# The Multi-round Process Matrix

Timothée Hoffreumon<EMAIL_ADDRESS>Centre for Quantum
Information and Communication (QuIC), École polytechnique de Bruxelles, CP
165, Université libre de Bruxelles, 1050 Brussels, Belgium.  Ognyan Oreshkov
<EMAIL_ADDRESS>Centre for Quantum Information and Communication (QuIC),
École polytechnique de Bruxelles, CP 165, Université libre de Bruxelles, 1050
Brussels, Belgium.

###### Abstract

We develop an extension of the process matrix (PM) framework for correlations
between quantum operations with no causal order that allows multiple rounds of
information exchange for each party compatibly with the assumption of well-
defined causal order of events locally. We characterise the higher-order
process describing such correlations, which we name the multi-round process
matrix (MPM), and formulate a notion of causal nonseparability for it that
extends the one for standard PMs. We show that in the multi-round case there
are novel manifestations of causal nonseparability that are not captured by a
naive application of the standard PM formalism: we exhibit an instance of an
operator that is both a valid PM and a valid MPM, but is causally separable in
the first case and can violate causal inequalities in the second case due to
the possibility of using a side channel.

##  1 Introduction

There has recently been significant interest in quantum processes in which the
operations performed by separate parties exhibit ‘indefinite causal order’
[25, 17, 38, 5, 12, 43, 6, 37, 20, 9, 42, 21, 23, 33, 45, 3, 31, 47, 2, 52,
46, 22, 39, 11, 50, 51, 44, 49]. A formal definition of this feature, termed
causal nonseparability [38, 6, 37, 50], has been given in the process matrix
(PM) framework [38], which describes correlations between elementary quantum
experiments, each defined by a pair of input and an output system, also
referred to as a quantum node [3, 7, 8], over which an agent could apply
different operations, without presuming the existence of global causal order
among the separate operations but only the validity of causal quantum theory
[16] for their local description. Causally nonseparable processes have been
shown to accomplish informational tasks that are not possible via processes in
which the operations are used in a defined order [14, 5, 20, 23, 44]. They
have been conjectured to be potentially relevant in quantum gravity scenarios
[25, 17, 38, 52] where the causal structure of spacetime may be subject to
quantum indefiniteness, as well as in the presence of closed timelike curves
[17, 38, 12, 4, 10, 49]. But some of these processes, such as the quantum
SWITCH [17] for which most of the examples of advantage have been found, also
admit realisations within standard quantum mechanics on time-delocalized
systems [35] via coherent control of the order of operations. This has been
demonstrated in several experimental setups [43, 46, 45, 33, 22, 51], offering
blueprints for possible applications.

In view of developing potential applications of indefinite causal order, the
standard PM framework appears limited by the fact that each party is assumed
restricted to a single round of information exchange, where from the local
causal perspective of that party [24], information is received once via the
input system and subsequently sent out once via the output system. Practical
communication protocols and distributed computing algorithms generally involve
multiple rounds of information exchange between separate parties that can use
local memory and condition the operations applied at a given time on
information obtained at other times. It is therefore natural to ask whether
the PM framework can be extended to allow for multiple rounds of communication
and whether such an extension would contain any new possibilities that are not
captured by the standard PM framework.

In this paper, we formulate a multi-round extension of the PM framework in
which each party can perform operations over an ordered sequence of quantum
nodes, assuming standard causality locally but not globally. The formalism is
analogous to the PM formalism, except that the local operations are now
generalised to quantum networks [15]—the most general causal operations that
can be applied over a given sequence of input and output systems, and which
are described by the theory of quantum combs [15]—while the process matrix is
replaced by an operator that we call the multi-round process matrix (MPM). The
MPM is a specific higher-order process in the hierarchy of higher order
processes classified by Perinotti and Bisio [39, 11]. We derive handy
necessary and sufficient conditions for an operator to be a valid MPM, which
are expressed via a generalisation of the projector techniques introduced for
PMs in Ref. [6]. Given a set of parties and an ordered set of nodes for each
party, a valid MPM is in particular a valid PM on all nodes, if each node is
regarded as belonging to a separate party. However, it respects additional
constraints that ensure the possibility of using side channels for causal
communication between the nodes within the laboratory of each actual party. As
we show, these constraints amount to the condition that the operator of a
valid MPM is an affine combination of deterministic quantum combs that are
compatible with the local orders of the nodes assumed for each different
party.

We further formulate a notion of causal separability in the MPM framework,
building upon the one defined for PMs [37, 50]. The key insight behind this
definition is that since every quantum comb can be implemented as a sequence
of independent operations connected via side channels, by viewing these side
channels as ‘external’ to the parties’ operations, we can reduce the problem
of defining causal separability to that for standard PMs. Remarkably, we show
that the possibility of using side channels can make a radical difference on
the causal (non)separability of the process: we describe an example of a
bipartite MPM with two nodes for Alice and one node for Bob, which is such
that, when viewed as a standard PM on three nodes it is causally separable
(and admitting a simple physical realisation), but when the possibility of
using a side channel in the presumed direction between the nodes of Alice is
considered, this opens up the possibility of violating causal inequalities.
This shows that the standard PM framework, applied naively on the nodes over
which the parties can operate, does not suffice to capture the causal
nonseparability of the process on those nodes: the in-principle possibility of
using side channels, which underlies the MPM concept, can have nontrivial
consequences.

