, but always with the
complete measurement sets. This maximisation is subject to the constraints
that $W_{\text{worst}}$ is a physical process matrix and that the predictions
of $W_{\text{worst}}$ match the experiment within some error $\epsilon$:

$\displaystyle\sum_{abcxyzw}\absolutevalue{\Tr(S_{abc|xyzw}W)-p_{\text{exp}}(abc|xyzw)}\leq\epsilon.$
(44)

Here $\epsilon$ is closely related to the residuals defined in Eq. 29. In
particular, if $\epsilon<r$ the maximization will fail, as there is no
physical process matrix compatible with this constraint. Thus, we perform
worst case process tomography for the four witnesses discussed above starting
from $\epsilon=r_{\text{exp}}=0.0089$ and increasing to $\epsilon=0.015$. The
results of this analysis are plotted in Fig. 7a and b. We find that the
generalized robustness witnesses are more tolerant $\epsilon$, finding that
our data is only consistent with causally separable process matrices for
$\epsilon\lesssim 0.012$, while the white noise witnesses require
$\epsilon\lesssim 0.0105$. Although this analysis suggests that the causal
non-separability is rather fragile, we stress the worst-case nature of this
treatment: if a single causally separable process matrix is compatible to our
data within $\epsilon$ it will be returned, even if a causally non-separable
data fits our data better. In any case, we see that for a range of
experimentally relevant errors our data are only compatible with a causally
non-separable process matrix.

##  V Discussion

In this work, we have presented a protocol to perform process tomography on a
higher-order quantum operation, the quantum SWITCH. We discussed how to
construct a complete set of measurements. The requirements for this go beyond
standard quantum process tomography, wherein one must “only” send a complete
set of input states through the process, and perform a complete set of
measurements after the process. In particular, because HOQOs take quantum
channels as inputs, we must also implement a complete set of quantum channels
for each input channel. This can be achieved using measure-and-reprepare
instruments. Since this procedure scales even worse than standard process
tomography, we implement it using a new phase-stable architecture of the
quantum SWITCH, allowing for long integration times.

Our photonic quantum SWITCH uses a time-bin qubit as the control system. By
recombining time-bin qubits using a passively-stable interferometer, we were
able to keep our experiment stable indefinitely. We believe this technique
will be beneficial for various time-bin quantum information experiments and
may even be scaleable to high dimensional time-bin qudits. This would enable
the construction of a multi-party quantum SWITCH. The results of performing
quantum process matrix tomography on our experiment show that we have indeed
implemented a high-fidelity quantum SWITCH, with a fidelity of $F=0.920\pm
0.001$. We then used our results to verify the causal non-separability of our
experiment, designing causal witnesses specifically for our experimental
process matrix. Finally, we implemented a worst-case process tomography,
searching for a causally-separable process that could also describe our
measurement. To find such a process, we had to allow for a $\approx 1.5$ times
larger disagreement between our measurements and our causally separable model.

Although our protocol was presented for the quantum SWITCH, it could be
adapted to general HOQOs in a straight-forward manner. In our present work, we
performed an over complete set of measurements, but it should be possible to
implement a reduced set of measurements by taking into account the constraints
on the space of physical process matrices. We also point out that many
complexity-reducing techniques from standard state and process tomography,
including compressed sensing [86], shadow tomography [87], adaptive tomography
[88], etc., should apply to our protocol equally well. But we leave these as
topics for future work.

## Data Availability

All the data that are necessary to replicate, verify, falsify and/or reuse
this research is available online at [79].

## Acknowledgements

This project was funded in whole, or in part, from the European Union’s
Horizon 2020 research and innovation programme under grant agreement No 820474
(UNIQORN) and No 899368) (EPIQUS), the Marie Skłodowska-Curie grant agreement
No 956071 (AppQInfo), and the QuantERA II Programme under Grant Agreement No I
6002 - N (PhoMemtor); the Austrian Science Fund (FWF) through [F7113]
(BeyondC), and [FG5] (Research Group 5); the AFOSR via FA8655-20-1-7030
(PhoQuGraph), and FA9550-21- 1-0355 (QTRUST); the Austrian Federal Ministry
for Digital and Economic Affairs, the National Foundation for Research,
Technology and Development and the Christian Doppler Research Association. For
the purpose of open access, the author has applied a CC BY public copyright
licence to any Author Accepted Manuscript version arising from this
submission.

