# Mapping indefinite causal order processes to composable quantum protocols in
a spacetime

Matthias Salzger  International Centre for Theory of Quantum Technologies,
University of Gdańsk, 80-309 Gdańsk, Poland  Institute for Theoretical
Physics, ETH Zurich, 8093 Zürich, Switzerland
<EMAIL_ADDRESS>V. Vilasini  Institute for Theoretical
Physics, ETH Zurich, 8093 Zürich, Switzerland<EMAIL_ADDRESS>
(April 2024)

###### Abstract

Formalisms for higher order quantum processes provide a theoretical
formalisation of quantum processes where the order of agents’ operations need
not be definite and acyclic, but may be subject to quantum superpositions.
This has led to the concept of indefinite causal structures (ICS) which have
garnered much interest. However, the interface between these information-
theoretic approaches and spatiotemporal notions of causality is less
understood, and questions relating to the physical realisability of ICS in a
spatiotemporal context persist despite progress in their information-theoretic
characterisation. Further, previous work suggests that composition of
processes is not so straightforward in ICS frameworks, which raises the
question of how this connects with the observed composability of physical
experiments in spacetime. To address these points, we compare the formalism of
quantum circuits with quantum control of causal order (QC-QC), which models an
interesting class of ICS processes, with that of causal boxes, which models
composable quantum information protocols in spacetime. We incorporate the set-
up assumptions of the QC-QC framework into the spatiotemporal perspective and
show that every QC-QC can be mapped to a causal box that satisfies these set
up assumptions and acts on a Fock space while reproducing the QC-QC’s
behaviour in a relevant subspace defined by the assumptions. Using a recently
introduced concept of fine-graining, we show that the causal box corresponds
to a fine-graining of the QC-QC, which unravels the original ICS of the QC-QC
into a set of quantum operations with a well-defined and acyclic causal order,
compatible with the spacetime structure. Our results also clarify how the
composability of physical experiments is recovered, while highlighting the
essential role of relativistic causality and the Fock space structure.

###### Contents

  1. 1 Introduction
  2. 2 Review of higher order quantum processes
    1. 2.1 General higher order quantum processes
    2. 2.2 Subset of processes modelled by generalised quantum circuits
  3. 3 Review of the causal box framework
    1. 3.1 Messages and Fock spaces
    2. 3.2 Definition of causal boxes
    3. 3.3 Representations of causal boxes
    4. 3.4 Composition of causal boxes
  4. 4 Extending QC-QCs to causal boxes
    1. 4.1 Overview of the results
    2. 4.2 When does a causal box model a QC-QC?
    3. 4.3 Photonic causal box realisation of the Grenoble process
    4. 4.4 Isometric extension
    5. 4.5 Projective extension
  5. 5 Insights on physical realisability in a spacetime
    1. 5.1 Fine-graining and spacetime realisations
    2. 5.2 All QC-QCs admit an acyclic fine-grained causal structure
  6. 6 Insights on composability
  7. 7 Conclusions and outlook
  8. A Mathematical tools
    1. A.1 Choi isomorphism
    2. A.2 Link product
    3. A.3 Symmetric tensor product and inner product in Fock spaces
    4. A.4 Isometries
  9. B More detailed review of QC-QCs
  10. C A general extension of QC-QCs inspired by photonic experiments
  11. D Fine-graining
  12. E Proofs of results
    1. E.1 Proofs for the main part
    2. E.2 Proofs for the appendix

##  1 Introduction

In recent years, significant progress has been made in understanding causal
influence in quantum theory, particularly regarding its deviation from
classical intuitions due to phenomena like superpositions and entanglement.
Various approaches to quantum causality have emerged, such as frameworks for
quantum causal models [1, 2, 3, 4, 5, 6], that provide causal explanations for
practical quantum experiments such as Bell scenarios, as well as theoretical
formalisms for higher order quantum processes which lead to more exotic
notions such as indefinite causal structures [7, 8, 9]. While the former
entails quantum operations of agents occurring in a definite and acyclic
order, the latter involves more general, abstract quantum protocols where the
order of operations is no longer fixed and definite.111We do not necessarily
endorse the terminology ”indefinite” causal structures for such protocols in
general but adhere to it in this paper for consistency with the literature.

These indefinite causal structures have been extensively studied for their
intriguing theoretical possibilities which extend beyond the standard quantum
circuit paradigm, and the potential applications they may offer for
information processing [10, 11, 12]. However, the physical realisability of
these theoretical processes, as well as what constitutes a “faithful”
realisation thereof, remains a highly discussed open problem in the field [13,
14, 15, 16, 17].

The framework of _quantum circuits with quantum control of causal order (QC-
QC)_ [16] describes a subset of processes which can be interpreted in terms of
generalised quantum circuits. These allow for processes which are associated
with an indefinite causal structure, and involve a quantum superposition of
the order of agents’ operations, which can be coherently controlled by the
state of some quantum system, or where one agent’s quantum/classical output
may dynamically determine the order in which future agents act. QC-QCs are
widely regarded as being physically realisable (in principle), but it is yet
to be formalised in what precise sense. Moreover, a crucial question is
whether this class encompasses _all_ physically realisable processes.

A related question concerns the composability of physical processes, due to
indications that composing even simple processes in typical frameworks for
indefinite causality is not so straightforward [18, 19]. Composability is
central to our understanding of physics, as the composition of two physical
experiments is another physical experiment. How does this potential difficulty
with compositions in the process framework reconcile with the observed
composability of real-world experiments? Previous approaches [19, 20] have
proposed consistent rules for composing (higher order) quantum processes.
However, the implications of these abstract rules for the composability of
quantum experiments in space and time have not been previously considered.

The aforementioned approaches operate within an information-theoretic
understanding of causality, based on the flow of information between systems.
This is a priori distinct from relativistic notions of causality related to
spacetime [21, 22, 23]. To formalise physical realisability and address
related questions, it is necessary to link such information-theoretic
structures to space and time and account for relativistic principles of
causality. In [23], a top-down framework was proposed for making precise the
link between these notions and formalising relativistic principles for general
quantum protocols in a _fixed and acyclic background spacetime_. This led to a
formalisation of the concept of realisation of a quantum process in such a
spacetime, allowing for scenarios where quantum messages exchanged between
agents222The agents and their labs are regarded as classical, although they
can perform quantum operations on quantum systems entering their labs. This
distinguishes the approach of [23] from frameworks for quantum reference
frames where the reference frame/observer is itself regarded as a quantum
system. can take superpositions of trajectories in the spacetime. There are
two important insights derived from this approach which are relevant for the
physicality question: the first concerns the concept of _fine-graining_ and
the second concerns the relation to the previously known framework of _causal
boxes_ [13]. We describe these two aspects in turn.

The concept of fine-graining of causal structures introduced in [23]
demonstrates that an abstract information-theoretic causal structure, which is
not acyclic, can unravel into an acyclic causal structure at a fine-grained
level once realised in space and time, without violating relativistic
causality principles. A simple and intuitive example of this phenomenon is the
following: if the demand $D$ and price $P$ of a commodity causally influence
each other, we have a cyclic causal structure between $D$ and $P$, although
the fine-grained description of the physical process is an acyclic one where
demand $D_{1}$ at time $t_{1}$ influences price $P_{2}$ at time $t_{2}>t_{1}$
which influences demand $D_{3}$ at time $t_{3}>t_{2}$ and so on. This enables
information-theoretic and spatiotemporal causality notions to be consistently
reconciled for quantum experiments in spacetime, both at a formal and a
conceptual level (see [23] and Section 5 for details).

The causal box framework [13] describes composable information processing
protocols within fixed acyclic spacetimes, allowing for scenarios where
quantum states may be sent or received at a superposition of different
spacetime locations. Originally developed for studying security notions in
relativistic quantum cryptography which remain stable under composition of
protocols [24], the formalism guarantees that composition of two or more
causal boxes results in yet another causal box. It has been shown through the
top-down approach of [23] that the most general protocols that can be realised
without violating relativistic causality in a background spacetime are those
that can be described as causal boxes.333This statement applies to all
protocols involving finite-dimensional quantum systems and a finite number of
information processing steps, not to the infinite-dimensional case in its
current form.

Here we focus on the question of physical realisability of processes in a
fixed background spacetime— arguably the regime in which current-day
experiments operate. The above-mentioned results of [23] enable the question
of physical realisability of processes in a fixed spacetime to be reduced to
asking which subset of processes can be modelled as causal
boxes.444Explicitly, since causal boxes describe the most general protocols in
a fixed spacetime, a necessary condition for a process to be realisable in a
fixed spacetime is that it admits a causal box model. This motivates us to
compare and map between the causal boxes and QC-QC frameworks. However, these
two frameworks, while sharing some common features, are quite different. For
example, causal boxes are composable and allow multiple rounds of information
processing and superpositions of different numbers of messages while
processes/QC-QCs are not immediately composable as suggested by [18], and only
consider agents in closed labs acting once on a single message. Thus, mapping
between the formalisms requires a careful analysis of the underlying objects,
state spaces, and spacetime information. We undertake such an analysis here,
and our results (which we summarise below) formally ground this important
class of processes within a spatiotemporal and operational perspective, and
shed light on their fine-grained causal structure as well as the composability
of physical experiments.

Summary of contributions Starting with a review of higher-order quantum
processes (including QC-QCs) in Section 2 and the causal box formalism in
Section 3, Section 4.2 delves into a detailed analysis of state spaces and
operations within the QC-QC and causal box frameworks. Incorporating the set-
up assumptions of the process formalism (such as the fact that each party can
act exactly once) in terms of the spacetime picture, we formally address the
question: what does it mean for a causal box to model/behave like a QC-QC
under these assumptions? We term such a causal box an extension of the
associated QC-QC. In Section 4, we demonstrate the existence of a causal box
extension for each QC-QC and construct multiple extensions. In Section 5.2, we
show the causal box extension of a QC-QC is a fine-graining of the QC-QC
according to the concept introduced in [23]. Importantly, although the QC-QC
may be associated with an indefinite causal structure, we show that the
resulting causal box always exhibits a definite and acyclic fine-grained
causal structure. This holds even as the causal box reproduces the QC-QC’s
action on relevant states, permitting each party to act once on a physical
system. In Section 6, we discuss how one can reconcile the suggested
difficulties in composition within the process framework [18] with the
observation that physical experiments in spacetime are composable, emphasizing
the interplay of relativistic causality principles, the Fock space structure
and the set-up assumptions of the process/QC-QC frameworks in this regard.
This work sets the foundation for a follow-up paper, where we explore the
reverse mapping, from general causal boxes to QC-QCs, to provide a tighter
characterisation of processes realisable in a fixed background spacetime.

##  2 Review of higher order quantum processes

###  2.1 General higher order quantum processes

The standard quantum circuit paradigm describes protocols with a well-defined
acyclic ordering between the different operations (or quantum gates). This is
consistent with a clear arrow of time. Recently, frameworks have been proposed
which go beyond this to define abstract information-processing protocols
without assuming the existence of a definite acyclic order between the
different operations, or the existence of a background spacetime structure [5,
8, 9, 16].

In general, we consider agents which apply quantum instruments
$\mathcal{M}_{A}^{a}=\\{\mathcal{M}_{A}^{x|a}\\}_{x}$ where
$\mathcal{M}_{A}^{x|a}:\mathcal{L}(\mathcal{H}^{A^{I}})\rightarrow\mathcal{L}(\mathcal{H}^{A^{O}})$
[25] where $\mathcal{H}^{A^{I}}$ is the agent’s input Hilbert space while
$\mathcal{H}^{A^{O}}$ is their output Hilbert space, $a$ is the measurement
setting and $x$ the measurement outcome. However, for our purposes, it will
actually be enough to just consider generic CP maps $\mathcal{M}_{A}$ as we
will be interested in generic outcome probabilities. We will refer to such
operations as local operations.

Figure 1: A higher order process (pink) taking two local operations (blue) as
input. The result is a map from some global past $P$ to some global future
$F$.

We can then view a general quantum process as a higher order map $\mathcal{W}$
which acts on $N$ local operations and maps them to another operation (CP map)
$\mathcal{M}:\mathcal{L}(\mathcal{H}^{P})\rightarrow\mathcal{L}(\mathcal{H}^{F})$,
where $\mathcal{H}^{P/F}$ correspond to the Hilbert spaces associated with the
global past/future (relative to the remaining operations)

$\mathcal{W}:\\{\mathcal{M}_{A_{1}},...,\mathcal{M}_{A_{N}}\\}\mapsto\mathcal{M}.$
(1)

This is depicted diagrammatically in Fig. 1 for the case of two agents $A$ and
$B$. If $\mathcal{H}^{P/F}$ are trivial (one-dimensional), $\mathcal{W}$
encodes the joint probability associated with the CP maps of the local
operations.

The Choi representation of such a higher order process is also known as the
process matrix [8],
$W\in\bigotimes_{i=1}^{N}\mathcal{L}(\mathcal{H}^{A_{i}^{I}A_{i}^{O}})\otimes\mathcal{L}(\mathcal{H}^{PF})$,
which lives in the joint space of all the agents’ in and outputs and the
global past and future. The action of the supermap on the local operations can
be written in terms of the process matrix. The result is the Choi
representation of the map $\mathcal{C}$ which lives in
$\mathcal{L}(\mathcal{H}^{PF})$,

$W(\mathcal{M}_{A_{1}},...,\mathcal{M}_{A_{N}})\coloneqq(M_{A_{1}}\otimes...\otimes
M_{A_{N}})*W\in\mathcal{L}(\mathcal{H}^{PF})$ (2)

where $M_{A_{i}}$ denotes the Choi matrix of the local operation
$\mathcal{M}_{A_{i}}$ and $*$ is the link product. For a review of the link
product we refer to Section A.2. The outcome probabilities for a given state
$\rho^{P}\in\mathcal{H}^{P}$, and local operations associated with particular
outcomes and settings can be obtained via the generalised Born rule,

$P(x_{1},...x_{N}|a_{1},...,a_{N})=\operatorname{tr}_{F}[(\rho^{P}\otimes
M_{A_{1}}^{x_{1}|a_{1}}\otimes...\otimes M_{A_{N}}^{x_{N}|a_{N}})*W].$ (3)

For the remainder of this work we will work in the process matrix formulation.
However, one could equivalently work in the supermap formulation of [9] and as
such our results apply to both equally.