##  2 The framework

Consider a set of parties, $\mathfrak{N}\equiv\left\\{A,B,C,\ldots\right\\}$,
of finite cardinality $\left|\mathfrak{N}\right|$. In the original process
matrix (PM) framework [38], each party $X$ is assumed associated with a pair
of finite-dimensional input and output quantum systems, $X_{in}$ and $X_{out}$
with respective Hilbert spaces $\mathcal{H}^{X_{in}}$ and
$\mathcal{H}^{X_{out}}$, hereby referred to as a (quantum) node [3, 7, 8]. The
party $X$ can perform an arbitrary causal quantum operation (also called
quantum instrument) from the input to the output. This is described by a
collection of completely positive (CP) maps [34] corresponding to the
different possible outcomes $i_{X}$ of the operation,
$\left\\{\mathcal{M}_{i_{X}}^{X_{in}\rightarrow X_{out}}\right\\}$,
$\mathcal{M}^{X}_{i_{X}}:\mathcal{L}(\mathcal{H}^{X_{in}})\rightarrow\mathcal{L}(\mathcal{H}^{X_{out}})$,
where $\mathcal{L}(\mathcal{H})$ denotes the space of linear operators over
$\mathcal{H}$. These CP maps obey the constraint that
$\mathcal{M}^{X_{in}\rightarrow
X_{out}}\equiv\sum_{i_{X}}\mathcal{M}_{i_{X}}^{X_{in}\rightarrow X_{out}}$ is
a completely positive and trace-preserving (CPTP) map.

In the PM formalism, CP maps are represented via positive semidefinite
operators using the Choi-Jamiołkowski (CJ) isomorphism [29, 18]. Here, we take
a basis-independent version of the isomorphism following [3, 7, 8] (mudulo an
overall transposition) for which a CP map $\mathcal{M}^{X_{in}\rightarrow
X_{out}}$ is represented by a CJ operator
$M^{X_{in}X^{*}_{out}}\in\mathcal{L}\left(\mathcal{H}^{X_{in}}\otimes(\mathcal{H}^{X_{out}})^{*}\right)$,
where $(\mathcal{H}^{X_{out}})^{*}$ is a copy of the dual of
$\mathcal{H}^{X_{out}}$ 111Note that this copy of the dual of the output
Hilbert space can be seen related to the time-reversed version of the output
Hilbert space [36] via a concrete linear map, which was incorporated in the
time-neutral generalisation of the formalism developed in Ref. [36]. We do not
invoke this here for simplicity., via

$M^{X_{in}X_{out}^{*}}\equiv\\\ \left(\sum_{i,j}\
\mathcal{M}^{X_{in}\rightarrow
X_{out}}(\ket{i}\bra{j}^{X_{in}})\otimes\ket{i}\bra{j}^{X^{*}_{in}}\right)^{\textrm{T}}\,,$
(1)

where $\left\\{\ket{i}^{X_{in}}\right\\}$ is an orthonormal basis of
$\mathcal{H}^{X_{in}}$, $\left\\{\ket{i}^{X^{*}_{in}}\right\\}$ the
corresponding dual basis, and we identify $(\mathcal{H}^{*})^{*}$ with
$\mathcal{H}$ via the canonical isomorphism. For a CPTP map
$\mathcal{M}^{X_{in}\rightarrow X_{out}}$,
$\Tr_{X_{out}^{*}}[M^{X_{in}X_{out}^{*}}]=\mathds{1}^{X_{in}}$.

Given a choice of instrument at each node, the joint probabilities for the
outcomes of all parties are then given by the ‘generalised Born rule’

$\displaystyle p(i_{A},i_{B},\ldots|\\{\mathcal{M}^{A_{in}\rightarrow
A_{out}}_{i_{A}}\\},\\{\mathcal{M}^{B_{in}\rightarrow
B_{out}}_{i_{B}}\\},\ldots)=$
$\displaystyle\textrm{Tr}\left[W^{A_{in}A^{*}_{out}B_{in}B^{*}_{out}\ldots}\left(M^{A_{in}A^{*}_{out}}_{i_{A}}\otimes
M^{B_{in}B^{*}_{out}}_{i_{B}}\ldots\right)\right],$ (2)

where
$W^{A_{in}A^{*}_{out}B_{in}B^{*}_{out}\ldots}\in\mathcal{L}(\mathcal{H}^{A_{in}}\otimes(\mathcal{H}^{A_{out}})^{*}\otimes\mathcal{H}^{B_{in}}\otimes(\mathcal{H}^{B_{out}})^{*}\ldots)$.
This follows simply from the assumption that, as in standard quantum theory,
the probabilities are linear functions of the CP maps corresponding to the
outcomes. The operator $W^{A_{in}A^{*}_{out}B_{in}B^{*}_{out}\ldots}$ is
called the process matrix (PM) (also process operator [7, 8]). The only
conditions it has to satisfy are
$W^{A_{in}A^{*}_{out}B_{in}B^{*}_{out}\ldots}\geq 0$ (coming from the
requirement of non-negativity of the probabilities, assuming the parties’
operations can be extended to act on local input ancillas prepared in
arbitrary joint quantum states), and
$\textrm{Tr}\left[W^{A_{in}A^{*}_{out}B_{in}B^{*}_{out}\ldots}\left(M^{A_{in}A^{*}_{out}}\otimes
M^{B_{in}B^{*}_{out}}\ldots\right)\right]=1$ on all CPTP maps
$M^{A_{in}A^{*}_{out}},M^{B_{in}B^{*}_{out}},\ldots$ (coming from the
requirement that probabilities sum up to 1). Practical necessary and
sufficient conditions for an operator to be a valid PM have been formulated
based on the types of nonzero terms appearing in the expansion of the operator
in a Hilbert-Schmidt basis [38, 37], as well as based on a superoperator
projector [6]. In the next section, we will use the latter, of which we will
derive a generalisation. A review of it can be found in Appendix A.1.