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##  Appendix A Experimental techniques

In this section, the used SPDC photon pair source is described and the
technique implemented to compensate for unwanted polarization transformations
induced by the birefringence of the optical fibers is explained. Moreover, the
implemented experimental instruments and measurements are shown in detail.

###  A.1 Photon Source

To generate photon pairs at a wavelength of $\lambda=1550$ nm we use a 775 nm,
CW laser beam to pump a 30mm long ppKTP crystal. A dichroic mirror reflects
775 nm light and transmits photons at 1550 nm. The signal and idler photon are
then separated on a PBS and coupled over optical fibers into the setup.

###  A.2 Polarization Compensation

In order to ensure that the quantum SWITCH performs the desired
transformations on the photon’s polarization, it is important to correct for
the birefringent behaviour of the fibers that connect Alice and Bob’s
laboratories; i.e. to ensure that the fibers do not change the polarization
state of the photon. Hence, each fiber link has to perform an identity
operation. To this end, each fiber is equipped with a 3-loop fiber
polarization controller, which allows us to implement any unitary polarization
transformation in the fiber. In order to implement a true identity operation,
we must check that the correct transformation is applied in two different
bases. We use the computational and diagonal bases. A convenient way to do
this, is to send classical light at the same wavelength as the single photons
through the fibers and detect the polarization with a polarimeter at the fiber
output. To ensure the identity transformation, we switch the light’s
polarization state at the target preparation stage(see Fig. 4) between
horizontal $\ket{H}$ and diagonal $\ket{D}$, while adjusting the polarization
controller until it converges to the correct setting in both bases. To correct
the polarization for the second trip through the channels, we place a
polarizer in one of the channels (without additional waveplates). This
decouples the compensation from the previous fiber. Then, we follow the same
procedure and alternate the polarizer to transmit $\ket{H}$ and then
$\ket{V}$. In order for this procedure to work properly, it is essential that
wavelengths of the classical light and the single photons is matched.

###  A.3 Experimental Instruments and Measurements

Here we explain in detail the measurements and instruments we implement in the
lab, and how they relate the ideal settings discussed in the main text.

####  A.3.1 Future Control Measurement

Our control qubit is the time-bin qubit. Its past state is fixed to
$\ket{y-}$. Ideally, we would measure the future control in three different
bases described by Eqs. 21 \- 23. However, due to experimental limitations we
only measure the future control in two bases

$\displaystyle\mathcal{\tilde{C}}:=\\{\tilde{C}_{c|z}\\}_{c=1,z=1}^{c=2,z=2},$
(45)

where

(52)

Experimentally, the measurement outcomes $c=1$ and $c=2$ correspond to finding
the photon exiting different ports of the beamsplitter (labeled FDC in Fig. 4
). Note that the order of the output indices has swapped compared to Eqs.
21-23 in order to be consistent with our experimental convention.

####  A.3.2 Past Target States

Our target system is encoded as in the polarization degree of freedom of the
photon. We thus prepare its state by sending the photon to a polarizer set to
transmit horizontal polarization, which we define to be the logical $\ket{0}$
state. We then set its state using a quarter waveplate, followed by a half
waveplate. We can thus prepare the set of states given by

$\displaystyle\mathcal{\tilde{S}}:=\\{\outerproduct{\tilde{\psi_{w}}}{\tilde{\psi}_{w}}\\}_{w=1}^{4},$
(53)

where

(60)

####  A.3.3 Alice and Bob’s Instruments

As our target system is encoded in a polarization state, Alice and Bob must
implement measure and re-prepare channels on this degree of freedom. To do the
measurement, they use a fixed polarizer to project onto horizontal
polarization. Using a quarter and a half waveplate before the polarizer, they
can then set the measurement basis. Since this only provides one outcome
(either the photon is transmitted or not, but they cannot detect the cases
when a photon is absorbed by the polarizer) they must also explicitly set the
waveplates to perform the orthogonal measurement in order to normalize the
data to compute a probability. These measurements defined by the set

$\displaystyle\mathcal{\tilde{M}}:=\\{\tilde{M}_{i|j}\\}_{i=1,j=1}^{i=2,j=3},$
(61)

where the waveplates angles and resulting measurement operators are given by

(70)