It is often much easier to work with process vectors [26] instead of process
matrices. While the process matrix can be viewed as the Choi matrix of the
environment, the process vector is essentially the corresponding Choi vector.
If the process vector is given by $\ket{w}$, then the process matrix is simply
$W=\ket{w}\bra{w}$. Instead of Eq. 2, we can then use

$(\mathinner{|{A_{1}}\rangle\\!\rangle}\otimes...\otimes\mathinner{|{A_{N}}\rangle\\!\rangle})*\ket{w}.$
(4)

where $\mathcal{M}_{A_{k}}(\rho)=A_{k}\rho A_{k}^{\dagger}$ and
$\mathinner{|{A_{k}}\rangle\\!\rangle}$ is the Choi vector of $A_{k}$ (cf.
Section A.1). In such cases, we will also refer to $A_{k}$ as the local
operation.555Note that we use $A_{k}$ to refer to both the local agent and the
single Kraus operator describing the pure operation that this agent applies.
This will allow us to keep equations compact while it should be clear from
context whether the agent or the Kraus operator is meant.

###  2.2 Subset of processes modelled by generalised quantum circuits

In general, interpreting the causal structures described by process matrices
is difficult. As discussed above, the framework can model very general
scenarios as it does not assume a background spacetime and a long-standing
open question is to understand which process matrices can be physically
realised and under what assumptions and physical regimes. In particular, there
are so-called non-causal processes that produce correlations which violate
bounds known as causal inequalities666These bounds on correlations are set by
the causal processes, i.e. those that are compatible with some definite
acyclic causal order or a convex combination of several such orders.. Such
processes can provide an advantage over more conventional processes (e.g.,
those with a definite, acyclic order) in certain information-theoretic games
[8].

The framework of quantum circuits with quantum control of causal order (QC-QC)
[16], adopts a bottom-up approach to this problem and defines a broad class of
process matrices that can be represented in terms of generalised quantum
circuits. As we will see, they can be viewed as circuits in the sense that the
local operations can be “plugged in” with their causal order being quantum
coherently or classically controlled. They are, however, more general than
standard quantum circuits as they include processes which are regarded as
having an indefinite causal structure. Nevertheless, QC-QCs have been shown to
not violate any causal inequalities (analogous to entangled states that do not
violate Bell inequalities).

A framework with similar results was developed in [27], we will, however,
focus on the QC-QC framework in this work. In this section, we will only
briefly summarise the main features of the QC-QC framework in words. This
should be enough to understand the core concepts and arguments presented in
the main text. A more mathematical summary can be found in Appendix B, whereas
for the full description we refer to the original paper [16].

(a) Quantum circuits with fixed order (QC-FO) are the standard quantum
circuits where the order of operations is fixed and determined.

(b) Quantum circuits with classical control of causal order (QC-CC) apply a
measurement during each time step (to the target or to some ancilla or to both
together). Each possible measurement outcome corresponds to an agent that has
not acted so far (therefore, if there are $N$ agents, in the $n$-th time step
there will be up to $N-n+1$ possible outcomes).

(c) Instead of applying a measurement as is the case for QC-CCs, a quantum
circuit with quantum control of causal order (QC-QC) sends the target system
in coherent superposition to (a subset of) all agents that have not acted so
far. Unlike in the case of QC-CCs, where the causal order is not predetermined
but still well defined, the target and the control are thus in a superposition
of different “paths”.

Figure 2: The three types of classifications for objects in the QC-QC
framework. The figures are taken from [28], where they are listed Fig. 4, Fig.
7 and Fig. 10

QC-QCs can be divided into three categories. Quantum circuits with fixed order
(QC-FO) are essentially standard quantum circuits with “slots” where an
external operation can be plugged in and the order in which operations are
applied by the circuit is fixed and acyclic. These objects are also referred
to as quantum combs [29]. Quantum circuits with classical control of causal
order (QC-CC) allow for classical mixtures, including in a dynamical fashion,
of definite acyclic orders. Here, the circuit applies a measurement during
each time step (to the target, an ancilla or to both together) and depending
on the classical measurement outcome, sends the target system to an agent who
has not yet acted so far. Thus at each time step, each possible measurement
outcome corresponds to an agent that has not acted so far (therefore, if there
are $N$ agents, in the $n$-th time step there will be up to $N-n+1$ possible
outcomes). The order is thus ultimately classical, but is not predetermined
and may be established dynamically during run time. Note that the applied
measurement must in general depend on which agents already acted. This can be
achieved by recording the agents that already acted in a control system which
is carried along and updated in the circuit. QC-FOs are a subset of QC-CCs as
they can be viewed as QC-CCs with a single measurement outcome in each time
step.

Finally, there is general QC-QCs of which QC-CCs (and thus also QC-FOs) are a
subset. The main difference is that instead of applying a measurement, the
circuit sends the target in coherent superposition to the agents. This is
controlled via a control system $\ket{\\{k_{1},...,k_{n}\\},k_{n+1}}$ where
$k_{1},...,k_{n}$ refers to the agents that acted previously (without
revealing the order in which they acted) and $k_{n+1}$ refers to the agent the
circuit sends the target to. We will usually write this in the more compact
form $\ket{\mathcal{K}_{n},k_{n}}$ with
$\mathcal{K}_{n}=\\{k_{1},...,k_{n}\\}$. Unlike in the case of QC-CCs, which
have only classical uncertainty in the causal order, QC-QCs enable a quantum
uncertainty in the order of the operations.

We note that which agent receives the target at a given time step can also
depend dynamically on the state of the target (in addition to depending on the
state of the control system). A simple example from the physical world is
given by a circuit consisting of a beam splitter: Alice sends a photon to the
circuit, and depending on the polarization of Alice’s photon, the beam
splitter then reflects it to Bob or transmits it to Charlie. Additionally, the
circuit may apply some transformation to the target before sending it to the
next agent.

##  3 Review of the causal box framework

The causal box framework [13] models information processing protocols
satisfying relativistic causality in a fixed background spacetime (such as
Minkowski spacetime), where quantum messages may be exchanged in
superpositions of different spacetime locations. The framework is, however,
very different from the process matrix or QC-QC approaches as it allows for
multiple rounds of information processing, is closed under arbitrary
composition and does not partition protocols into local operations of agents
and processes describing an inaccessible (to the agents) environment.
Additionally, it explicitly models sending “nothing” with a vacuum state
$\ket{\Omega}$. The existence of such a state and the possibility of sending
superpositions of “something” and “nothing”,
$\alpha\ket{\psi}+\beta\ket{\Omega}$, has physical relevance. For example, the
coherently controlled application of an unknown unitary to a target system was
shown to be impossible in theoretical formalisms that do not model the vacuum,
but has been experimentally realised due to the physical possibility of such
“vacuum” superpositions [30, 31].

###  3.1 Messages and Fock spaces

We now review the formal aspects of the causal box framework. Causal boxes are
defined on a background spacetime which is modelled as a partially ordered set
$\mathcal{T}$, capturing the light cone structure of the spacetime. However,
for our purposes it will suffice to consider finite and totally ordered sets
(in which case $\mathcal{T}$ can be interpreted as a set of time stamps). For
simplicity, we will therefore restrict to this case in this review as well. We
refer to [13] for the general case. A message corresponds to a Hilbert space
$\mathcal{H}^{A}$ together with a time stamp encoded in the sequence space
$l^{2}(\mathcal{T})$ of $\mathcal{T}$ with bounded 2-norm. We can write the
state of an arbitrary message as
$\ket{\psi}^{A}\otimes\ket{t}\in\mathcal{H}^{A}\otimes l^{2}(\mathcal{T})$,
where the content of the message is encoded in $\ket{\psi}^{A}$, while
$t\in\mathcal{T}$ contains the time information of when the message is sent or
received. We will frequently write $\ket{\psi,t}^{A}$ instead of
$\ket{\psi}^{A}\otimes\ket{t}$.

For process matrices and QC-QCs, the state space of a wire is a finite-
dimensional Hilbert space. As mentioned earlier, the causal box framework
allows the sending of multiple messages or, in other words, there can be any
number of messages on a wire. To capture this, the state space of the wire in
the causal box framework is modelled as the symmetric Fock space of the
single-message space

$\mathcal{F}(\mathcal{H}^{A}\otimes
l^{2}(\mathcal{T}))=\bigoplus_{n=0}^{\infty}\vee^{n}(\mathcal{H}^{A}\otimes
l^{2}(\mathcal{T}))$ (5)

where $\vee^{n}(\mathcal{H}^{A}\otimes l^{2}(\mathcal{T}))$ is the symmetric
subspace of $(\mathcal{H}^{A}\otimes l^{2}(\mathcal{T}))^{\otimes n}$. The
one-dimensional space $(\mathcal{H}^{A}\otimes l^{2}(\mathcal{T}))^{\otimes
0}$ corresponds to the vacuum state $\ket{\Omega}$. For more details on the
symmetric tensor product, see Section A.3. The notation in Eq. 5 is quite
cumbersome so we will often abbreviate it by writing
$\mathcal{F}^{\mathcal{T}}_{A}:=\mathcal{F}(\mathcal{H}^{A}\otimes
l^{2}(\mathcal{T}))$.

#### Wire isomorphisms:

There exist two useful isomorphisms regarding the splitting of wires. For any
$\mathcal{H}^{A},\mathcal{H}^{B}$ and any
$\mathcal{T}^{\prime}\subseteq\mathcal{T}$, we have

$\displaystyle\begin{aligned}
\mathcal{F}^{\mathcal{T}}_{C}&\cong\mathcal{F}^{\mathcal{T}}_{A}\otimes\mathcal{F}^{\mathcal{T}}_{B}\\\
\mathcal{F}^{\mathcal{T}}_{A}&\cong\mathcal{F}^{\mathcal{T}^{\prime}}_{A}\otimes\mathcal{F}^{\mathcal{T}\backslash\mathcal{T}^{\prime}}_{A}\end{aligned}$
(6)

where $\mathcal{H}^{C}=\mathcal{H}^{A}\oplus\mathcal{H}^{B}$.

The first isomorphism implies that two wires, one carrying messages from one
Hilbert space $\mathcal{H}^{B}$ and the other carrying messages from another
Hilbert space $\mathcal{H}^{C}$, are equivalent to a single wire carrying
messages from the direct sum of the two Hilbert spaces. This will allow us to
formally define causal boxes with just a single input and a single output
wire.

The second isomorphism applied recursively implies that there is an
equivalence between one wire carrying the messages from all times
$t\in\mathcal{T}$ and having a separate wire for each $t\in\mathcal{T}$.
Throughout this paper, we will treat messages with different time stamps are
being associated with different wires, such that states of multiple messages
associated with distinct time stamps can be written using the regular tensor
product (rather than a symmetrised product). Furthermore, for any
$\mathcal{T}^{\prime}\subseteq\mathcal{T}$, there is a natural embedding of
$\mathcal{F}^{\mathcal{T}^{\prime}}_{A}$ in $\mathcal{F}^{\mathcal{T}}_{A}$
given by appending the vacuum state on
$\mathcal{T}\backslash\mathcal{T}^{\prime}$.

$\mathcal{F}^{\mathcal{T}^{\prime}}_{A}\cong\mathcal{F}^{\mathcal{T}}_{A}\otimes\ket{\Omega,\mathcal{T}\backslash\mathcal{T}^{\prime}}.$
(7)

###  3.2 Definition of causal boxes

We give now a simplified definition of causal boxes that captures those causal
boxes defined on a finite and totally ordered $\mathcal{T}$.

######  Definition 1 (Causal boxes [13]).

A causal box is a system with an input wire $X$ and an output wire $Y$,
together with a CPTP map

$\mathcal{C}:\mathcal{L}(\mathcal{F}^{\mathcal{T}}_{X})\rightarrow\mathcal{L}(\mathcal{F}^{\mathcal{T}}_{Y})$
(8)

that fulfills for all $t\in\mathcal{T}$

$\operatorname{tr}_{>t}\circ\mathcal{C}=\operatorname{tr}_{>t}\circ\mathcal{C}\circ\operatorname{tr}_{>t}$
(9)

where $\operatorname{tr}_{>t}$ corresponds to tracing out all messages with
time stamps larger than $t$.

Eq. 9 encodes the causality condition, i.e. that we can calculate the outputs
up to time $t$ from the inputs up to $t$ which is equivalent to saying that
inputs after $t$ cannot influence outputs up to $t$. Note that from this
definition it directly follows that a causal box restricted to some subset of
timestamps less than $t$ is once again a causal box.

###  3.3 Representations of causal boxes

Causal boxes admit two alternative representations [13]. One of these is the
Choi representation. For infinite-dimensional Hilbert spaces, the usual Choi
operator can be unbounded and an alternate definition has to be used in those
cases [32]. For the definition in this general case, see [13]. However, we
note that the space $\bigoplus_{n=0}^{N}\vee^{n}(\mathcal{H}^{A}\otimes
l^{2}(\mathcal{T}))$ of up to $N$ messages is finite-dimensional for all $N$.
By restricting the causal box to this domain, we can thus use the normal Choi
operator. This restriction will not lead to any loss of generality in our
results because defining the causal box for every $N$ uniquely determines the
causal box on the full Fock space where we have a direct sum from $N=0$ to
$N=\infty$, as explained in Remark 1.

The other representation is the sequence representation, which is essentially
a consequence of the Stinespring dilation [33] of a CPTP map. As causal boxes
are CPTP maps, they also admit Stinespring dilations [13]. One can use this
fact to decompose a causal box into a sequence of isometries, each of which
describes the behaviour of the causal box during some disjoint subset of
$\mathcal{T}$ (see Fig. 3). This is called a sequence representation of the
causal box. This representation can thus be viewed as a causal unraveling of
the causal box. On the one hand, this is a useful way to visualise the
behaviour of a causal box and on the other hand, we can also use this to
construct causal boxes. A sequence of isometries $V_{n}$ with appropriate
input and output spaces will always yield a causal box.

Figure 3: Graphical representation of the sequence representation of a causal
box. The causal box is decomposed into a sequence of isometries $V_{n}$ (in
this case there are four such isometries). Each isometry acts on messages
during a specific time slice (named $\mathcal{T}_{n}$ for $n=1,2,3,4,5$ in the
figure) and outputs messages during the next time slice. A causal box can thus
be understood via its action during each time slice.

######  Remark 1.

Note that defining a linear map on each $n$-message subspace does not a priori
mean it is well-defined on the full Fock space. This is because an infinite-
dimensional Hilbert space is not simply the (finite) linear span of some
orthonormal basis ${\ket{j}}_{j\in\mathbb{N}}$ but the metric completion of
such a span (which contains also any elements corresponding to the limits of
Cauchy sequences). A counter-example is the linear map $V\ket{j}=j\ket{j}$. We
then have, for example,
$V\sum_{j\in\mathbb{N}}1/j\ket{j}=\sum_{j\in\mathbb{N}}\ket{j}$ which is
divergent in its norm. We additionally need that the map is continuous. That
this then suffices follows straightforwardly from the fact that the action of
a continuous function $f$ on a limit $a$, $f(a)$, must be the limit of
$f(a_{n})$ for a sequence $\\{a_{n}\\}_{n\in\mathbb{N}}$ that converges
towards $a$. In the case of causal boxes, we can always consider a
purification, which is an isometry and thus continuous. Therefore, it suffices
to work with maps on the $n$-message subspaces. Since Choi vectors and
matrices uniquely determine the corresponding maps (once we fix a basis) the
same argument justifies working with the finite-dimensional $n$-message Choi
representations as well.