Figure 1: Graphical representation of a quantum network with 3 nodes (plain
circles), and its associated quantum 3-instrument obtained through CJ
isomorphism $\mathfrak{C}$ applied on the composition of the operations at
each node.

Formula (2) can be interpreted as the result of composing in a loop the local
operations of the parties with a channel $\tilde{\mathcal{W}}$ from the output
systems of all nodes to the inputs systems of all nodes: the PM is the
transpose of the CJ operator of $\tilde{\mathcal{W}}$, where the transposition
simply reflects the link product [15] for composing channels in the CJ
representation.

Here, we generalise the framework by relaxing the assumption that each party
is restricted to a single round of receiving a system in and sending a system
out. We now assume that each party is associated with an ordered sequence of
quantum nodes over which they can operate in a causal fashion. Let $n_{X}$
denote the number of nodes for party $X$. There are then $2n_{X}$ systems
associated with $X$. We will label them by $X_{j}$, $j=0,\ldots 2n_{X}-1$,
with $X_{0}$ and $X_{1}$ being respectively the input system and the dual of
the output system of the first node, $X_{2}$ and $X_{3}$ the input system and
the dual of the output system of the second node, and so on, with even
(respectively, odd) numbering referring to an input (dual of output) system.
The $i$-th node of $X$ will be compactly denoted by $X^{(i)}$. In other words,
for $i=1,\ldots,n_{X}$, with $X_{i,in}$ (respectively, $X_{i,out}$) referring
to the input (output) of the $i$-th node we define

$\displaystyle\mathcal{H}^{X_{2i-2}}\equiv\mathcal{H}^{X_{i,in}}\quad,$ (3a)
$\displaystyle\mathcal{H}^{X_{2i-1}}\equiv\left(\mathcal{H}^{X_{i,out}}\right)^{*}\quad,$
(3b)
$\displaystyle\mathcal{H}^{X^{(i)}}\equiv\mathcal{H}^{X_{i,in}}\otimes\left(\mathcal{H}^{X_{i,out}}\right)^{*}\quad.$
(3c)

The most general causal quantum operation on a sequence of $n$ nodes is a non-
deterministic quantum network [15], which can be implemented as a sequence of
$n_{X}$ instruments acting on the given nodes plus ancillary systems connected
via side channels, as illustrated on the left-hand side of Fig. 1 for the case
of 3 nodes. Such a network amounts to implementing a quantum operation from
the joint input system of all nodes to the joint output system of all nodes,
called an n-instrument [15]. In the CJ representation, an $n_{X}$-instrument
is described by a collection of positive semidefinite (PSD) operators
$\\{M^{X_{0}X_{1}\cdots X_{2n_{X}-1}}_{i}\\}$, defined on
$\mathcal{L}(\mathcal{H}^{X_{0}}\otimes\mathcal{H}^{X_{1}}\ldots\otimes\mathcal{H}^{X_{2n_{X}-1}})$.
We will shorten the notation using $X_{0}X_{1}\cdots X_{2n_{X}-1}\equiv X$ to
lessen clutter when there is no ambiguity, writing
$\left\\{M^{X}_{i}\right\\}\in\mathcal{L}\left(\mathcal{H}^{X}\right)$. Each
$M^{X}_{i}$ is labelled by the outcome index $i$, which in this case is a
poly-index corresponding to the collection of outcomes at the different steps
in the network, $i\equiv(i_{X^{(1)}},\ldots,i_{X^{(n_{X})}})$. The operators
$M^{X}_{i}$ are the CJ operators of CP maps from the joint input system of all
nodes to the joint output system of all nodes, called probabilistic quantum
($n_{X}$-)combs [15]. These operators must satisfy the condition that
$M^{X}\equiv\sum_{i}M^{X}_{i}$ is a deterministic quantum ($n_{X}$-)comb [15].
The latter is the CJ operator of a deterministic $n_{X}$-instrument, i.e. the
most general CPTP map from the joint input system of the nodes to the joint
output system of the nodes that can be implemented via a network of channels
as in the left-hand side of Fig. 1. An operator $M^{X}$ is a deterministic
quantum $n_{X}$-comb if and only if it obeys the following constraints [15]:

$\displaystyle M^{X_{0}\cdots X_{2n_{X}-1}}\geq 0\,,$ (4a)
$\displaystyle\begin{split}\forall j\,:\,2\leq j\leq n_{X},&\\\
\mathrm{Tr}_{X_{2j-1}}\left[M^{X_{0}\cdots
X_{2j-1}}\right]&=\mathds{1}^{X_{2j-2}}\otimes M^{X_{0}\cdots X_{2j-3}},\\\
\mathrm{Tr}_{X_{1}}\left[M^{X_{0}X_{1}}\right]&=\mathds{1}^{X_{0}}\,,\end{split}$
(4b)

where $\mathds{1}^{Y}$ denotes the unit matrix on subsystem $Y$.