Following their measurements, Alice and Bob reprepare the target state in one
of 4 different options

$\displaystyle\mathcal{\tilde{P}}:=\\{\outerproduct{\tilde{\phi_{k}}}{\tilde{\phi}_{k}}\\}_{k=1}^{4}.$
(71)

This is again accomplished with a quarter and half wavplate. Since the their
polarizers transmit horizontal polarization, the horizontally polarized post-
measurement photon is then rotated to one of the following states:

(78)

The net action of their instruments is then given by all combinations of the
sets $\mathcal{\tilde{M}}$ and $\mathcal{\tilde{P}}$, that is

$\displaystyle\mathcal{\tilde{R}}:=\\{\tilde{R}_{i|(j,k)}\\}_{i=1,j=1,k=1}^{i=2,j=3,k=4},$
(79)

where

$\displaystyle\tilde{R}_{i|(j,k)}:=\tilde{M}_{i|j}\otimes\outerproduct{\tilde{\phi}_{k}}{\tilde{\phi}_{k}}^{T}.$
(80)

We then have the following experimental setting operator

$\displaystyle\tilde{S}_{abc|xyzw}$
$\displaystyle:={\outerproduct{\tilde{\psi}_{w}}{\tilde{\psi}_{w}}^{T}}^{P_{\text{t}}}\otimes\tilde{A}_{a|x}^{A_{I}A_{O}}\otimes\tilde{B}_{b|y}^{B_{I}B_{O}}\otimes\tilde{C}_{c|z}^{F_{\text{c}}},$
(81)

##  Appendix B Additional Measurements

Figure 8: Commutation Game. This figure shows the experimentally estimated
probabilities for the commutation / anti-commutation game. The green bars
represent the probability, that the photon leaves port 0, leading the
conclusion that the unitaries commute. The blue bars show the probability for
the photon to exit port 1, and thus the conclusion that the gates anti-
commute. The x-axis shows which gates have been implemented for
$\hat{U}_{A}$-$\hat{U}_{B}$.

In this section, we show additional measurements taken on this setup.

###  B.1 Commutation / Anti-Commutation Game

Shortly after it was proposed that the quantum SWITCH offers a computational
advantage over conventional quantum circuits for certain tasks [29, 13], one
such task—the Commutation / Anti-Commutation Game—was experimentally realized
with a two-party SWITCH [35]. This task takes the form of a game, in which a
referee provides a player with two unitary gates $\hat{A}$ and $\hat{B}$. The
referee states that these two unitaries gates either commute or anti-commute,
and the player’s goal is to decide which statement is true. However, the
player is only permitted to use each gate once. When the player has access to
a quantum SWITCH, they can win the game perfectly. However, When the player is
constrained to use a conventional (causally-ordered) quantum circuit, they
would need to query one gate at least twice. The SWITCH result can be easily
understood by looking at the output state of the SWITCH:

$\ket{\Psi}=\frac{1}{2}\ket{0}_{C}\anticommutator{\hat{A}}{\hat{B}}\ket{\Psi}_{T}+\frac{1}{2}\ket{1}_{C}\commutator{\hat{A}}{\hat{B}}\ket{\Psi}_{T}\,.$
(82)

Since the player is sure that either the commutator or anti-commutator is
zero, by measuring the control and finding it in $\ket{0}_{C}$ they are sure
the gates commute, while if they find it in $\ket{1}_{C}$ they can conclude
that the gates anti-commute.

Before performing higher-order process tomography, we first verified the
correct performance of the new architecture of our quantum SWITCH by
implementing this game. To do so, Alice and Bob could set the Pauli operators
$X=\sigma_{X}$, $Y=\sigma_{Y}$, $Z=\sigma_{Z}$ and the identity operator
$I=\sigma_{0}$ with the well known commutation relation
$\commutator{\hat{\sigma}_{i}}{\hat{\sigma}_{j}}=2i\epsilon_{ijk}\hat{\sigma}_{k}$
and anti-commutation relation
$\anticommutator{\hat{\sigma}_{i}}{\hat{\sigma}_{j}}=2\delta_{i}j\mathbb{1}_{2x2}$
[70]. Fig. 8 shows the probabilities for measuring the control either in state
$\ket{0}_{C}$ or $\ket{1}_{C}$.