###  3.4 Composition of causal boxes

Causal boxes can be composed in the following ways: _parallel composition_ of
two causal boxes is simply given by their tensor product, _sequential
composition_ involves connecting an output of a causal box at some time to the
input of a causal box at a later time, and _loop composition_ is an operation
on a single causal box which feeds back an output of the box to an input of
the same box associated with a later time. Sequential composition of two
causal boxes can be expressed in terms of the other two by first taking the
parallel composition of the two boxes and then performing an appropriate loop
composition [13, 23]. It can be shown that the set of valid causal boxes are
closed under arbitrary compositions of these types.

We give the definition of loop composition below. As for the Choi
representation, we will once again only consider the finite-dimensional case.
This is again sufficient to determine the behaviour on the full Fock space
because of the reasoning given in Remark 1.

######  Definition 2 (Loop composition [13]).

Consider a CP map
$\mathcal{M}:\mathcal{L}(\mathcal{H}^{AB})\rightarrow\mathcal{L}(\mathcal{H}^{CD})$
with input systems $A$ and $B$ and output systems $C$ and $D$ with
$\mathcal{H}^{B}\cong\mathcal{H}^{C}$. Let $\\{\ket{k}^{C}\\}_{k}$ be any
orthonormal basis of $\mathcal{H}^{C}$, and denote with
$\\{\ket{k}^{B}\\}_{k}$ the corresponding basis of $\mathcal{H}^{B}$ i.e. for
all $k$, $\ket{k}^{C}\cong\ket{k}^{B}$. The new system resulting from looping
the output system $C$ to the input system $B$, $\mathcal{M}^{C\hookrightarrow
B}$ is given as

$\mathcal{M}^{C\hookrightarrow
B}(\ket{\psi}^{A}\bra{\phi}^{A})=\sum_{k,l}\bra{k}^{C}\mathcal{M}(\ket{\psi}^{A}\ket{k}^{B}\bra{l}^{B}\bra{\phi}^{A})\ket{l}^{C}.$
(10)

The sequential composition of two CP maps
$\mathcal{M}_{A}:\mathcal{L}(\mathcal{H}^{A})\rightarrow\mathcal{L}(\mathcal{H}^{C})$
and
$\mathcal{M}_{B}:\mathcal{L}(\mathcal{H}^{B})\rightarrow\mathcal{L}(\mathcal{H}^{D})$
is then given by $(\mathcal{M}_{A}\otimes\mathcal{M}_{B})^{C\hookrightarrow
B}$.

######  Remark 2 (Basis dependence of the loop composition).

Note that the composition is basis-dependent in the same sense that the Choi
isomorphism and the link product are basis-dependent. The bases we use for
composition should thus be the same bases that we use to calculate Choi
matrices and link products to obtain consistent results.

##  4 Extending QC-QCs to causal boxes

###  4.1 Overview of the results

Here, we outline the ingredients behind the main theorems of this paper, the
statement of the results along with their relevant implications for causality
and composability. As highlighted in the introduction, characterising the
relation between QC-QCs and causal boxes is important for understanding the
physicality and composability of abstract higher order quantum processes, in
the context of their realisations in a background spacetime. The frameworks
are rather different a priori, and it is necessary to identify the subset of
protocols described by causal boxes that satisfy the set-up assumptions of the
process matrix/QC-QC frameworks [34]. We describe these assumptions below.

######  Assumption 1 (Acting once and only once).

Whenever a party in a protocol described by a QC-QC acts on a $d$-dimensional
system, then in any causal box description of the protocol, the party must act
on exactly one non-vacuum message, and this non-vacuum message must also be of
$d$-dimensions.

######  Assumption 2 (Order of local in/output events).

Whenever a party in a protocol described by a QC-QC has non-trivial in and
output spaces, then in any causal box description of the protocol, that party
must receive a non-vacuum input to their lab before they send out any non-
vacuum output.

We refer to the conjunction of these two assumptions as the _spatiotemporal
closed labs assumption_. These assumptions and consequently the following
theorem, which is a main result of this paper, will be mathematically
formalised in Section 4.2. The proof of the theorem is a consequence of Prop.
2 in Section 4.4.

######  Theorem 1.

Every protocol described by a QC-QC can be mapped to a protocol described by
valid causal boxes in Minkowski spacetime, which reproduce the action of the
QC-QC on a well-defined subspace and also respect the spatiotemporal closed
labs assumption.

As a consequence of this theorem, the subset of causal boxes in the image of
our mapping have certain natural and operationally motivated properties which
we consider necessary (making no claims about sufficiency)777Satisfying these
assumptions, while necessary for faithfully realising an indefinite causal
order process in spacetime, is not sufficient for regarding the realisation as
a genuinely indefinite causal structure. Even when these properties are
satisfied, such as in the experimental realisations of the quantum switch,
there has been a long-standing debate about whether these implement or
simulate indefinite causality. In fact, the results of [23] and Section 5 show
that such spacetime realisations will always, as a consequence of relativistic
causality, admit an explanation in terms of a definite acyclic causal
structure, lending support to the side of the debate that they are simulations
of ICO. These conditions ensure that regardless of whether these realisations
are regarded as simulations or implementations, they are faithful to the set
up assumptions respected by the original abstract process. for regarding the
causal box as a faithful spatiotemporal realisation of the QC-QC. Moreover,
the causal boxes in the image of our mapping also encode the spatiotemporal
degrees of freedom in a minimal manner, and satisfy the following property
that simplify their mathematical representation. We map $N$-partite QC-QCs to
causal boxes where each party can act at $N$ distinct pairs of in/output times
$\\{t^{I}_{1},t^{O}_{1},...,t^{I}_{N},t^{O}_{N}\\}$ such that every non-vacuum
input at $t_{i}^{I}$ yields a non-vacuum output at the corresponding output
time $t_{i}^{O}>t_{i}^{I}$.

#### Fine-grained causal structure of QC-QCs

When considering the realisation of indefinite causal order processes in a
fixed spacetime, one is faced with a fundamental question: how can an
indefinite information-theoretic causal structure be consistent with a
definite spacetime causal structure? This question has been resolved in [23]
where it is shown that a realisation of any (possibly indefinite causal order)
process satisfying relativistic causality in a fixed spacetime will ultimately
admit a fine-grained description in terms of a fixed and acyclic causal order
process which is compatible with the light cone structure of the spacetime.
Intuitively, the picture painted by this result is similar to the use of
cyclic information-theoretic causal structures in classical statistics to
describe physical situations with feedback (e.g., the demand and price of a
commodity influence each other), but one is aware that this is a coarse-
grained description of an acyclic fine-grained causal structure (where demand
at a given time influences price at a later time and vice-versa). In this
previous work, the set of processes that can indeed be realised in a spacetime
in this manner, was not characterised. Here we link the causal boxes in the
image of the mapping provided in Theorem 1 to the definition of spacetime
realisation of a process proposed in [23], which allows us to regard the
causal box description as a fine-graining of the QC-QC description, and
consequently as a fixed spacetime realisation of the QC-QC in the formal sense
defined in [23].

######  Theorem 2.

Every QC-QC can be mapped to a causal box which reproduces the action of the
QC-QC on a subspace, satisfies the spatiotemporal closed labs assumption, is a
fine-graining of the QC-QC, where the fine-grained causal order is definite
and acyclic and consistent with relativistic causality principles in
spacetime.

We give an outline of the proof of this theorem in Section 5.2, while the
detailed proof can be found in the appendix, Appendix D.

Our results imply that the set of processes realisable in a spacetime without
violating relativistic causality, must be at least as large as QC-QCs. This
also establishes that for each QC-QC, the indefinite causal structure (if
present) can be explained in terms of a fine-grained acyclic causal structure
(that of the causal box in the image of our mapping). Showing that a causal
box is a fine-graining of a QC-QC also provides an explicit mapping back from
the causal box (fine-grained picture) to the QC-QC (coarse-grained picture)
through an encoding-decoding scheme, which is analogous to the mapping from
physical layer operations and corresponding logical layer description of a
quantum computation.

#### Insights on composability of QC-QCs

The notion of composition as well as the closedness of the set of physical
operations under composition is absolutely central to physical protocols and
experiments (and is respected by causal boxes). However, there are no-go
theorems indicating apparent limitations for the composability of process
matrices, even those which have a fixed causal order (i.e which are QC-FOs).
Our mapping from QC-QCs to causal boxes sheds new light on how process
matrices, despite this apparent composability issue, can consistently recover
our standard intuitions about composability when physically realised in a
spacetime. This connects to a previous resolution of the issue proposed within
a purely informational and abstract approach [19, 20], while offering new
dimensions to the solution coming from relativistic causality, the use of Fock
spaces, as well as the interplay between the sets of causal boxes that do or
do not respect the set up assumptions of the process matrix and QC-QC
frameworks. We discuss this in Section 6.

###  4.2 When does a causal box model a QC-QC?

As a first step towards our goal of mapping QC-QCs to causal boxes, we need to
ask when can we regard a causal box as reproducing the behaviour of a QC-QC.
As we have already pointed out, the two frameworks differ in a number of
points. In particular, causal boxes account for spacetime information and use
Fock spaces as their state spaces, neither of which are explicitly modelled in
the QC-QC framework. However, when we map a QC-QC to a causal box we want the
two to behave in “the same way” at least “in situations of interest”. The aim
of this section is to formalise this statement. More specifically, we want a
mapping from local operations in the QC-QC framework to the causal box
framework and of the QC-QC supermap to the causal box supermap such that when
composing the local operations with the supermap in the respective frameworks
yields the same transformation.

In the following, let us assume that $\mathcal{T}=\\{1,2,...,2N+2\\}$ for some
$N\in\mathbb{N}$ as this will simplify our discussion. Additionally, let us
split $\mathcal{T}$ into the set of input times
$\mathcal{T}^{I}=\\{2,4,...,2N+2\\}$ and the set of output times
$\mathcal{T}^{O}=\\{1,3,...,2N+1\\}$888These assumptions and others we will
make in this section may seem too restrictive at a glance. One may think that
they come with some loss of generality. However, in the end, for each QC-QC,
we only wish to find one corresponding causal box. So, in principle, we can
make arbitrary restrictions to the kind of causal boxes we consider as long as
we can still achieve this goal. However, naturally, these restrictions make it
impossible for us to find all causal box extensions of a given QC-QC. One
could also say that the QC-QC describes a general and abstract information-
theoretic protocol whereas the causal box describes a specific implementation
in spacetime of an experiment, of which there may be many..

We now model 1. The QC-QC picture involves agents $A_{k}$ with input and
output spaces $\mathcal{H}^{A^{I}_{k}}$ and $\mathcal{H}^{A^{O}_{k}}$. It
models superpositions of orders by identifying these spaces with the generic
spaces $\mathcal{H}^{\tilde{A}^{I}_{n}}$ and $\mathcal{H}^{\tilde{A}^{O}_{n}}$
which are labelled by the index $n$. The label $n$ of the generic space
assigned to an agent $A_{k}$ specifies the order in which the agent acts, and
is determined (possibly coherently) by a control system (cf. Fig. 2(c)). Thus,
it is natural to map these generic spaces to the state space in the causal box
picture at a specific time, i.e. $n\leftrightarrow t=2n$ or $n\leftrightarrow
t=2n+1$ depending on whether we are looking at outputs or inputs.

(a) In the QC-QC framework each agent is initially associated with a single
input and output wire in between which they apply their operation (left). The
output spaces are then identified with generic spaces, which specify the order
in which the agent acts, and we can think of there being one generic space for
each time step, yielding the picture to the right. Superpositions of orders
are modelled by assigning generic spaces to agents in a coherently controlled
manner. In a given branch of the superposition, the agent acts during one time
step (specified by the label $n$ of the generic space assigned to them) and is
inactive during all other times.

(b) In the causal box framework the agent can be viewed as having an input and
output wire for each time step (left) and acting during all time steps which
yields the picture to the right. Superpositions of orders are achieved through
superpositions of vacuum and non-vacuum states, in each branch of the
superposition, the agent acts on a non-vacuum state at one time and the vacuum
state at all other times (as ensured by Assumptions 1 and 2). On a non-vacuum
input, the original QC-QC operation $A_{k_{n}}$ is applied while a vacuum
state input is mapped to a vacuum output at the next time step. Thus, under
these assumptions, at each time step, the agent effectively applies the same
operation $*$, which stands for
$A_{k_{n}}\otimes\ket{t+1}\bra{t}+\ket{\Omega,t+1}\bra{\Omega,t}$. This is
precisely a single tensor factor of Eq. 17, with
$A_{k_{n}}\otimes\ket{t+1}\bra{t}$ taking the place of $\bar{A}^{x,t}$ for all
times $t$, capturing the time-independence of the local operations as in the
QC-QC picture.

Figure 4: Local operations in the QC-QC vs causal boxes pictures, and
different ways to capture superpositions of orders.

(a) The QC-QC is a supermap acting on the channels corresponding to the local
operations of the agents. It can be equivalently represented as a channel in
itself, a simplified picture of the statespace of a QC-QC is shown here (in
its channel representation) where it is a map from the agents’ outputs to
their inputs. There is a single input/output wire for each agent.

(b) In the causal box framework, the causal box modelling the supermap can be
seen as having wires not just for each agent but also for each time stamp.
Once we impose that agents act once and only once (1) and that agents only
send a non-vacuum message after receiving a non-vacuum message (2), we obtain
the picture on the right, where we can view the causal box as having only a
single wire for each agent, which is the wire along which they receive a non-
vacuum state in a given “branch” of the superposition (hence the sum on the
right).

Figure 5: The supermap (represented as a channel) in the QC-QC and causal
boxes pictures, which would act on the local operations of Figs. 4(a) and 4(b)
respectively through composition of systems with the same labels.