Analogously to the PM framework, we assume that the joint probabilities for
the outcomes of the generalised instruments performed by the separate parties
in this multi-round setting are linear functions of the probabilistic quantum
combs associated with the outcomes, as would be the case in standard quantum
mechanics if the nodes are associated with quantum systems at definite times.
This means that we can write these probabilities in the ‘generalised Born
rule’ form

$\displaystyle
p(i_{A},i_{B},\ldots|\\{M^{A}_{i_{A}}\\},\\{{M}^{B}_{i_{B}}\\},\ldots)$
$\displaystyle=\textrm{Tr}\left[W^{AB\ldots}\cdot\left(M^{A}_{i_{A}}\otimes
M^{B}_{i_{B}}\ldots\right)\right],$ (5)

where for the superscripts that label the systems over which the operators are
defined we have used the short-hand notation $X\equiv X_{0}X_{1}\ldots
X_{2n_{X}-1}$, $X\in\mathfrak{N}$. We call the operator $W^{AB\ldots}$ the
multi-round process matrix (MPM) (see Fig. 2) .

Figure 2: Graphical representation of an MPM for 2 parties A and B. A has 3
rounds and B has 2. Here, the name of the subsystems associated with each wire
is apparent. By convention, the ordering of the indices goes from bottom to
top

As in the PM framework, the only property we demand from $W^{AB\ldots}$ is
that it yields valid probabilities through Eq. (5) for all generalised
instruments that can be applied by the parties, including when the instruments
are extended to act on ancillary input systems in arbitrary quantum states. It
is straightforward to see in analogy to the argument in Ref. [38] that this is
equivalent to the following constraints:

$\displaystyle W^{AB\ldots}\geq 0\,,$ (6)

and

$\displaystyle\textrm{Tr}\left[W^{AB\ldots}\cdot\left(M^{A}\otimes
M^{B}\ldots\right)\right]=1$ (7)

for all deterministic quantum combs $M^{A},M^{B},\ldots$.

Figure 3: Graphical representations of an MPM $W$ for: (left) 3 parties with a
single node each [$W$ is a PM in this case]; (center) a party $A$ having a
single round and $B$ having two rounds; (right) a single party $A$ with 3
rounds [$W$ is a deterministic quantum comb in this case].

Remark 1. In the case when the number of nodes per party is 1, $W^{AB\ldots}$
reduces to a standard process matrix, whereas when the number of parties is 1
and that party has $n$ nodes, $W^{A}$ reduces to (the transpose of) a
deterministic quantum $(n+1)$-comb with a trivial first input system and a
trivial last output system [15]. See Fig. 3.

Remark 2. Note that if an operator $W^{AB\ldots}$ is a valid MPM, it is in
particular a valid PM if each different node is interpreted as belonging to a
separate party. This is because the probabilities must be equal to 1 when the
parties perform independent CPTP maps in their different nodes, which is a
special case of a deterministic network. The converse, however, is obviously
not true since a PM is not normalised on deterministic quantum combs on
arbitrary ordered subsets of its nodes (e.g. if the PM is equivalent to a
deterministic quantum comb for a specific order of the nodes, it would
generally not admit compositions with deterministic quantum combs in the
opposite order).

##  3 Characterisation of the set of valid MPMs

We now derive necessary and sufficient conditions for an operator to be a
valid MPM that are easier to handle. They can be seen as a generalisation of
the projective characterisation of the set of PMs obtained in Ref. [6], where
the authors showed that the set of valid PMs belongs to a subspace of the
space of operators, the projector on which can be deduced from the one on
valid quantum channels.

Our approach goes as follows. Given the projective characterisation of the
space of valid quantum channels in the CJ picture [6], we can iteratively
infer a similar projective characterisation of the space of deterministic
quantum combs. This is done in Appendix A.2, and we find that all the
deterministic quantum combs $M^{X}$, no matter their number of nodes (or
‘teeth’), have the same algebraic structure:

$\displaystyle M^{X}\geq 0\quad,$ (8a)
$\displaystyle\mathcal{P}^{X}_{n_{X}}\left[M^{X}\right]=M^{X}\quad,$ (8b)
$\displaystyle\mathrm{Tr}\left[M^{X}\right]=\prod_{j=0}^{n-1}d_{X_{2j}}\equiv
d_{X_{in}}\,.$ (8c)

That is, they are elements of a positive and trace-normalised subset within a
subspace of the space of operators on the full system. The subspace is defined
through a projector $\mathcal{P}^{X}_{n_{X}}$, whose superscript refers to the
system it acts upon (we will refine the signification of the superscript in
the next section) and whose subscript indicates the total number of nodes. As
a function of $X$, the exact form of the projector is given by the recursive
relation

$\mathcal{P}_{n_{X}}^{X}=\mathcal{I}^{X}-\prescript{}{X_{2n_{X}-1}}{\left(\cdot\right)}+\prescript{}{X_{2n_{X}-2}X_{2n_{X}-1}}{\left(\mathcal{P}^{X^{\prime}}_{n_{X}-1}\right)}\,,$
(9)

expressed in terms of the mapping
$\prescript{}{X_{i}}{\left(\cdot\right)}\equiv\frac{\mathds{1}^{X_{i}}}{d_{X_{i}}}\otimes\Tr_{X_{i}}\left[\cdot\right]$
introduced in Ref. [6] (see Appendix A.1) and the identity projector
$\mathcal{I}^{X}\left[O\right]=O\,,\forall\;O\in\mathcal{L}\left(\mathcal{H}^{X}\right)$.
$\mathcal{P}^{X^{\prime}}_{n_{X}-1}$ is the $(n_{X}-1)$-comb projector that
acts on subsystems $X^{\prime}\equiv X_{0}\ldots X_{2n_{X}-3}$ (implicitly
understood extended by a tensor product with the identity on
$X_{2n_{X}-2}X_{2n_{X}-1}$). The recursion starts with the 0-comb projector
being 1, $\mathcal{P}_{0}\equiv 1$, because the space of 0-combs is the real
numbers.