The probabilities were calculating via measuring coincidence rates between the
heralding detector and both output ports of the MZI with a coincidence window
of $5$ ns between. Data was acquired for a total of $10$ s for each setting.
The visibility of the interferometer was measured as $v^{2}=0.97$. We
calculated the success probability for the photon leaving the correct port as

$\begin{split}p_{\text{succ}}&=\frac{1}{2}(\bar{p}(0|[\cdot,\cdot])+\bar{p}(1|\\{\cdot,\cdot\\}))\\\
&=0.974\pm 0.18\,.\end{split}$ (83)

Thus, our new architecture achieves a similar performance as previous
implementations [35], but now in a phase-stable manner.

##  Appendix C The definition of causal non-separability

| ${G}_{\text{y-,res}}^{\text{GR}}$ | ${G}_{\text{exp,res}}^{\text{GR}}$ | ${G}_{\text{y-,all}}^{\text{GR}}$ | ${G}_{\text{exp,all}}^{\text{GR}}$
---|---|---|---|---
$W^{y-}_{s}$ | $-0.500$ | $-0.483$ | $-0.500$ | $-0.484$
$W_{\text{exp}}$ | $-0.346\pm 0.003$ | $-0.361\pm 0.003$ | $-0.323\pm 0.003$ | $-0.363\pm 0.003$
Table 3: Generalized Robustness Alternative Definition Witness Analysis. A summary of the different generalized robustness witnesses constructed using the definition of [37], presented in Eq. (91). The witnesses $G_{i,j}$ are labelled by two subscripts. The first indicates if the witness was designed for the ideal process matrix $W^{y-}$ (subscript ”$y-$”) or the experimental process matrix $W_{\text{exp}}$ (subscript ”exp”). The second subscript indicates if the restricted measurement set (subscript ”res”) or the complete measurement set (subscript ”all”) was used for the witness. The first row shows the value of the witness for $W^{y-}$, and the second row shows the experimental values, which were evaluated as $\Tr[G_{i,j}W_{\text{exp}}]$.  | ${G}_{\text{y-,res}}^{\text{WN}}$ | ${G}_{\text{exp,res}}^{\text{WN}}$ | ${G}_{\text{y-,all}}^{\text{WN}}$ | ${G}_{\text{exp,all}}^{\text{WN}}$
---|---|---|---|---
$W^{y-}_{s}$ | $-0.828$ | $-0.800$ | $-1.000$ | $-0.965$
$W_{\text{exp}}$ | $-0.572\pm 0.005$ | $-0.614\pm 0.006$ | $-0.689\pm 0.007$ | $-0.739\pm 0.008$
Table 4: White Noise Alternative Definition Witness Analysis. A summary of the
different white noise witnesses constructed using the definition of [37],
presented in Eq. (91). The witnesses $G_{i,j}$ and process matrix labels are
labelled by two subscripts described in the caption of Tab. 3.

Let $W\in\leavevmode\nobreak\
\mathcal{L}\left(\mathcal{H}_{P}\otimes\mathcal{H}_{A_{\text{in}}}\otimes\mathcal{H}_{A_{\text{out}}}\otimes\mathcal{H}_{B_{\text{in}}}\otimes\mathcal{H}_{B_{\text{out}}}\otimes\mathcal{H}_{F}\right)$
be a bipartite process matrix which a common past and common future. A process
$W^{A>B}$ is said to be causally ordered from $A$ to $B$ if it can be
constructed by beans of a sequential quantum circuit [6, 13], also called a
quantum comb [2, 8]). More explicitly, a process matrix
$W^{A>B}\in\leavevmode\nobreak\
\mathcal{L}\left(\mathcal{H}_{P}\otimes\mathcal{H}_{A_{\text{in}}}\otimes\mathcal{H}_{A_{\text{out}}}\otimes\mathcal{H}_{B_{\text{in}}}\otimes\mathcal{H}_{B_{\text{out}}}\otimes\mathcal{H}_{F}\right)$
is causally ordered from $A$ to $B$ if it respects