Figures 4(a) and 4(b) illustrate the correspondence. The input space
$\mathcal{H}^{A^{I}_{k}}$ is isomorphic to all $N$ generic input spaces
$\mathcal{H}^{\tilde{A}^{I}_{n}}$, but in a given branch of the temporal
superposition, the QC-QC framework only identifies it with one of them. This
is because of the assumption that each agent only acts once during a run of
the experiment modelled by the QC-QC. The same holds for the output spaces. On
the other hand, we model the agent $\bar{A}_{k}$ (where from here on out we
will put a bar over objects in the causal box framework to distinguish them
from the corresponding objects in the QC-QC framework) in the causal box
picture as another causal box with an input wire for each
$t\in\mathcal{T}^{I}$ and an output wire for each $t\in\mathcal{T}^{O}$. 1
then corresponds to imposing that in a given branch of the temporal
superposition, only one of these wires will contain a non-vacuum state of the
dimension given by the corresponding in/output system of the QC-QC999To be
clear, this assumption is not satisfied when composing the agent with
arbitrary causal boxes. This is thus an imposition on the whole set-up, and
not just the agent alone. For example, consider the causal box with a single
output wire and trivial input wire that outputs a qubit at $t=1$ and another
one at $t=2$. If we compose this causal box with an agent, the agent receives
multiple messages.

$\displaystyle\begin{aligned}
\mathcal{H}^{\tilde{A}^{I}_{n}}&\leftrightarrow\mathcal{H}^{\bar{A}^{I}_{k}}\otimes\ket{t=2n}\cong\mathcal{H}^{\bar{A}^{I,t=2n}_{k}}\\\
\mathcal{H}^{\tilde{A}^{O}_{n}}&\leftrightarrow\mathcal{H}^{\bar{A}^{O}_{k}}\otimes\ket{t=2n+1}\cong\mathcal{H}^{\bar{A}^{O,t=2n+1}_{k}},\end{aligned}$
(11)

where $\mathcal{H}^{\tilde{A}^{I}_{n}}$ and $\mathcal{H}^{\bar{A}^{I}_{k}}$
are isomorphic and similar for the output spaces, and there is an equivalence
between writing timestamps in kets (middle) and writing them as system labels
(right). On the level of states, we can then make the following identification

$\displaystyle\begin{aligned}
\ket{\psi}^{\tilde{A}^{I}_{n}}\otimes\ket{\mathcal{K}_{n-1},k_{n}}&\leftrightarrow\ket{\psi,t=2n}^{\bar{A}^{I}_{k_{n}}}\ket{\Omega,\mathcal{T}^{I}\backslash
2n}^{\bar{A}^{I}_{k_{n}}}\ket{\mathcal{K}_{n-1},k_{n}}^{C_{n}}\\\
\ket{\psi}^{\tilde{A}^{O}_{n}}\otimes\ket{\mathcal{K}_{n-1},k_{n}}&\leftrightarrow\ket{\psi,t=2n+1}^{\bar{A}^{O}_{k_{n}}}\ket{\Omega,\mathcal{T}^{O}\backslash
2n+1}^{\bar{A}^{O}_{k_{n}}}\ket{\mathcal{K}_{n-1},k_{n}}^{C_{n}}\end{aligned}$
(12)

where we now also added the control system which allows us to make the
previous correspondence one-to-one.

Given these correspondences, let us now consider how the agents apply their
local operations in each framework. In terms of these time stamps, in the QC-
QC framework, the agent $A_{k_{n}}$ applies their local operation on an input
system only at the time step $t=2n$ (and outputting the result at $t=2n+1$),
associated with the generic space labelled by $n$, which can be determined
coherently by a control. The agent remains fully inactive during all other
time steps, as depicted in Fig. 4(a). In the causal box picture on the other
hand, each agent’s operation acts between every pair of input/output time
stamps but it may act on a vacuum or non-vacuum state depending coherently on
a control system. This gives us the picture in Fig. 4(b) for causal boxes.

2 is then modelled by imposing that the agents’ operations leave vacuum states
invariant. To formalise this, we consider the Kraus operators associated with
the local operations. Let
$A_{i}^{x}:\mathcal{H}^{A^{I}_{i}}\rightarrow\mathcal{H}^{A^{O}_{i}}$ denote
the Kraus operator associated with an outcome $x$ obtained by the $i$-th agent
in the QC-QC picture, where we take $A^{I}_{i}$ and $A^{O}_{i}$ to be
$d$-dimensional spaces.101010Here, $d$ is arbitrary and if the two spaces are
not of the same dimension the smaller one can be trivially enlarged so that
the dimensions match, as is also done in the QC-QC framework [16]. As noted
above, in the causal box picture the corresponding operation acts at each time
step, and we consider the operation of the agent associated with obtaining the
outcome $x$ when acting on a $d$-dimensional non-vacuum input at time $t$,
$\bar{A}_{i}^{x,t}:\mathcal{H}^{\bar{A}^{I}_{i}}\otimes\ket{t}\rightarrow\mathcal{H}^{\bar{A}^{O}_{i}}\otimes\ket{t+1}$.
As the local operation $A_{i}^{x}$ in the QC-QC picture is defined
independently of the order in which it is applied by the QC-QC, it is natural
to replicate this in the causal box picture by requiring
$\bar{A}_{i}^{x,t}:=A_{i}^{x}\otimes\ket{t+1}\bra{t}$ for all
$t\in\mathcal{T}^{I}$.

For vacuum states, we then need to impose that $\ket{\Omega,t}$ maps to
$\ket{\Omega,t+1}$ under the local operation in the causal box picture. This
ensures that a non-vacuum output at time $t+1$ must be preceded by a non-
vacuum input at $t$, thereby imposing the local order condition of 2. If the
agent obtains the outcome $x$ by measuring a non-vacuum message at time $t$,
then 1 and 2 impose that at all other times, they receive and send vacuum
states. Hence, we have the map

$\bar{A}^{x,t}_{i}\otimes\ket{\Omega,(t+1)^{c}}\bra{\Omega,t^{c}}$ (13)

where $\ket{\Omega,(t+1)^{c}}\bra{\Omega,t^{c}}=\bigotimes_{t^{\prime}\neq
t}\ket{\Omega,t^{\prime}+1}\bra{\Omega,t^{\prime}}$, and
$\bar{A}^{x,t}_{i}\ket{\Omega,t}=0$.

Generally, the agent may receive a non-vacuum state at any time
$t\in\mathcal{T}_{i}^{I}$ (possibly in a superposition of different arrival
times). Thus, the operation associated with obtaining $x$ (at any $t$) is the
sum over all $t\in\mathcal{T}^{I}$ of Eq. 13,

$\bar{A}^{x}_{i}\coloneqq\sum_{t\in\mathcal{T}^{I}_{i}}\bar{A}^{x,t}_{i}\otimes\ket{\Omega,(t+1)^{c}}\bra{\Omega,t^{c}}.$
(14)

This is a complete set of Kraus operators on the one-message subspace whenever
the operators $\bar{A}^{x,t}_{i}$ form a complete set for each time $t$, and
this is the case for us as we construct these operators from the QC-QC
operators $A^{x}_{i}$ for each $t$. We can explicitly check that
$\sum_{x}(\bar{A}^{x}_{i})^{\dagger}\bar{A}^{x}_{i}$ is equal to the identity
on the one message space,

$\sum_{x}\Bigg{(}\sum_{t\in\mathcal{T}^{I}_{i}}(\bar{A}^{x,t}_{i})^{\dagger}\otimes\ket{\Omega,(t+1)^{c}}\bra{\Omega,t^{c}}\Bigg{)}\Bigg{(}\sum_{t^{\prime}\in\mathcal{T}^{I}_{i}}\bar{A}^{x,t^{\prime}}_{i}\otimes\ket{\Omega,(t^{\prime}+1)^{c}}\bra{\Omega,t^{\prime
c}}\Bigg{)}$ (15)

Then denoting the identity operator on the $m$-message space at time $t$, up
to shifting $t$ to $t+1$, as $\mathbb{1}_{m}^{t}$, we see that
$\ket{\Omega,(t+1)}\bra{\Omega,t}=\mathbb{1}_{0}^{t}$. We will use the natural
short form
$\ket{\Omega,(t+1)^{c}}\bra{\Omega,t^{c}}=\mathbb{1}_{0}^{t^{c}}=\bigotimes_{t^{\prime}\neq
t}\mathbb{1}_{0}^{t}$. Then the above simplifies as follows, when we also note
that $\bar{A}^{x,t}\ket{\Omega,t}=0$ and $\langle
t|t^{\prime}\rangle=\delta_{t,t^{\prime}}$,

$\sum_{t\in\mathcal{T}^{I}_{i}}\sum_{x}(\bar{A}^{x,t}_{i})^{\dagger}\bar{A}^{x,t}_{i}\otimes\mathbb{1}_{0}^{t^{c}}=\sum_{t\in\mathcal{T}^{I}_{i}}\mathbb{1}_{1}^{t}\otimes\mathbb{1}_{0}^{t^{c}},$
(16)

which is precisely the identity operator on the overall one-message space
(over all times). Here we used the normalisation of the one-message Kraus
operators $\bar{A}^{x,t}$ at each time.

Further, Eq. 14 preserves the coherence of temporal superpositions, in
particular, if we have $A^{x}_{i}\ket{x}=\ket{x}$ for the original QC-QC
operators, then an eigenstate $\alpha\ket{x,t_{1}}+\beta\ket{x,t_{2}}$ of an
outcome $x$ arriving at a superposition of distinct times $t_{1}$ and $t_{2}$
(for any non-zero amplitudes $\alpha$ and $\beta$) will remain unaltered by
Kraus operators constructed above for the causal box picture.

We can also obtain an alternative representation of the local map in the
causal box picture, where we do not start by assuming 1, but when we do, we
recover the above operators. The map $\bar{A}_{i}^{x,t}$ acts on
$d$-dimensional non-vacuum messages arriving at time $t$. We can extend this
to a map on the $d+1$-dimensional space, with trivial action on vacuum (as
required by 2), by considering
$\bar{A}_{i}^{x,t}+\ket{\Omega,t+1}\bra{\Omega,t}$. As this map acts at each
time step, we have the following overall map in the causal box picture,

$\bigotimes_{t\in\mathcal{T}^{I}_{i}}(\bar{A}^{x,t}_{i}+\ket{\Omega,t+1}\bra{\Omega,t}).$
(17)

We can see that the maps defined in Eq. 14 and Eq. 17 are equivalent on the
one-message state space of each agent (i.e. when imposing 1) by direct
calculation,

$\displaystyle\begin{aligned}
\Bigg{(}\sum_{t\in\mathcal{T}^{I}_{i}}&\bar{A}^{x,t}_{i}\otimes\ket{\Omega,(t+1)^{c}}\bra{\Omega,t^{c}}\Bigg{)}\ket{\psi,t_{i}}^{\bar{A}^{I}_{i}}\ket{\Omega,t_{i}^{c}}^{\bar{A}^{I}_{i}}\\\
=&\bar{A}^{x,t_{i}}_{i}\ket{\psi,t_{i}}^{A^{I}_{i}}\otimes\ket{\Omega,(t_{i}+1)^{c}}\bra{\Omega,t_{i}^{c}}\ket{\Omega,t_{i}^{c}}^{\bar{A}^{I}_{i}}\\\
=&(\bar{A}^{x,t_{i}}_{i}+\ket{\Omega,t_{i}+1}\bra{\Omega,t_{i}})\ket{\psi,t_{i}}^{\bar{A}^{I}_{i}}(\bar{A}^{t_{i}^{c},x}_{i}+\ket{\Omega,(t_{i}+1)^{c}}\bra{\Omega,t_{i}^{c}})\ket{\Omega,t_{i}^{c}}^{\bar{A}^{I}_{i}}\\\
=&\bigotimes_{t\in\mathcal{T}^{I}_{i}}(\bar{A}^{x,t}_{i}+\ket{\Omega,t+1}\bra{\Omega,t})\ket{\psi,t_{i}}^{\bar{A}^{I}_{i}}\ket{\Omega,t_{i}^{c}}^{\bar{A}^{I}_{i}}\end{aligned}$
(18)

where we used that $\braket{\psi}{\Omega}=0$ and that
$\bar{A}^{x,t}_{i}\ket{\Omega,t}=0$ for all $t\in\mathcal{T}^{I}_{i}$. By
linearity of the two maps, it is clear that this equivalence extends to
superpositions and mixtures of states living in the one-message space.

Finally, we note that it suffices to consider agents with a single non-vacuum
outcome and thus a single Kraus operator $A_{i}$ (or $\bar{A}_{i}$ depending
on the framework). Any statement that holds for such agents also holds for the
general case via linearity. Based on the above, we define an extension of a
local operation in the QC-QC framework to one in the causal box framework as
follows.

######  Definition 3 (Extension of local operations).

Let $A_{i}$ be a local agent in the QC-QC framework who applies a local
operation $A_{i}:\mathcal{H}^{A^{I}_{i}}\rightarrow\mathcal{H}^{A^{O}_{i}}$.
We define the extension of this local agent $\bar{A}_{i}$ in the causal box
framework as a causal box with input/output Fock space
$\mathcal{F}(\mathcal{H}^{\bar{A}_{i}^{I/O}}\otimes
l^{2}(\mathcal{T}_{i}^{I/O}))$. We then call the local operation $\bar{A}_{i}$
in the causal box framework the extension of the local operation $A_{i}$ in
the QC-QC framework if
$\bar{A}_{i}=\bigotimes_{t\in\mathcal{T}^{I}_{i}}(A_{i}\otimes\ket{t+1}\bra{t}+\ket{\Omega,t+1}\bra{\Omega,t})$
on zero and one message states where the action of $A_{i}$ on
$\mathcal{H}^{\bar{A}^{I}_{i}}$ is defined via the trivial isomorphism.

A slightly modified version of this definition could also be applied to the
global past. We restrict the possible output times to $t=1$ so that the global
past is indeed in the past of all other agents. Since it has a trivial input,
i.e. corresponds to a state $\ket{\psi}^{P}$, the corresponding version in the
causal box picture is then simply the state $\ket{\psi}^{P}\otimes\ket{t=1}$.

######  Remark 3 (A full set of Kraus operators).

As previously mentioned, the Kraus operators Eq. 13 are only a complete set
(i.e., normalised) on the one-message subspace. As this is the case relevant
for modelling QC-QCs, it is not necessary to define the action of the agents
on multi-message states. However, a rather trivial extension to the full Fock
space can be achieved by adding (1) a single additional Kraus operator
$A^{m>1}_{i}$ (for each agent $A_{i}$) which is the projection onto the space
$\bigoplus_{m>1}\vee^{m}(\mathcal{H}^{A_{i}^{I}}\otimes l^{2}(\mathcal{T}))$,
i.e. the space of states consisting of more than one message (up to
relabelling the input system $A_{i}^{I}$ to output system $A_{i}^{O}$ and
shifting the time-stamp $t$ to $t+1$), and (2) the all-time vacuum projector
$A^{m=0}_{i}=\bigotimes_{t\in\mathcal{T}_{i}^{I}}\ket{\Omega,t+1}\bra{\Omega,t}$
for the zero-message space. $A^{m>1}_{i}$ and $A^{m=0}_{i}$ act as the
identity on the $m>1$ and $m=0$ message subspaces respectively, and together
with the complete set we have constructed above for $m=1$, these yield a
complete set of Kraus operators for the full space.