Here, we are interested in characterising the set $\\{W\\}$ of valid MPMs as
defined in Eqs. (6) and (7). Note that condition (7), which states that an MPM
is normalised on all tensor products of deterministic quantum combs, is
equivalent to the condition that an MPM is normalised on all affine
combinations of tensor product of such combs:

$\mathrm{Tr}\left[W^{AB\ldots}\cdot\sum_{i}q_{i}\left(M^{A}\otimes
M^{B}\ldots\right)_{i}\right]=1\,,$ (10)

with $\sum_{i}q_{i}=1$ while $\left(M^{A}\otimes M^{B}\ldots\right)_{i}$ is a
tensor product of deterministic quantum combs that may be different for each
index $i$. The fact that (10) is necessary for (7) follows from the linearity
of the trace, while its sufficiency follows from the fact that (7) is a
special case of (10).

In Appendix B, we show that an operator $M$ is an affine combination of tensor
product of $|\mathfrak{N}|$ deterministic combs if and only if it respects the
conditions

$\displaystyle\mathcal{P}\left[M\right]\equiv\left(\bigotimes_{X\in\mathfrak{N}}\mathcal{P}^{X}_{n_{X}}\right)[M]=M\,,$
(11a)
$\displaystyle\mathrm{Tr}\left[M\right]=\prod_{X\in\mathfrak{N}}d_{X_{in}}\equiv
d_{in}\,,$ (11b)

where we have defined the notations $\mathcal{P}$ for the overall projector
and $d_{in}$ for the input dimension of the full Hilbert space. In the
following, $\mathrm{Span}\left\\{\bigotimes M\right\\}$ will be used to refer
quickly to the subspace of operators spanned by a tensor product of
deterministic combs, i.e. operators satisfying (11a).

With these considerations, the rationale in Ref. [6] of inferring PM validity
conditions from the projector on the space of quantum channels and the
normalisation can be generalised to MPMs using the projective characterisation
(11) of the affine combination of tensor products of deterministic combs. This
result is systematised as a theorem:

######  Theorem 1.

Let
$\mathcal{L}\left(\mathcal{H}\right)\equiv\mathcal{L}\left(\mathcal{H}^{in}\otimes\left(\mathcal{H}^{out}\right)^{*}\right)$
be a linear space of operators defined on a Hilbert space of dimension
$d\equiv d_{in}d_{out}$. This Hilbert space admits a tensor factorisation into
$2|\mathfrak{N}|$ subsystems associated with the inputs and outputs of
$|\mathfrak{N}|$ parties. Let
$\\{W\\}\subset\mathcal{L}\left(\mathcal{H}\right)$ be the set of valid MPMs
shared between the parties. Let $\mathcal{P}^{X}_{n_{X}}$ be the projector
characterising the validity subspace of the deterministic combs that can be
applied by party $X$. Let
$\mathcal{P}\equiv\bigotimes_{X\in\mathfrak{N}}\mathcal{P}^{X}_{n_{X}}$ be the
tensor product of the different such projectors of all parties. Then, an
operator $W$ belongs to the set $\\{W\\}$ if and only if it satisfies

$\displaystyle W\geq 0\,,$ (12a)
$\displaystyle\prescript{Q}{}{\mathcal{P}}[W]\equiv\left(\mathcal{I}-\mathcal{P}+\mathcal{D}\right)[W]=W\,,$
(12b) $\displaystyle\mathrm{Tr}\left[W\right]=d_{out}\,,$ (12c)

where $\mathcal{I}$ is the identity projector, and $\mathcal{D}$ is the
projector on the span of the unit matrix,
$\mathcal{D}[O]=\frac{\mathds{1}}{d}\otimes\Tr\left[O\right]$.

The projector $\prescript{Q}{}{\mathcal{P}}$ defined in Eq. (12b) will be
referred to as the quasiorthogonal projector to $\mathcal{P}$. The proof of
this theorem is presented in Appendix B. It can be understood in a more
mathematical terminology as “the CJ representation of the generalised
instruments is a subset of a unital subalgebra which is the quasiorthogonal
complement to the subalgebra in which lies the set of valid processes
$\\{W\\}$” [40, 27]. Equivalently, it says that the traceless part of an MPM
is orthogonal (with respect to the Hilbert-Schmidt inner product) to the
traceless part of the CPTP maps that are plugged into it [11].

We note that an equivalent characterisation has been independently explored in
[39, 11], but without using projective methods. In particular, Theorem 1
corresponds to the Lemma 4 of [11]. Indeed, in Perionotti and Bisio’s
classification of higher-order quantum processes, the MPM is described by a
type
$\left(\left(\mathbf{n}_{A}\otimes\mathbf{n}_{B}\otimes...\right)\rightarrow
1\right)$. That is, a structure taking in a tensor product of quantum combs of
types $\mathbf{n}_{A},\mathbf{n}_{B},\ldots$ and sending them onto the trivial
type 1. Theorem 1 is therefore a direct way of finding the constraints to
apply on an operator so that it is the representation of such a type. In a
subsequent work, we will present a systematic way of associating a projector
to a type structure.