$\displaystyle\tr_{F}(W^{A>B})=$
$\displaystyle\tr_{B_{O}F}(W^{A>B})\otimes\frac{\mathds{1}_{B_{O}}}{d_{B_{O}}}$
(84) $\displaystyle\tr_{B_{I}B_{O}F}(W^{A>B})=$
$\displaystyle\tr_{A_{O}B_{I}B_{O}F}(W^{A>B})\otimes\frac{\mathds{1}_{A_{O}}}{d_{A_{O}}}$
(85) $\displaystyle\tr_{A_{I}A_{O}B_{I}B_{O}F}(W^{A>B})=$
$\displaystyle\tr_{PA_{I}A_{O}B_{I}B_{O}F}(W^{A>B})\otimes\frac{\mathds{1}_{P}}{d_{p}}.$
(86)

Analogously, a process matrix $W^{B>A}\in\leavevmode\nobreak\
\mathcal{L}\left(\mathcal{H}_{P}\otimes\mathcal{H}_{A_{\text{in}}}\otimes\mathcal{H}_{A_{\text{out}}}\otimes\mathcal{H}_{B_{\text{in}}}\otimes\mathcal{H}_{B_{\text{out}}}\otimes\mathcal{H}_{F}\right)$
is causally ordered from $B$ to $A$ if it respects

$\displaystyle\tr_{F}(W^{B>A})=$
$\displaystyle\tr_{A_{O}F}(W^{B>A})\otimes\frac{\mathds{1}_{A_{O}}}{d_{A_{O}}}$
(87) $\displaystyle\tr_{A_{I}A_{O}F}(W^{B>A})=$
$\displaystyle\tr_{B_{O}A_{I}A_{O}F}(W^{B>A})\otimes\frac{\mathds{1}_{B_{O}}}{d_{B_{O}}}$
(88) $\displaystyle\tr_{A_{I}A_{O}B_{I}B_{O}F}(W^{B>A})=$
$\displaystyle\tr_{PA_{I}A_{O}B_{I}B_{O}F}(W^{B>A})\otimes\frac{\mathds{1}_{P}}{d_{p}}.$
(89)

In the main part of this work, we follow the definition of Ref. [43], where a
bipartite process matrix $W\in\leavevmode\nobreak\
\mathcal{L}\left(\mathcal{H}_{P}\otimes\mathcal{H}_{A_{\text{in}}}\otimes\mathcal{H}_{A_{\text{out}}}\otimes\mathcal{H}_{B_{\text{in}}}\otimes\mathcal{H}_{B_{\text{out}}}\otimes\mathcal{H}_{F}\right)$
with common past and common future is said to be causally separable if it can
be written as a convex sum of causally-ordered process matrices. That is, if
we can write

$\displaystyle W=pW^{A>B}+(1-p)W^{B>A},$ (90)

where $p\in[0,1]$, and $W^{A>B}$ and $W^{B>A}$ are causally-ordered process.
However, as discussed earlier, there exists a non-equivalent definition, which
go beyond simple convex combination and allow incoherent classical control of
causal orders. This definition is presented in Ref. [84], and it is proven to
be equivalent to the notion of extensible causal, presented in Ref. [11]. In
the bipartite scenario with a common past and common future, Ref. [84] states
that a process matrix $W\in\leavevmode\nobreak\
\mathcal{L}\left(\mathcal{H}_{P}\otimes\mathcal{H}_{A_{\text{in}}}\otimes\mathcal{H}_{A_{\text{out}}}\otimes\mathcal{H}_{B_{\text{in}}}\otimes\mathcal{H}_{B_{\text{out}}}\otimes\mathcal{H}_{F}\right)$
is causally separable when there exists causally ordered processes $W^{A>B}$
and $W^{B>A}$ such that,

$\displaystyle\frac{\mathds{1}_{P}}{d_{P}}\otimes\tr_{P}(W)=pW^{A>B}+(1-p)W^{B>A}.$
(91)

Notice that the definition of Ref. [84] presented in Eq. 91 is more relaxed
than the one of Ref. [43], and presented in (90). Indeed, as we show latter,
there are processes which are causally separable following the definition of
[84], but are causally non-separable following the definition of [43].

In the definition of Ref. [84], in order for a causal witness to be valid, one
should include an extra constraint, which reads as

$\displaystyle G=\frac{\mathds{1}_{P}}{d_{P}}\otimes\tr_{P}(G).$ (92)

We can then re-calculate the numbers presented in Table 1 and Table 2 from the
main text, but following the definition of Ref. [84]. These results are
presented in Table 3 and Table 4, and we notice that although there are some
difference in the obtained numbers, the qualitative result remains the same.