######  Remark 4 (Vacuum extensions).

We note that our definition of Kraus operators can be viewed as a vacuum
extension [29]. In the framework of [29], the vacuum extension of a channel
$A^{x}$ defined on a non-vacuum space, takes the form
$\bar{A}^{x}=A^{x}\oplus\gamma^{x}\ket{\Omega}\bra{\Omega}$ with
$\sum_{x}|\gamma^{x}|^{2}=1$. Equation 13 (or rather the operators
$\bar{A}^{x,t}_{i}$ which are related to Eq. 13 via the vacuum isomorphism Eq.
7) together with vaccuum projector
$\bar{A}^{\Omega,t}_{i}:=\ket{\Omega,t+1}\bra{\Omega,t}$ thus yield a vacuum
extension, with $\gamma^{x}=0$ for all $x\neq\Omega$ and $\gamma^{\Omega}=1$.
Physically, this corresponds to the idea that if agents were to implement a
measurement with Kraus operators of this form, they can always accurately
distinguish whether there are messages on the wire ($x\neq\Omega$) or not
($x=\Omega$). Note however that when this measurement is not performed, any
initial superposition in the temporal order in which non-vacuum states arrive
to a party is preserved coherently (as explained before). Moreover, other
choices of Kraus operators in the vacuum-extended picture are possible which
will correspond to different causal boxes. Indeed, [29] provides a different
vacuum extension of the quantum SWITCH, which can also be considered a causal
box. Furthermore, while Eq. 17 may look like a vacuum extension (or a tensor
product of vacuum extensions), it is only an effective version of Eq. 13 on
the one-message subspace. The normalization condition
$\sum_{x}|\gamma^{x}|^{2}=1$ is not fulfilled as $\gamma^{x}=1$ for all $x$.
Nevertheless, these operators are normalised on the one-message subspace which
follows from the fact that they are equivalent to the original definition on
this subspace. Note however that the equivalence breaks when we include states
from the zero-message (vacuum at all times) subspace or $n>1$ message
subspace.

We can now formulate what it means for a causal box to be an extension of a
QC-QC. This will be easier to do if we first show that sequential composition
in the causal box framework is the same as the link product.

######  Lemma 1 (Composition is equivalent to the link product).

Let
$\mathcal{M}_{A}:\mathcal{L}(\mathcal{H}^{A})\rightarrow\mathcal{L}(\mathcal{H}^{C})$
and
$\mathcal{M}_{B}:\mathcal{L}(\mathcal{H}^{B})\rightarrow\mathcal{L}(\mathcal{H}^{D})$
be two CP maps and denote their parallel composition as
$\mathcal{M}=\mathcal{M}_{A}\otimes\mathcal{M}_{B}$. Then, the Choi matrix of
their sequential composition $\mathcal{M}^{C\hookrightarrow
B}=(\mathcal{M}_{A}\otimes\mathcal{M}_{B})^{C\hookrightarrow B}$ can be
expressed with the link product as

$M^{C\hookrightarrow B}=M_{A}*M_{B}$ (19)

if one takes $\mathcal{H}^{B}=\mathcal{H}^{C}$ for the purposes of the link
product.

This allows us to write the composition of a causal box $\mathcal{C}$ with
local agents as

$(M_{\bar{A}_{1}}\otimes...\otimes M_{\bar{A}_{N}})*C$ (20)

or in terms of the Choi vector

$(\bar{A}_{1}\otimes...\otimes\bar{A}_{N})*\mathinner{|{\bar{V}}\rangle\\!\rangle},$
(21)

which are very reminiscent of the general Born rules Eqs. 3 and 4. Finally, we
define what it means for a causal box to be an extension of a QC-QC.

######  Definition 4 (Extensions of QC-QCs to causal boxes).

Let $\ket{w}$ be the process vector of an $N$-partite QC-QC and
$\mathinner{|{\bar{V}_{w}}\rangle\\!\rangle}$ the Choi vector of an
$N$-partite causal box with a totally ordered set of positions (of the form
depicted on the left side of Fig. 5(b)). We say that the causal box is an
extension of a QC-QC if for any set of local operations of the QC-QC
$\\{A_{1},...,A_{N}\\}$ and any set of local operations of the causal box
$\\{\bar{A}_{1},...,\bar{A}_{N}\\}$ such that $\bar{A}_{n}$ is an extension of
$A_{n}$ (Def. 3) for all $n\in\\{1,...,N\\}$ and any
$\ket{\psi}^{P}\in\mathcal{H}^{P}$, it holds that

$(\mathinner{|{A_{1}}\rangle\\!\rangle}\otimes....\otimes\mathinner{|{A_{N}}\rangle\\!\rangle}\otimes\ket{\psi}^{P})*\ket{w}\otimes\ket{t=2N+2}=(\mathinner{|{\bar{A}_{1}}\rangle\\!\rangle}\otimes...\otimes\mathinner{|{\bar{A}_{N}}\rangle\\!\rangle}\otimes\ket{\psi,t=1}^{P})*\mathinner{|{\bar{V}_{w}}\rangle\\!\rangle}.$
(22)

For simplicity, we just consider pure processes in the above definition. This
is not a problem as all QC-QCs and all causal boxes are purifiable [13, 16].

######  Remark 5 (Extensions vs. realisations).

In [23], a formal definition of what it means to realise an abstract QC-QC
protocol in spacetime was proposed, using the concept of fine-graining
introduced there. In Section 5.2, we connect the mathematical notion of
extensions defined here to the more physical concepts of spacetime
realisations and fine-graining, showing that the causal box extensions that we
construct for QC-QCs can be understood as fine-grained descriptions of their
spacetime realisations.

######  Remark 6 (Hilbert spaces vs. Fock spaces).

The in/output state spaces of QC-QCs/process matrices are finite-dimensional
Hilbert spaces of a fixed dimension, while that of causal boxes is potentially
an infinite-dimensional Fock space. Mathematically speaking, Fock spaces are a
specific type of Hilbert spaces. However, each finite-dimensional Hilbert
space of dimension $d$ can be seen as the one-message subspace of an infinite-
dimensional Fock space of $d$-dimensional messages. This is the viewpoint we
take in our mapping.

###  4.3 Photonic causal box realisation of the Grenoble process

As a warm up to the general result, in this section, we consider the specific
example of the Grenoble process [16] and provide a causal box description for
it. Unlike the frequently discussed quantum switch, the Grenoble process
features dynamical control of causal order. This means that the causal order
depends not just on some fixed control bit in the global past but on the
outputs of the agents produced during the run time of the protocol.
Additionally, the Grenoble process is not causally separable.

The Grenoble process is a process between three agents. In a first time step,
a qubit is sent in superposition to all three agents. Dynamical control of the
causal order happens when the agents send the qubit back to the circuit. In
this second time step, the qubit acts as a quantum control for which agent
receives it next (so for example, if the circuit receives the qubit from
$A_{1}$, it sends it to $A_{2}$ if the state of the qubit is $\ket{0}$ and to
$A_{3}$ if the state is $\ket{1}$). In the final step, the circuit sends the
qubit to the last remaining agent, attaching an additional ancillary state and
applying a CNOT gate to the target-ancilla pair.

A proposed experimental realisation of this QC-QC using photons [16] is
depicted in Fig. 6. This realisation uses well-known components like coherent
COPY and CNOT gates and polarizing beam splitters (PBS). The COPY gates copy
the photon’s state onto its polarization which the PBS then use to achieve the
dynamical control of the causal order.

Figure 6: Schematic of an experimental implementation of the Grenoble process
using photons. The figure is taken from [16]. Initially, the photon is sent to
all agents in superposition with the path acting as the control system. In the
next step, the state of the target is copied onto the polarization via the
COPY gates. This then allows the polarizing beam splitters (PBS) to guide the
photon to the correct agent. Finally, PBS along with CNOT gates acting jointly
on the target and the polarization implement the final internal operation.

Let us now consider how we obtain a causal box from this experimental set-up.
For this, start with the action of the gates on non-vacuum, qubit states. The
COPY gate in Fig. 6 implements the following operation on an incoming qubit
state $V_{\text{Copy}}=\sum_{i=0,1}\ket{i}^{O}\ket{i}^{\alpha}\bra{i}^{I}$
while the CNOT gates apply the following operation on the two input qubits,
$V_{\text{CNOT}}=\sum_{ij}\ket{i}^{O}\ket{i\oplus
j}^{\alpha}\bra{i}^{I}\bra{j}^{\alpha}$. It can be easily checked that both of
these are isometries. A beam splitter consists of two input wires $I^{0}$ and
$I^{1}$ and two output wires $O^{0}$ and $O^{1}$, each of which can carry two-
dimensional messages. The action of the beam splitter is that it reflects one
type of polarization (say photons in state $\ket{0}$) and transmits those of
the orthogonal polarization ($\ket{1}$). We can thus say that a polarizing
beam splitter sends a photon in state $\ket{i}$ arriving on wire $I^{j}$ to
the output wire $O^{i\oplus j}$. If there is only one photon, it is clear that
the beam splitter acts as an isometry.

For a causal box representation, we also need to consider the possibility of
multiple (qubit) messages as well as no messages on each of the wires. There
are many ways in which one can extend the action of these gates to the Fock
space. One natural extension involves having it act on each photon
independently essentially. To do this we define orthonormal occupation number
bases for qubits and pairs of qubits (to model the joint system of target and
ancilla). A basis for qubits is given by $\ket{m,n}$ for $m,n\in\mathbb{N}$,
which is the state with $m$ zero-qubits and $n$ one-qubits. For the ququarts
we similarly have $\ket{m,n,k,l}$ for $m,n,k,l\in\mathbb{N}$, which is the
state with $m$ qubit pairs in the state $00$, $n$ $01$-pairs, $k$ $10$-pairs
and $l$ $11$-pairs. The action of the aforementioned gates is then

$\displaystyle\begin{aligned}
V_{\text{Copy}}(\ket{m,n}^{I})&=\ket{m,0,0,n}^{O\alpha}\\\
V_{\text{CNot}}(\ket{m,n,k,l}^{I\alpha})&=\ket{m,n,l,k}^{O\alpha}\\\
V_{\text{PBS}}(\ket{m,n}^{I_{0}}\ket{k,l}^{I_{1}})&=\ket{m,l}^{O_{0}}\ket{k,n}^{O_{1}}\end{aligned}$
(23)

The extended gates are then also isometries since they map an orthonormal
basis to an orthonormal basis in an injective manner. They also reproduce the
correct behaviour in the one-message subspace and are thus indeed Fock space
extensions of the corresponding gates after which they are named. The one-
message subspace is spanned by the states $\ket{1,0}\equiv\ket{0}$ and
$\ket{0,1}\equiv\ket{1}$ for qubits and
$\ket{1,0,0,0}\equiv\ket{00},\ket{0,1,0,0}\equiv\ket{01},\ket{0,0,1,0}\equiv\ket{10},\ket{0,0,0,1}\equiv\ket{11}$
for pairs of qubits. Using these equivalences we see that $V_{\text{Copy}}$ in
Eq. 23 correctly maps $\ket{1,0}^{I}\equiv\ket{0}^{I}$ to
$\ket{1,0,0,0}^{O\alpha}\equiv\ket{0}^{O}\ket{0}^{\alpha}$ and
$\ket{0,1}^{I}\equiv\ket{1}^{I}$ to
$\ket{0,0,0,1}^{O\alpha}\equiv\ket{1}^{O}\ket{1}^{\alpha}$. Correct behavior
for the CNOT gate and the polarising beam splitter can be shown similarly.

Finally, we need to add the time stamps. We can do so after composing the
gates and beam splitters into the various isometries that take the outputs of
the local agents to their inputs. If $\bar{V}$ is such an isometry we can
replace it with $\bar{V}\otimes\mathcal{O}$ where $\mathcal{O}$ simply
increases the time by 1, $\mathcal{O}\ket{t}=\ket{t+1}$. The operator
$\mathcal{O}$ is an isometry and therefore $\bar{V}\otimes\mathcal{O}$ is an
isometry as well.

In conclusion, arbitrary compositions and tensor products of COPY and CNOT
gates and polarizing beam splitters are isometries. Any sequence
representation made up of these components is thus a causal box. Further, if
these components are composed in some way to form a QC-QC, then the causal box
obtained via the same composition is an extension. This is because when we
restrict the components to the zero- and one-message space (at each time),
they must behave the same (in the sense of Def. 4) as in the original QC-QC
picture by construction. At the same time, the components conserve the number
of messages which means that once we compose the causal box with local agents
as defined in Def. 3, which do the same, the number of messages in the circuit
never changes. We can thus make the restriction to the zero- and one-message
space because we start out with a single photon (this is necessarily the case
because otherwise the composition would not define a QC-QC). This models 1 as
discussed in Section 4.2.

In particular, this shows that the resultant causal box satisfies 1 and 2 (as
formalised in Section 4.2) and we have found a causal box extension of the
Grenoble process which respects the requirements of Theorem 1.

Figure 7: Schematic depiction of the sequence representation causal box
extension of the Grenoble process. The internal operations $\bar{V}_{n}$ are
themselves compositions of other operations, namely the COPY and CNOT gates
and polarizing beam splitters. The control and ancilla are part of the target
system and as such are sent to the agents together with the target. However,
we assume that the agents do not act on them. Once this causal box is composed
is composed with agents’ local operations only the single-message sector is
relevant, yielding behaviour that can be seen as equivalent to that of the QC-
QC.

###  4.4 Isometric extension

We now construct a general scheme to extend QC-QCs to causal boxes. However,
this will not be a direct generalization of the extension provided for the
Grenoble process in the previous section. One reason is that the latter was
based on a photonic experimental realisation proposed for the Grenoble process
while no such explicit proposal exists for general abstract QC-QCs
(nevertheless, we give an alternative general extension inspired by such
photonic experiments in Appendix C, which uses more abstract operations of the
QC-QC framework instead of COPY and CNOT gates and polarizing beam splitters).
A key difference arising between the photonic realisation and the abstract QC-
QC description is that in the latter, the ancilla and the control are “inside”
the QC-QC, whereas for the former, these systems and the target were simply
different degrees of freedom of the same photon and thus travel “outside” the
QC-QC and into the local operations (although the devices implementing the
local operations only act on the target degree of freedom). Thus, in the
photonic case, a new control and ancilla are automatically introduced,
whenever a new photon is introduced in the experiment. On the other hand, when
keeping these systems internal to the QC-QC, there will only be one control
and ancilla.