##  4 Link between comb and MPM subspaces

Using the above projective characterisation, we are able to highlight the link
between the validity subspace of deterministic combs with trivial first input
and last output and that of MPMs. We will need to introduce the symbol $\prec$
to mark the causal relation “is before”, indicating that a node is in the
causal past of another. The projector on the subspace of valid 3-combs with
teeth named $A,B,C$ so that $A\prec B\prec C$ will be denoted
$\mathcal{P}^{A\prec B\prec C}_{3}$, with the total number of teeth indicated
as a subscript and the causal ordering as a superscript. Permutations of the
ordering will be denoted $\pi_{i}\left(A,B,C\right)$, with the subscript
referring to a particular permutation, e.g. $\pi_{0}\left(A,B,C\right)=A\prec
B\prec C\,,\pi_{1}\left(A,B,C\right)=B\prec A\prec C,\ldots$ . When the
letters refer to an ordered set of teeth, for example
$A=\left\\{A^{(1)}\right\\}$ and $B=\left\\{B^{(1)}\prec B^{(2)}\right\\}$,
the permutations are limited to those that are compatible with the the
ordering between the elements within each set. So for this particular example,
compatibility means that permutations containing a causal relation where
$B^{(2)}$ appears before $B^{(1)}$ are forbidden, giving
$\\{\pi_{i}(A,B)\\}=\\{A^{(1)}\prec B^{(1)}\prec B^{(2)},B^{(1)}\prec
A^{(1)}\prec B^{(2)},B^{(1)}\prec B^{(2)}\prec A^{(1)}\\}$.

Observe that Theorem 1 states that the projector on the subspace of valid MPMs
is quasiorthogonal to the projector on $\mathrm{Span}\left\\{\bigotimes
M\right\\}$, where deterministic combs in the tensor product are associated
with the different parties. For a single party $A$, it is expected that the
MPM is a deterministic comb [15] (see Fig. 3, rightmost case). Indeed, it can
be verified using Eq. (9) that the quasiorthogonal projector to a single comb
projector is a comb projector itself; as expected, it sends to the validity
subspace of deterministic $(n_{A}+1)$-combs with trivial first input and last
output, and with node ordering compatible with the ordering of the teeth of
$A$. By “compatible”, we mean here that the nodes in the $(n_{A}+1)$-comb,
which are now the gaps between its teeth, are properly ordered, i.e.
$A^{(1)}\prec A^{(2)}\prec\ldots$ . Since we are working with orthogonal
projectors, equivalence between projectors is sufficient to prove the
equivalence between subspaces.

We can actually use the algebra of these projectors described in Appendix A.1
to generalise this observation to any number of parties. For $|\mathfrak{N}|$
parties sharing an MPM with $n=n_{A}+n_{B}+\ldots$ nodes, what we find is that
the MPM validity subspace is the span of all the deterministic $(n+1)$-combs
with trivial first input and last output such that their node ordering is
compatible with the local ordering of each of the $|\mathfrak{N}|$ combs that
are to be plugged into it.

For example, if Alice has one operation and Bob two, there are three possible
such 4-combs compatible with $B^{(1)}\prec B^{(2)}$, having projectors
$\prescript{Q}{}{\mathcal{P}}^{A^{(1)}\prec B^{(1)}\prec B^{(2)}}_{3}$,
$\prescript{Q}{}{\mathcal{P}}^{B^{(1)}\prec A^{(1)}\prec B^{(2)}}_{3}$, and
$\prescript{Q}{}{\mathcal{P}}^{B^{(1)}\prec B^{(2)}\prec A^{(1)}}_{3}$ sending
to three different subspaces. The validity subspace of the MPM is then spanned
by the union of these three subspaces.

To make this claim explicit, we need to introduce the notion of union of
projectors [41]. For two arbitrary linear projectors $\mathcal{P}$ and
$\mathcal{P}^{{}^{\prime}}$ acting on the same space, the projector on the
span of the union of their subspaces is the union of the projectors given by

$\mathcal{P}\cup\mathcal{P}^{{}^{\prime}}=\mathcal{P}+\mathcal{P}^{{}^{\prime}}-\mathcal{P}\mathcal{P}^{{}^{\prime}}\,,$
(13)

provided they commute. Thus, for
$\mathfrak{M},\mathfrak{N}\subset\mathcal{L}\left(\mathcal{H}\right)$ such
that $\mathfrak{M}=\left\\{M|\mathcal{P}\left[M\right]=M\right\\}$ and
$\mathfrak{N}=\left\\{N|\mathcal{P}^{{}^{\prime}}\left[N\right]=N\right\\}$,
we have
$O\in\text{Span}\left\\{\mathfrak{M}\cup\mathfrak{N}\right\\}\iff\left(\mathcal{P}\cup\mathcal{P}^{{}^{\prime}}\right)[O]=O$.