A concrete issue that arises when trying to formulate a causal box extension
where the control is internal is that we can have states with a mismatch
between the control and target degrees of freedom, which by construction never
arises in the QC-QC picture. That is the control system is in a state
$\ket{\mathcal{K}_{n},k_{n+1}}$ (which indicates that agent $k_{n+1}$ was the
last to perform a non-trivial action), but there is a vacuum state on the wire
of the agent $k_{n+1}$ and a non-vacuum state on the wire of a different
party. Consider a causal box that during the first time step simply forwards
the target system in a superposition to every agent. The control system is
then $\sum_{k_{1}\in\mathcal{N}}\ket{\emptyset,k_{1}}$. If we then input a
non-vacuum state $\ket{\psi}$ on a single wire (say wire 1, for example) in
the next time step, the isometry $\bar{V}_{2}$ will receive
$\sum_{k_{1}\in\mathcal{N}}\ket{\psi,t=3}^{\bar{A}^{O}_{1}}\otimes\ket{\emptyset,k_{1}}$.
Only the term corresponding to $k_{1}=1$ is valid. The way we will account for
this is by noting that “correct” messages and “mismatched” messages lie in
different, orthogonal Hilbert spaces. We will thus define the action of the
causal box on each subspace separately, and as long as each defines an
isometry on its own domain, the full map will also be an isometry. Moreover,
on the correct one-message subspace, it will reproduce the action of the QC-
QC.

Figure 8: Schematic depiction of the sequence representation causal box
extension of a QC-QC, here for three agents. The internal operations
$\bar{V}_{n}$ conserve the number of message coming from the agents. The
control and ancilla are internal wires and are marked in red in the figure.
Once this causal box is composed is composed with agents’ local operations
only the single-message sector is relevant, yielding behaviour that can be
seen as equivalent to that of the QC-QC.

A similar procedure can be used to deal with multiple message. We define the
internal operation of the causal box as a direct sum

$\bar{V}_{n+1}=\bigoplus_{m=0}^{\infty}\bar{V}_{n+1}^{m}$ (24)

where $\bar{V}_{n+1}^{m}$ acts on states with a total of $m$ messages on the
agents’ output wires at time $t=2n+1$ and outputs to an $m$-messages state
over the input wires of the agents at time $t=2n+2$111111Note that previously
when we talked about an $m$-message state, in the context of local operations,
we were referring to the number of messages at all times but on the wire of a
single agent whose operation was being considered (Fig. 4(b)). Here, we are
considering the internal operations of the causal box modelling the supermap,
where each operation acts on $m$ messages at a fixed time but spread over the
wires of all agents (Fig. 8). Note that 1 and 2 enforce that in the former
case $m=1$ but not in the latter case. However, by construction in the QC-QC
framework (and w.l.o.g), at each (time) step, only one message is received by
its internal operation, which is why we recover the original QC-QC action on
the one-message space here as well.  , i.e.

$\displaystyle\begin{aligned}
\bar{V}_{1}^{m}:&\vee^{m}(\mathcal{H}^{P}\otimes\ket{t=1})\rightarrow\Big{(}\bigoplus_{\begin{subarray}{c}m_{1},...,m_{N}:\\\
\sum_{i}m_{i}=m\end{subarray}}\bigotimes_{k_{1}}\vee^{m_{k_{1}}}(\mathcal{H}^{\bar{A}^{I}_{k_{1}}}\otimes\ket{t=2})\Big{)}\otimes\mathcal{H}^{C_{1}}\otimes\mathcal{H}^{\alpha_{1}}\\\
\bar{V}_{n+1}^{m}:&\Big{(}\bigoplus_{\begin{subarray}{c}m_{1},...,m_{N}:\\\
\sum_{i}m_{i}=m\end{subarray}}\bigotimes_{k_{n}}\vee^{m_{k_{n}}}(\mathcal{H}^{\bar{A}^{O}_{k_{n}}}\otimes\ket{t=2n+1})\Big{)}\otimes\mathcal{H}^{C_{n}}\otimes\mathcal{H}^{\alpha_{n}}\\\
&\rightarrow\Big{(}\bigoplus_{\begin{subarray}{c}m_{1},...,m_{N}:\\\
\sum_{i}m_{i}=m\end{subarray}}\bigotimes_{k_{n+1}}\vee^{m_{k_{n+1}}}(\mathcal{H}^{\bar{A}^{I}_{k_{n+1}}}\otimes\ket{t=2n+2})\Big{)}\otimes\mathcal{H}^{C_{n+1}}\otimes\mathcal{H}^{\alpha_{n+1}}\\\
\bar{V}_{N+1}^{m}:&\Big{(}\bigoplus_{\begin{subarray}{c}m_{1},...,m_{N}:\\\
\sum_{i}m_{i}=m\end{subarray}}\bigotimes_{k_{N}}\vee^{m_{k_{N}}}(\mathcal{H}^{\bar{A}^{O}_{k_{N}}}\otimes\ket{t=2N+1})\Big{)}\otimes\mathcal{H}^{C_{N}}\otimes\mathcal{H}^{\alpha_{N}}\rightarrow\vee^{m}(\mathcal{H}^{F}\otimes\ket{t=2N+2})\otimes\mathcal{H}^{\alpha^{\prime}_{F}}\end{aligned}$
(25)

where $\alpha^{\prime}_{F}$ is an ancilla such that the corresponding ancilla
$\alpha_{F}$ from the QC-QC is a subspace. Note that Remark 1 ensures that
defining the operations for each $m$ as above is sufficient to have well-
defined causal box on the full Fock space (assuming, of course, that we are
actually dealing with isometries, which we prove below).

Let us now deal with $\bar{V}_{n+1}^{1}$, which acts on the one-message
subspace, by formalising what we outlined earlier. We define for all
$n\in\mathcal{N}=\\{1,...,N\\}$ two subspaces,

$\mathcal{H}^{O_{n}}_{\text{corr}}=\text{span}\\{\mathcal{H}^{\bar{A}^{O}_{k_{n}}}\otimes\ket{t=2n+1}\otimes\ket{\mathcal{K}_{n-1},k_{n}}\otimes\mathcal{H}^{\alpha_{n}}\\}_{k_{n},\mathcal{K}_{n-1}}$
(26)

and its orthogonal complement (relative to the one-message subspace)

$\mathcal{H}^{O_{n}}_{\text{mis}}=(\mathcal{H}^{O_{n}}_{\text{corr}})^{\perp}=\text{span}\\{\mathcal{H}^{\bar{A}^{O}_{k_{n}}}\otimes\ket{t=2n+1})\otimes\ket{\mathcal{K}_{n-1},k^{\prime}_{n}}\otimes\mathcal{H}^{\alpha_{n}}\\}_{k_{n}\neq
k^{\prime}_{n},\mathcal{K}_{n-1}}.$ (27)

For the edge case of the global past, we simply have
$\mathcal{H}^{O_{0}}_{\text{corr}}=\mathcal{H}^{P}\otimes\ket{t=1}$ and thus
$\mathcal{H}^{O_{0}}_{\text{mis}}$ is trivial.

For the agents’ input spaces, we similarly define

$\displaystyle\begin{aligned}
\mathcal{H}^{I_{n+1}}_{\text{corr}}&=\text{span}\\{\mathcal{H}^{\bar{A}^{I}_{k_{n+1}}}\otimes\ket{t=2n+2})\otimes\ket{\mathcal{K}_{n},k_{n+1}}\otimes\mathcal{H}^{\alpha_{n}}\\}_{k_{n+1},\mathcal{K}_{n}}\\\
\mathcal{H}^{I_{n+1}}_{\text{mis}}&=\text{span}\\{\mathcal{H}^{\bar{A}^{I}_{k_{n+1}}}\otimes\ket{t=2n+2})\otimes\ket{\mathcal{K}_{n},k^{\prime}_{n+1}}\otimes\mathcal{H}^{\alpha_{n}}\\}_{k_{n+1}\neq
k^{\prime}_{n+1},\mathcal{K}_{n}}.\end{aligned}$ (28)

For the edge case of the global future, we define

$\displaystyle\begin{aligned}
\mathcal{H}^{I_{N+1}}_{\text{corr}}&=\mathcal{H}^{F}\otimes\ket{t=2N+2}\otimes\mathcal{H}^{\alpha_{F}}\\\
\mathcal{H}^{I_{N+1}}_{\text{mis}}&=\mathcal{H}^{F}\otimes\ket{t=2N+2}\otimes(\mathcal{H}^{\alpha_{F}})^{\perp}.\end{aligned}$
(29)

The spaces with the subscript corr can be seen as messages where the control
system is in the correct state, whereas the spaces with the subscript mis
correspond to states where the control is mismatched. In order to reproduce
the action of the internal operations $V_{n+1}$ of the QC-QC, the action on
$\mathcal{H}^{O_{n}}_{\text{corr}}$ must then necessarily be

$\bar{V}_{n+1}^{1}\ket{\psi,t=2n+1}^{\bar{A}^{O}_{k_{n}}}\ket{\mathcal{K}_{n-1},k_{n}}\ket{\alpha}^{\alpha_{n}}=V_{n+1}(\ket{\psi}^{\bar{A}^{O}_{k_{n}}}\ket{\mathcal{K}_{n-1},k_{n}}\ket{\alpha}^{\alpha_{n}})\ket{t=2n+2}\in\mathcal{H}^{I_{n+1}}_{\text{corr}}$
(30)

where $\ket{\psi}\neq\ket{\Omega}$ and we interpret
$V_{n+1}(\ket{\psi}^{\bar{A}^{O}_{k_{n}}}\ket{\mathcal{K}_{n-1},k_{n}}\ket{\alpha}^{\alpha_{n}})$
via the obvious isomorphism between the spaces $\mathcal{H}^{A^{I/O}_{k_{n}}}$
and $\mathcal{H}^{\bar{A}^{I/O}_{k_{n}}}$. The restrictions to the respective
“correct” spaces of the edge cases $\bar{V}_{1}^{1},\bar{V}_{N+1}^{1}$ are
defined analogously.

On the other hand, we simply demand that the restriction of
$\bar{V}_{n+1}^{1}$ to $\mathcal{H}^{O_{n}}_{\text{mis}}$ is an isometry and
that it is a map of the form

$\bar{V}_{n+1}^{1}|_{\mathcal{H}^{O_{n}}_{\text{mis}}}:\mathcal{H}^{O_{n}}_{\text{mis}}\rightarrow\mathcal{H}^{I_{n+1}}_{\text{mis}}$
(31)

For $m\neq 1$, it similarly suffices for $\bar{V}_{n+1}^{m}$ to be an
arbitrary isometry.

######  Proposition 1 (Sequence representation for QC-QCs).

There exist maps $\bar{V}_{1},...,\bar{V}_{N+1}$ which are isometries and
fulfill all the conditions of Section 4.4.

From this it immediately follows that we have constructed a sequence
representation of a causal box. We can then show that this causal box is also
an extension of the original QC-QC.

######  Proposition 2 (Causal box description of QC-QCs).

The causal box with sequence representation given by any set of isometries
$\bar{V}_{1},...,\bar{V}_{N+1}$ satisfying the conditions in Section 4.4 is an
extension of the QC-QC described by the process vector

$\ket{w}=\sum_{(k_{1},...,k_{N})}\mathinner{|{V^{\rightarrow
k_{1}}_{\emptyset,\emptyset}}\rangle\\!\rangle}*...*\mathinner{|{V^{\rightarrow
k_{N}}_{\\{k_{1},...,k_{N-2}\\},k_{N-1}}}\rangle\\!\rangle}*\mathinner{|{V^{\rightarrow
F}_{\mathcal{N}\backslash k_{N},k_{N}}}\rangle\\!\rangle}.$

###  4.5 Projective extension

The extension that we just constructed allows for multiple messages to be sent
to the causal box at the same time. However, since the extension is equivalent
to the QC-QC, once it is actually composed with local agents, there is at any
time at most one non-vacuum state on the input wires of the causal box. An
experimentalist interested in simulating just the usual action of the QC-QC
might therefore not care about the multi-message space and build their device
in such a way that it simply aborts the procedure if an unexpected (meaning
unobtainable in the QC-QC framework) input is encountered. We can capture this
idea by applying a projective measurement
$\\{\mathcal{P}^{I_{n}}_{accept},1-\mathcal{P}^{I_{n}}_{accept}\\}$ on the
inputs and $\\{\mathcal{P}^{O_{n}}_{accept},1-\mathcal{P}^{O_{n}}_{accept}\\}$
on the outputs, each with outcomes $\\{accept,abort\\}$, before and after
every internal operation. Here, $\mathcal{P}^{I_{n}/O_{n}}_{accept}$ projects
on $\mathcal{H}^{I_{n}/O_{n}}_{\text{corr}}$ defined in the previous section.
If the outcome is $abort$, the circuit outputs the state $\ket{abort}$. If the
outcome is $accept$, the circuit applies the isometry $\bar{V}_{n+1}$.

When we compose this with local agents satisfying 1 and 2, this extension
never aborts.

######  Lemma 2 (No aborts).

When composing the projective extension with local agents satisfying 1 and 2,
the projective measurements yield the outcome $accept$ with unit probability.

Further, note that a projective measurement which always yields the same
outcome on some space acts as the identity on that space. Therefore, this
causal box acts equivalently to the one from Section 4.4 when composed with
local operations which are extensions of the QC-QC operations and acts as a
trace preserving map in this context. However, on the full Fock space, this
causal box extension can decrease the trace unlike the extension of Section
4.4 which remains an isometry (and hence trace preserving) on the full space.

Figure 9: Schematic depiction of the sequence representation causal box
extension of a QC-QC, here for three agents, which uses projective
measurements to check that there is only a single message on the wires and the
control matches the wire. The wires carrying the ancilla and control are
marked in red. Once this causal box is composed with agents’ local operations
the projective measurements always yield the outcome $accept$.

An additional consequence is that whenever we consider causal boxes which
fulfill the requirements laid out in Section 4.2 and only care about their
composition with local agents, we can replace the Choi vector with an
effective Choi vector [34],

$\bigotimes_{n=0}^{N}\mathinner{|{\mathcal{P}^{I_{n}}_{accept}}\rangle\\!\rangle}*\mathinner{|{\mathcal{P}^{O_{n}}_{accept}}\rangle\\!\rangle}*\mathinner{|{\bar{V}_{w}}\rangle\\!\rangle}.$
(32)

Naturally, an analogous statement can be made for Choi matrices.