This allow us to formally state the result as a theorem:

######  Theorem 2.

For a set of parties $\\{A,B,\ldots\\}$ with each having possibly more than
one node, let $\\{\pi_{i}\\}$ denote the set of valid permutations between the
nodes as defined above. The projector on the linear subspace of valid MPMs
shared between the parties is then equivalent to the union of all the
quasiorthogonal projectors to the projectors on the linear subspaces of the
deterministic quantum combs that respect the partial ordering of the teeth for
each party:

$\prescript{Q}{}{\left(\mathcal{P}^{A}_{n_{A}}\otimes\mathcal{P}^{B}_{n_{B}}\otimes\ldots\right)}=\bigcup_{\pi_{i}}\prescript{Q}{}{\mathcal{P}}^{\pi_{i}(A,B,\ldots)}_{n_{A}+n_{B}+\ldots}\,.$
(14)

A proof of this theorem is given in Appendix C. It shows that any MPM, and as
a special case any PM, is a linear combination of deterministic quantum combs
that respect the local ordering of each party. The trace condition (12c)
further constrains the coefficients in the combination to sum up to one, hence
this is an affine combination.

##  5 (Non)causal correlations and causal (non)separability for MPMs

Now that the set of valid MPMs has been characterised, a natural ensuing
question is which ones describe causal correlations between the nodes. On a
purely theory-independent account, let the set of nodes
$A^{(1)},\ldots,A^{(n_{A})},B^{(1)},\ldots,B^{(n_{B})},C^{(1)},\ldots$ receive
settings $\vec{s}\equiv\left(s_{A^{(1)}},s_{A^{(2)}},\ldots\right)$ and
produce outcomes $\vec{o}\equiv\left(o_{A^{(1)}},o_{A^{(2)}},\ldots\right)$
after performing their operations. We can formulate a notion of the
correlations $p(\vec{o}|\vec{s})$ being causal, building upon the notion of
causal correlations introduced for the original process framework in Ref.
[37].

The idea of that definition is that the correlations are deemed casual if they
are compatible with a stochastic unravelling of the events at the nodes in a
causal sequence, which can be described as follows: first, according to some
probability distribution, one of all nodes is selected to come first; then,
depending on the setting chosen at that node, a specific outcome occurs at it
with a specific probability; after that, one of the remaining nodes is
selected to come second according to a probability distribution that may
generally depend on the setting and outcome that have occurred at the first
node; this continues until all nodes have been used, with the probability for
each subsequent event generally depending on all events that have occurred in
the past. In the multiround case, the same idea can be formalised as follows:

######  Definition 1 (Causal multi-round correlations).

We call the conditional probability distribution $p(\vec{o}|\vec{s})$ causal
if and only if it can be decomposed as

$\begin{split}&p(\vec{o}|\vec{s})\\\
&=\sum_{X\in\mathfrak{N}}q_{X^{(1)}}\;p(o_{X^{(1)}}|s_{X^{(1)}})p(\vec{o}_{\setminus
X^{(1)}}|o_{X^{(1)}},\vec{s})\,,\end{split}$ (15)

with $q_{X^{(1)}}\in[0;1]$, $\sum_{X\in\mathfrak{N}}q_{X^{(1)}}=1$, where
$p(o_{X^{(1)}}|s_{X^{(1)}})$ is a single-node distribution and
$p(\vec{o}_{\setminus X^{(1)}}|o_{X^{(1)}},\vec{s})$ is itself a conditional
causal probability distribution on the remaining nodes after the relabelling
$X^{(i)}\rightarrow X^{(i-1)}$ for $i=2,\ldots,n_{X}$ and $n_{X}\rightarrow
n_{X}-1$. We have used the shorthand notation $\vec{o}_{\setminus
X^{(1)}}\equiv\vec{o}\setminus\left\\{o_{X^{(1)}}\right\\}$ for the vector of
remaining outcomes.

This definition is similar to the one for the single-round case [37, 1], with
the difference that only unravellings compatible with the local causal orders
of the nodes of the parties are permitted. This extra condition is a
nontrivial addition: the fact that the overall correlations must be compatible
with no signalling from future nodes to past nodes does not guarantee that if
the correlations admit a causal unravelling in principle, that unravelling
would respect the local orders. Consider for example the correlations between
a party $A$ having two nodes and a party $B$ having one node. Let there be no
signalling from $A^{(2)}$ to $A^{(1)}$. Hence, the correlations are in
principle compatible with the assumed local causal ordering. However, with
this assumption alone, there can be perfect signalling from $A^{(2)}$ to $B$
and from $B$ to $A^{(1)}$, as long as $B$ does not forward any information
obtained from $A^{(2)}$ to $A^{(1)}$. In such a scenario, no unravelling is
compatible with the supplementary constraint that $A^{(1)}$ is before
$A^{(2)}$.

The new definition allows us to speak about causal inequalities for the MPM,
which are bounds on the set of joint probability distributions whose violation
by a specific distribution implies that the distribution is not causal [38].
Investigating the subject of causal inequalities for MPMs is left for future
work.

In the process formalism, a given PM is causally separable if and only if it
produces causal correlations in which the conditional probability
distributions appearing in the causal unravelling can themselves be seen as
arising from a quantum process [37] (see precise definition later in the case
of the MPM). Note that while causal separability implies causal correlations,
the converse is not true. In accordance with this definition, we want to
define causal separability for MPMs as the ability to unravel the causal
ordering of the operations. This unravelling should be compatible with the
partial ordering existing between the nodes associated to each party, and it
should also take into account the effects of the new means of communication
provided by the side channels between those nodes. As it turns out, this
requirement motivates non-trivial modifications to the current definition for
PMs [37, 50]. Noticing that every generalised instrument performed by a given
party can be thought of as consisting of independent single-node operations
connected via side channels (see below), a concise way of taking this into
account is to apply the PM definition of causal separability on an MPM
extended by identity side channels.