##  5 Insights on physical realisability in a spacetime

The previous sections focused on Theorem 1, which is about the mapping from
QC-QCs to causal boxes. Here, we describe the relevant concepts behind Theorem
2 and provide a proof sketch of the same, which enables us to regard the
causal box in the image of our mapping as a fine-grained description of the
QC-QC and consequently as a spacetime realisation of the given QC-QC.

In [23], a formal definition of a spacetime realisation of an abstract quantum
network, and a relativistic causality condition necessary for ensuring the
absence of superluminal causal influences were formalised. These quantum
networks include in particular, all protocols described by process matrices
and the definition applies to all globally hyperbolic spacetimes i.e. those
without closed timelike curves. This definition places minimal but essential
conditions that relate the abstract information-theoretic description of the
protocol to the description of its spacetime realisation. It shares analogies
with how we define the physical realisation of a quantum computation on $n$
logical qubits in terms of $m\geq n$ physical qubits. The concept of fine-
graining of causal structures was introduced to formalise this definition, and
allows one to interpret the spacetime realisation as a fine-grained
description of the abstract process matrix description, as the former can in
general involve a larger number of degrees of freedom. We review the core
aspects of this definition here, before moving on to the proof sketch of
Theorem 2. In the appendix we provide the full proof of the theorem.

###  5.1 Fine-graining and spacetime realisations

#### Fine-graining of quantum protocols

Suppose we have a quantum channel (a CPTP map)
$\mathcal{M}:\mathcal{L}(\mathcal{H}^{I})\rightarrow\mathcal{L}(\mathcal{H}^{O})$
which we wish to “implement” by some physical means where the input $I$ is
encoded in a set $\\{I_{1},...,I_{n}\\}$ of physical input systems and the
output $O$ is decoded from a set $\\{O_{1},...,O_{m}\\}$ of physical output
systems. For example, in quantum computing, $\mathcal{M}$ can be a quantum
gate such as the bit flip on a single logical qubit and it may be implemented
on a physical hardware where each logical qubit corresponds to $n$ physical
qubits (in this example, $n=m$). The physical layer map from inputs
$\\{I_{1},...,I_{n}\\}$ to outputs $\\{O_{1},...,O_{m}\\}$ is denoted as
$\mathcal{M}^{f}$.

To regard $\mathcal{M}^{f}$ as an “implementation” of $\mathcal{M}$, or in the
language we will use, a _fine-graining_ of $\mathcal{M}$, it is necessary that
there exists an encoding and decoding scheme between the coarse in/outputs $I$
and $O$ and fine in/outputs $\\{I_{1},...,I_{n}\\}$ and
$\\{O_{1},...,O_{m}\\}$ which can recover $\mathcal{M}$ from
$\mathcal{M}^{f}$. Formally, we require that there exists a pair of maps
$(\mathrm{Enc},\mathrm{Dec})$, where
$\mathrm{Enc}:\mathcal{L}(\mathcal{H}^{I})\rightarrow\bigotimes_{i=1}^{n}\mathcal{L}(\mathcal{H}^{I_{i}})$
is CPTP and
$\mathrm{Dec}:\bigotimes_{j=1}^{m}\mathcal{L}(\mathcal{H}^{O_{j}})\rightarrow\mathcal{L}(\mathcal{H}^{O})$
is CPTP on the image of $\mathcal{M}^{f}\circ\mathrm{Enc}$ such that

$\mathcal{M}=\mathrm{Dec}\circ\mathcal{M}^{f}\circ\mathrm{Enc}$ (33)

Figure 10: Illustration of Eq. 33. Figure taken from [23].

This is illustrated in Fig. 10. A second condition we will require is that the
causal properties of $\mathcal{M}$ are preserved in $\mathcal{M}^{f}$, where
the relevant property we care about are the signalling relations: the ability
to communicate information between in/output systems of the channel. Formally,
we say that $\\{I\\}$ signals to $\\{O\\}$ in $\mathcal{M}$ if there exists a
map
$\mathcal{N}^{I}:\mathcal{L}(\mathcal{H}^{I})\rightarrow\mathcal{L}(\mathcal{H}^{I})$
such that $\mathcal{M}\neq\mathcal{M}\circ\mathcal{N}^{I}$, which captures
that an agent with access to $I$ can communicate information to an agent with
access to $O$ by a suitable local operation $\mathcal{N}^{I}$ on $I$. More
generally, one can define signalling between arbitrary subsets of in and
output systems in CPTP maps with multiple in and output systems (hence the set
symbol for $I$ and $O$ here) [23] (see also [17] for equivalent definitions of
signaling). Then we will require that: $\\{I\\}$ signals to $\\{O\\}$ in
$\mathcal{M}$, then $\\{I_{i}\\}_{i=1}^{n}$ signals to $\\{O_{j}\\}_{j=1}^{m}$
in $\mathcal{M}^{f}$.

More generally, given a channel $\mathcal{M}$ with any number of inputs and
outputs, another map $\mathcal{M}^{f}$ is called a fine-graining of
$\mathcal{M}$ if each input $I$ of $\mathcal{M}$ can be identified with a
distinct set $\bar{I}$ of inputs of $\mathcal{M}^{f}$, and likewise for
outputs such that (1) there is an encoding-decoding scheme relating these
in/output systems which satisfies Eq. 33 and (2) the signalling relations of
$\mathcal{M}$ are preserved in $\mathcal{M}^{f}$. The concept has also been
extended to quantum networks formed by the composition of multiple CPTP maps,
which covers a large class of abstract quantum information protocols, even
those with a cyclic causal structure. For defining fine-graining for such
general networks of CPTP maps, one applies the above definition for each CPTP
map involved in the network. We refer to [23] for further details of this
definition.

#### Spacetime realisation of abstract quantum protocols

How does this concept of fine-graining relate to the spatiotemporal picture?
When we realise a CPTP map such as $\mathcal{M}$ above (or more generally a
network of CPTP maps) in a spacetime, we are assigning a spacetime region to
each in/output quantum system of the map (or network). In the most minimal,
yet operational sense, we can model an acyclic spacetime (such as Minkowski
spacetime) as a partially ordered set $\mathcal{T}$ and regard any subset of
points in the partially ordered set as defining a spacetime region.121212Note
that generally, if the subsets of $\mathcal{T}$ defining the regions are not
individual elements of $\mathcal{T}$, then the possibility for causal
influence between regions need not be transitive. However, in the case where
we choose regions that coincide with individual elements of $\mathcal{T}$ (or
are sufficiently localised around single points), we do have transitivity as
$\mathcal{T}$ is a poset. Suppose that we have a spacetime embedding
$\mathcal{E}$ which assigns to the in and output systems $I$ and $O$, the
spacetime regions $\mathcal{E}(I)=\\{P_{1},...,P_{n}\\}$ and
$\mathcal{E}(O)=\\{Q_{1},...,Q_{m}\\}$ consisting of some spacetime locations
$P_{i},Q_{j}\in\mathcal{T}$. Then, a spacetime realisation of $\mathcal{M}$
relative to this embedding would be a fine-graining
$\mathcal{M}^{f}:\bigotimes_{i=1}^{n}\mathcal{L}(\mathcal{H}^{I_{i}})\rightarrow\bigotimes_{j=1}^{m}\mathcal{L}(\mathcal{H}^{O_{j}})$
of
$\mathcal{M}:\mathcal{L}(\mathcal{H}^{I})\rightarrow\mathcal{L}(\mathcal{H}^{O})$
where each input $I_{i}$ of $\mathcal{M}^{f}$ is associated with a spacetime
location $P_{i}$ and each output $O_{j}$ is associated with a corresponding
spacetime location $Q_{j}$.

Generally, the $P_{i}$ and $Q_{j}$ can themselves be subregions rather than
individual spacetime events. The ability to partition a larger region into
subregions translates operationally into the ability of agents to perform more
fine-grained interventions, where they may perform a different operation
within different spacetime subregions of a larger protocol.

#### Relativistic causality

Since signalling in $\mathcal{M}$ is preserved in $\mathcal{M}^{f}$,
relativistic causality implies that whenever $\\{I\\}$ signals to $\\{O\\}$ in
$\mathcal{M}$, then the embedding must be such that at least one
$Q_{j}\in\mathcal{E}(O)$ must be in the future light cone of some
$P_{i}\in\mathcal{E}(I)$, denoted $P_{i}\prec Q_{j}$. There will in general be
additional relativistic causality conditions at the fine-grained level, for
instance if $\\{I_{i}\\}$ signals to $\\{O_{j},O_{k}\\}$ in fine-graining
$\mathcal{M}^{f}$, then the spacetime point $P_{i}$ associated with $I_{i}$
must be in the past light cone of at least one of the spacetime points $Q_{j}$
or $Q_{k}$ associated with the corresponding outputs. This captures that even
under the more fine-grained interventions, agents cannot signal outside the
future light cone. Again, most generally, we can extend this operational
definition of relativistic causality for spacetime realisations of arbitrary
networks of CPTP maps [23].

(a)

(b)

Figure 11: (a) The coarse-grained process matrix/QC-QC of the quantum switch,
which admits an indefinite causal order. Here the target takes the blue path
(Alice before Bob) when the control is $\ket{0}$ and the orange path (Bob
before Alice) when the control is $\ket{1}$ and a superposition of the two in
general. (b) A causal box model of the quantum switch, which corresponds to a
fine-grained description that is associated with a definite and acyclic causal
structure. The non-vacuum system takes the blue path (and the vacuum takes the
orange path) when the control is $\ket{0}$ and vice versa when the control is
$\ket{1}$. This figure is reproduced from [23]. In this work, we have shown
that every QC-QC admits a causal box extension with such an acyclic fine-
grained causal structure, providing a similar physical interpretation to all
QC-QCs in the context of a background space-time.

###  5.2 All QC-QCs admit an acyclic fine-grained causal structure

We sketch the proof of Theorem 2. The full proof can be found in Appendix D.

  1. 1.

_QC-QC to causal box mapping_ We consider the image of our mapping from QC-QCs
to causal boxes, given by the isometric extension (Section 4.4) and recall
that every protocol described by a QC-QC acting on some local operations can
be described as a network of CPTP maps formed through parallel or loop
composition, as defined and proven in [23].

  2. 2.

_Encoding-decoding scheme_ We then construct an explicit encoding-decoding
scheme which relates the causal box in the image of the above mapping to the
original QC-QC as per Eq. 33 where the causal box plays the role of the fine-
grained map $\mathcal{M}^{f}$ and the QC-QC plays the role of the coarse-
grained map $\mathcal{M}$.

  3. 3.

_Preserving signalling relations_ We show that the signalling relations of the
QC-QC are preserved in the causal box, as a consequence of the properties that
a causal box must satisfy to be regarded as an extension of the QC-QC
satisfying the spatiotemporal closed labs assumption (according to Theorem 1).

  4. 4.

_Extending the argument to the full network_ We also take into account the
local operations and show that the whole network formed by composing the
causal box modelling the QC-QC with corresponding local operations is a fine-
graining of the original QC-QC network.

  5. 5.

_Acyclic fine-grained causal structure_ The fact that the fine-grained network
is completely described by valid causal boxes with sequence representation in
terms of a totally ordered set $\mathcal{T}$ guarantees that its information-
theoretic causal structure is acyclic and compatible with the time ordering
given by $\mathcal{T}$.

We have shown that the causal box realises the action of the QC-QC in a
spatiotemporal setting while respecting the spatiotemporal closed labs
assumptions and relativistic causality. The causal box preserves the
signalling structure of the QC-QC as well as other relevant information
captured by Eq. 33. However, Theorem 2 highlights that this does not imply
that we have genuinely implemented the indefinite causal structure of the QC-
QC. Indefinite causal structures in the sense of the process matrix framework
can be viewed as special cases of cyclic information-theoretic causal
structures and cyclic causal structures can become acyclic under fine-graining
[23]. This accords with the physical interpretation of cyclic causal
structures describing feedback, such as mutual causal influence between demand
and price, where the fine-grained description accounting for spacetime degrees
of freedom is acyclic. The causal boxes in the image of our mapping from QC-
QCs provide an analogously clear and physical interpretation to the causal
structure of the QC-QC, at least in the context of their realisations in a
background spacetime. We already discussed the physical relevance of this
result in Section 4.1, and the intuition is illustrated in Fig. 11 for the
quantum switch process which is the simplest example of a QC-QC.

##  6 Insights on composability

Composability is central to our understanding of the physical world, as it
helps us comprehend how complex structures are built up from constituent
parts. Moreover, we expect the composition of any two physical experiments to
be yet another physical experiment, i.e. we expect physical processes to be
closed under composition. This is indeed the case for causal boxes. They
describe information processing protocols in spacetime such that we can
compose two protocols described by causal boxes to obtain a new protocol that
can also be described as a causal box.

On the other hand, previous work [18] suggests that composition in the process
matrix framework is not so straightforward, with no-go results pointing to
difficulties in composing even fixed order processes. In light of this,
further works have proposed consistent rules for composition of higher order
processes [20] as well as necessary and sufficient conditions for identifying
when the composition of two processes is valid [19]. This prior literature
focuses on abstract information-theoretic approaches, and it is not immediate
how this discussion relates to the observed composability of physical
experiments in spacetime, where both the information and spatiotemporal
aspects play a role. Here, we address this relation in light of our results.

We start by discussing an example from [18, 19], which was used to highlight
the difficulties in consistently composing process matrices. Consider two
bipartite processes, $W_{1}$ over the parties $A_{1}$ and $B_{1}$, and $W_{2}$
over the parties $A_{2}$ and $B_{2}$. Suppose $W_{1}$ is a fixed order process
with the order $A_{1}\prec B_{1}$ and $W_{2}$ is a fixed order process with
the opposite order, $B_{2}\prec A_{2}$. The parallel composition of the two
processes, namely $W_{1}\otimes W_{2}$, is no longer a valid bipartite process
over the parties $A_{1}A_{2}$ and $B_{1}B_{2}$. This is because $W_{1}\otimes
W_{2}$ involves a channel in both directions between the parties $A_{1}A_{2}$
and $B_{1}B_{2}$ which corresponds to a paradoxical causal loop, as
illustrated and explained in Fig. 12.

Figure 12: Composition of two process matrices $W_{1}$ and $W_{2}$. $W_{1}$
sends a state to Alice and then sends her output to Bob, while $W_{2}$ does
the same but with the order of Alice and Bob reversed. The composition then
includes a loop and if Alice applies the Pauli-$X$ unitary while Bob applies
the identity the generalised Born rule yields a probability of 0, even though
it should have yielded 1 if this composition were well defined.

However, notice that $W_{1}\otimes W_{2}$ is still a valid four-party process
over the parties $A_{1}$, $A_{2}$, $B_{1}$ and $B_{2}$. The difference being
that, when we treat it as a bipartite process, we allow for arbitrary
operations
$\mathcal{M}_{A_{1}A_{2}}:\mathcal{L}(\mathcal{H}^{A_{I}^{1}A_{I}^{2}})\rightarrow\mathcal{L}(\mathcal{H}^{A_{O}^{1}A_{O}^{2}})$
and
$\mathcal{M}_{B_{1}B_{2}}:\mathcal{L}(\mathcal{H}^{B_{I}^{1}B_{I}^{2}})\rightarrow\mathcal{L}(\mathcal{H}^{B_{O}^{1}B_{O}^{2}})$
and when we treat it as a four-party process, only product operations are
allowed,
$\mathcal{M}_{A_{1}A_{2}}=\mathcal{M}_{A_{1}}\otimes\mathcal{M}_{A_{2}}$ and
$\mathcal{M}_{B_{1}B_{2}}=\mathcal{M}_{B_{1}}\otimes\mathcal{M}_{B_{2}}$.
Generally, composition of processes allowing only product local operations and
composition allowing general local operations acting jointly on in/outputs
from the two processes can be formulated as two distinct types of composition
[20]. In particular, if we compose two $N$-party processes $W_{1}$ and $W_{2}$
always through the former type of composition (parallel composition), then
$W_{1}\otimes W_{2}$ is always a valid process, but over $2N$ parties.

The solution to only use product operations aligns with the motivation of the
process framework where each party is considered to be in a closed and
isolated lab which can communicate to the outside world only through the
in/output interface that they have with the process matrix. However, in
practice, proposed realisations of process matrices entail table top
experiments where the different parties do not physically correspond to closed
and isolated labs, and relativistic causality does permit non-product
operations in these scenarios. It remains unclear what physical principles
impose the restriction to parallel composition in physical experiments in
spacetime. Moreover, notice that even in the abstract framework, $W_{1}\otimes
W_{1}$ and $W_{2}\otimes W_{2}$ from the previous example are both valid
bipartite processes on the joint Alice and Bob systems, and are consistent
under composition with arbitrary joint operations of the two Alices and two
Bobs. Therefore, the restriction to parallel composition for all processes, is
not necessary for consistency (see also [19]).

We now discuss the insights provided by our work, regarding the relation to
the spatiotemporal picture. Our results which map QC-QCs to causal boxes shows
how composability of physical experiments is recovered, without restricting to
parallel composition by fiat. We illustrate this for the above example. We can
map the fixed order processes (and hence QC-QCs) $W_{1}$ and $W_{2}$ to the
causal box picture (which adds a (space-)time stamp to each message) and
consider general physical operations that $A_{1}A_{2}$ and $B_{1}B_{2}$ can
perform. The problematic composition in Fig. 12 is due to the loop from
$A^{O}_{1}$ to $B^{I}_{1}$ to $B^{O}_{2}$ to $A^{I}_{2}$ to $A^{O}_{1}$ again.
In the causal box description, we would have messages with (space-)time labels
going around the wires defining this loop. W.l.o.g. for this argument, we can
start on any of the systems in the loop, say $A^{O}_{1}$, and consider a
message, say some state $\ket{0}$, with a time stamp $t_{1}$ sent by Alice.
Due to relativistic causality, each channel in the loop can only send the
message forward in time, such that the message reaches $B^{I}_{1}$ only at
some $t_{2}>t_{1}$, then $B^{O}_{2}$ at some $t_{3}>t_{2}$, $A^{I}_{2}$ at
some $t_{4}>t_{1}$ (so far all are identity channels so the message is still
$\ket{0}$) and finally back to the wire $A^{O}_{1}$ at time $t_{5}>t_{4}$
after the bit flip, which gives $\ket{1,t_{5}}^{A^{O}}$, and the cycle can
continue indefinitely.131313Note that if the timestamps don’t satisfy the
right ordering, the loop cannot be formed in the first place. There is no
contradiction because the mismatched messages on $A^{O}_{1}$ are associated
with distinct times: $\ket{0,t_{1}}^{A^{O}_{1}}$ and
$\ket{1,t_{5}}^{A^{O}_{1}}$ with $t_{1}<t_{5}$, and this will always be the
case even if we continue the loop.

We thus see that the composition of causal boxes may also lead to cycles when
we compose with non-product operations on $A_{1}A_{2}$ and $B_{1}B_{2}$, but
these are well defined due to relativistic causality (indeed, such loops are
only a coarse-grained representation of a fine-grained acyclic causal
structure, as corroborated by Section 5). Generally, such compositions can
lead to scenarios with superpositions of receiving two messages (e.g.,
$\ket{0,t_{1}}\ket{1,t_{5}}$) and one message (e.g.,
$\ket{0,t_{1}}\ket{\Omega,t_{5}}$). Fully understanding the composability of
experimental realisations of QC-QCs therefore requires the use of Fock spaces
(or a similar space that models sending of multiple messages), which we have
carefully accounted for in our results and mapping of QC-QCs to causal boxes.

Observe that if such a loop occurs in the causal box picture, the associated
causal box protocol would no longer satisfy the set-up assumptions of the
process framework which requires in particular that each party acts exactly
once (which we referred to as the spatiotemporal closed lab assumption).
Therefore, while causal boxes (like the set of physical experiments) are
closed under arbitrary composition, the subset of causal boxes which model the
set-up assumptions of the process framework are not closed under composition
[34]. The image of our mapping from QC-QCs to causal boxes lies within this
subset as we have shown. This highlights that whenever the composition of the
causal boxes in the image of two QC-QCs yields a causal box that violates the
set-up assumptions (e.g., by allowing an agent who acts on a $d$-dimensional
system in the process picture to receive or send multiple $d$-dimensional
messages), then the corresponding composition of the QC-QCs would be invalid
in the QC-QC framework. In other words, invalid QC-QC compositions still map
to physical experiments, but these will be experiments that violate the set up
assumptions of the QC-QC/process matrix framework. This highlights the
important role played by the interplay of relativistic causality, Fock spaces
and the set up assumptions of the process framework in recovering the
composability of physical experiments.

##  7 Conclusions and outlook

We compared two quantum causality frameworks— QC-QCs [16], an abstract
information-theoretic model, and causal boxes [13], which involve both
information and spatiotemporal structures. Our findings reveal that every QC-
QC can be mapped to a causal box, adhering to the process/QC-QC framework’s
set-up assumptions (such as each party acting once) and operating on a Fock
space while replicating the QC-QC’s behaviour in the relevant subspace as
defined by these assumptions. The resulting causal box exhibits a definite,
acyclic fine-grained causal structure consistent with spacetime, even if the
QC-QC displays an indefinite causal structure (the physical interpretation of
which is discussed in Section 4.1). We established multiple such mappings,
illustrating that the same abstract protocol (described by a QC-QC) may have
multiple distinct physical realisations (described by causal boxes). Moreover,
our results demonstrate that the failure to consistently compose two processes
in the abstract framework does not lead to any paradoxes in the physical world
(as we should rightfully expect), but translates to a physical protocol that
violates the set up assumptions of the process framework, by producing
in/outputs in the larger Fock space with multiple messages. Our results offer
concrete reconciliations of the notions of causality and composability between
the information-theoretic and spatiotemporal viewpoints, providing a
foundation for further explorations of this interface.

There is a large volume of work (for example, [10, 11, 12]) suggesting
advantages of indefinite causal order processes over standard quantum circuits
for information processing tasks. However, there are important distinctions
between the abstract picture and the fine-grained spatiotemporal picture as
highlighted by our work. More specifically, a QC-QC is a supermap that accepts
as inputs arbitrary quantum operations as “black boxes” while in the spacetime
realisation, adherence to the set-up assumptions of acting once on a non-
vacuum system, constrains the local operations to leave the vacuum state
invariant, making them “gray boxes”. This distinction has also been crucial in
previous contexts where tasks that are theoretically impossible with black box
unitaries (such as coherently controlling them) become experimentally feasible
with “gray box” unitaries, where the action even on a one-dimensional subspace
is known [30, 31]. Together with previous works such as [23, 35], our results
can inform a systematic investigation of whether and how proposed theoretical
advantages from higher order processes translate into practical quantum
experiments in spacetime, while providing more fine-grained tools for this
purpose.

The original use of symmetric Fock spaces in the causal box framework aims to
encapsulate the fact that all ordering information is contained in the
position labels [13]. In the causal box extension of the photonic realisation
of the Grenoble process in Section 4.3 and another similar extension found for
arbitrary QC-QCs in Appendix C, the symmetrization aligns with the perspective
of viewing the QC-QC as a photonic (i.e. bosonic) circuit. There is scope for
further inquiry into whether causal boxes could be realised using
antisymmetric Fock spaces to describe fermionic circuits. The use of Fock
spaces in the causal box framework also motivates its utility in studying
quantum protocols in a quantum field theory (QFT) setting. In the context of
applying the process formalism in QFT, an issue that was noted in [36] is that
some QFT processes would require a variable number of operations (e.g.
radiative beta decay has a variable number of outgoing photons). As the causal
box framework does not assume that each agent acts exactly once and allows
superpositions of number of messages, it can be a promising candidate for
addressing such questions.

This work focused on the mapping from QC-QCs to causal boxes, which lays the
foundation for a follow-up work where we consider mappings in the reverse
direction. Specifically, we are interested in causal box protocols which
satisfy the set up assumptions of the process framework. These assumptions are
operationally relevant also in the context of so-called causal inequalities
which are known to be violated by certain theoretical processes, but
interestingly, not by QC-QC type processes. If these assumptions are not
respected in an experiment, this can open up loopholes which enable the
trivial violation of causal inequalities, akin to violating Bell inequalities
using classical communication [24]. Recalling the result of [23] that causal
boxes describe the most general finite quantum protocols in a fixed spacetime,
such a mapping will enable a characterisation of the largest set of processes
realisable in a fixed spacetime in accordance with the set-up assumptions. We
note that neither [23] nor our current or follow-up work implies that process
matrices should be a subset of causal boxes. Indeed, we expect neither
framework to be a subset of the other. On the one hand, there are causal boxes
which violate the set-up assumptions. They thus trivially cannot correspond to
an $N$-party process (although, they could be understood as an $M$-party
process for some $M>N$ by splitting agents such that the set-up assumptions
are respected). On the other hand, we expect that there exist at least some
processes which cannot be realised in a fixed background spacetime.
Consequently, these cannot be modelled by a causal box.

A related framework that accounts for vacuum states is that of routed quantum
circuits. The concept of routes gives a natural and general way to model
constraints on how unitaries act within orthogonal subspaces (such as vacuum
vs. non-vacuum spaces), but the formalism does not explicitly consider
relativistic causality in spacetime [17, 37, 35, 38]. An interesting open
conjecture in the formalism states that processes without weak loops (which
are cyclic causal influences relative to a more refined notion of causal
structure for processes proposed in [38]) are precisely those that do not
violate causal inequalities. Our results may be seen as further evidence
towards this conjecture. QC-QCs which do not violate causal inequalities have
an acyclic fine-grained causal structure and are thus realisable in an acyclic
background spacetime (in the sense of [23]) while our follow-up work mentioned
above, suggests a tighter relation between QC-QCs and the set of processes
realisable in an acyclic spacetime. This acyclicity bears similarities to the
absence of weak loops (in the sense of [38]). Further work is needed to make
precise the link between causal boxes and routed circuits. This can provide a
more comprehensive set of tools towards the characterisation of different
classes of quantum causal structures, their realisability in different
physical regimes/spacetimes, as well as the resource responsible for any
potential information-processing advantages that one class of processes may
provide over another.

Finally, we note that there is potential for generalising our results beyond
fixed background spacetimes, thanks to the general modelling of spacetime as a
partially ordered set. For quantum protocols in fixed spacetimes (with
classical reference frames/agents) this partial order as well as the
localisation of operational events in spacetime is reference frame independent
i.e. all agents will agree on whether or not a quantum message is localised in
spacetime. This may no longer be the case when quantum reference frames are
used, and it would be interesting to consider a generalisation of the causal
box approach and our results by allowing the partial order to be reference
frame or agent dependent. This can inform a better understanding of the
interplay of operational and spatiotemporal notions of causality and the role
of composability in quantum gravitational scenarios, where an absolute
background partial order may not exist. By connecting two distinct frameworks
featuring different causality notions, this work provides a step forward
towards the larger goal of build a more unifying understanding of quantum and
relativistic causality.

Acknowledgements We are thankful to Augustin Vanrietvelde for useful comments
regarding the composability of higher order processes, to Renato Renner for
insightful discussions on Fock spaces and to Časlav Brukner for interesting
discussions on the notion of events and fine-grained causal structures. The
authors also thank the anonymous referees for QPL 2022 and TQC 2023 for their
reviews, which have helped to improve this work. MS acknowledges support from
National Science Centre, Poland, grant number 2021/41/B/ST2/03149. MS thanks
the Quantum Information Theory group at the Institute of Theoretical Physics,
ETH Zurich, for financial support to present the results of this work at QPL
2022, Oxford, UK. VV’s research has been supported by an ETH Postdoctoral
Fellowship. VV also acknowledges support from the ETH Zurich Quantum Center,
the Swiss National Science Foundation via project No. 200021_188541 and the
QuantERA programme via project No. 20QT21_187724.

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##  Appendix A Mathematical tools

###  A.1 Choi isomorphism

Frequently, it is more convenient to express channels as matrices. For this
purpose, we use the Choi isomorphism.

######  Definition 5 (Choi matrix [39]).

Given some Hilbert spaces $\mathcal{H}^{X}$ with some basis
$\\{\ket{i}^{X}\\}_{i}$ and $\mathcal{H}^{Y}$ with some basis
$\\{\ket{i}^{Y}\\}_{i}$, let
$\mathcal{M}:\mathcal{L}(\mathcal{H}^{X})\rightarrow\mathcal{L}(\mathcal{H}^{Y})$
be some linear map. The Choi matrix of $\mathcal{M}$ is defined as

$M\coloneqq\mathcal{I}^{X}\otimes\mathcal{M}\mathinner{|{\mathbb{1}}\rangle\\!\rangle}\mathinner{\langle\\!\langle{\mathbb{1}}|}^{XX}$
(34)

where
$\mathinner{|{\mathbb{1}}\rangle\\!\rangle}^{XX}=\sum_{i}\ket{i}^{X}\ket{i}^{X}$
denotes the maximally entangled state.

If $\mathcal{M}$ is pure, i.e. there exists a linear operator
$V:\mathcal{H}^{X}\rightarrow\mathcal{H}^{Y}$ such that
$\mathcal{M}(\rho)=V\rho V^{\dagger}$ for any