######  Definition 2.

An MPM $W$ is causally separable if and only if the operator obtained by
extending the nodes of each party with identity side channels between
consecutive nodes is causally separable in the sense of a PM. In equation, let
there be an extension of all the parties’ subsystems in an MPM $W$ such that
$\bigotimes_{i=0}^{2n_{X}-1}\mathcal{H}^{X_{i}}\rightarrow\bigotimes_{i=0}^{2n_{X}-1}\mathcal{H}^{X_{i}}\otimes\bigotimes_{j=1}^{2n_{X}-2}\mathcal{H}^{X^{\prime}_{j}}$,
where $d_{X^{\prime}_{2k-1}}=d_{X^{\prime}_{2k}}$, $k=1,\ldots,n_{X}-1$, so
that a node $X^{(i)}$, $1<i<n_{X}-1$, is now defined on
$\mathcal{H}^{X_{2i-2}}\otimes\mathcal{H}^{X_{2i-1}}\otimes\mathcal{H}^{X^{\prime}_{2i-2}}\otimes\mathcal{H}^{X^{\prime}_{2i-1}}$,
$X^{(1)}$ is defined on
$\mathcal{H}^{X_{0}}\otimes\mathcal{H}^{X_{1}}\otimes\mathcal{H}^{X_{1}^{\prime}}$,
and $X^{(n_{X}-1)}$ is defined on
$\mathcal{H}^{X_{2n_{X}-2}}\otimes\mathcal{H}^{X_{2n_{X}-1}}\otimes\mathcal{H}^{X^{\prime}_{2n_{X}-2}}$.
We define a PM $W^{\prime}$ on the extended nodes as

$W^{\prime}=W\otimes\left(\bigotimes_{X\in\mathfrak{N}}\bigotimes_{i=1}^{n_{X}-1}Id^{X^{\prime}_{2i-1}X^{\prime}_{2i}}\right)\,,$
(16)

where $Id^{X^{\prime}_{2i-1}X^{\prime}_{2i}}$ is the transpose of the CJ
operator of an identity side channel from
$\left(\mathcal{H}^{X^{\prime}_{2i-1}}\right)^{*}$ to
$\mathcal{H}^{X^{\prime}_{2i}}$. Then, $W$ is a causally separable MPM if and
only if $W^{\prime}$ is a causally separable PM for all dimensions of the
identity side channels.

For completeness, we now show explicitly that the correlations obtained
through the ‘Born rule’ (5) for MPMs can be transformed into correlations
obtained with a PM extended by identity side channels. We will make use of a
version of the link product [15], which allows to merge combs together. In the
context of our basis-independent convention for the CJ isomorphism, we define
the link product between two operators $M^{AB^{*}}$ and $N^{BC^{*}}$, where
$A$ and $C^{*}$ are separate systems or the trivial system, and $B$ and
$B^{*}$ are mutually dual (i.e. $\mathcal{H}^{B^{*}}$ and $\mathcal{H}^{B}$
are Hilbert spaces dual to each other, assuming the canonical isomorphism
$(\mathcal{H}^{*})^{*}\cong\mathcal{H}$ in the finite-dimensional case), as

$\begin{gathered}L^{AC^{*}}=M^{AB^{*}}\ast N^{BC^{*}}:=\\\
\mathrm{Tr}_{BB^{*}}\left[\left(M^{AB^{*}}\otimes
N^{BC^{*}}\right)\left(\mathds{1}^{A}\otimes
Id^{BB^{*}}\otimes\mathds{1}^{C^{*}}\right)\right],\end{gathered}$ (17)

where $Id^{BB^{*}}\equiv\sum_{ij}^{d_{B}}\outerproduct{ii}{jj}^{BB^{*}}$.

Figure 4: Graphical interpretation of Eq.(21): an MPM (left) taking in a
2-comb is interpreted as a PM (right) extended by a side-channel (dashed part)
taking in two 1-combs.

Consider a quantum 2-comb $M^{A}\equiv M^{A_{0}A_{1}A_{2}A_{3}}$. The CP map
it describes can be seen as resulting from the composition of two CP maps with
an intermediate identity side channel. In the CJ representation, this is given
by the link product between the corresponding 1-combs,

$M^{A}=M^{A_{3}A_{2}L_{2}}\ast Id^{L_{2}^{*}L_{1}^{*}}\ast
M^{L_{1}A_{1}A_{0}}\quad,$ (18)

where $L_{1}^{*}$ is the ancillary output system of the first CP map (with CJ
operator $M^{L_{1}A_{1}A_{0}}$), $L_{2}$ is the ancillary input system of the
second CP map (with CJ operator $M^{A_{3}A_{2}L_{2}}$), and
$Id^{L_{2}^{*}L_{1}^{*}}$ is the CJ operator of the identity side channel
connecting these systems. This is explicitly given by

$M^{A}=\\\
\text{Tr}_{L_{2}L_{2}^{*}L_{1}L_{1}^{*}}\Big{[}\left(M^{A_{3}A_{2}L_{2}}\otimes
Id^{L_{2}^{*}L_{1}^{*}}\otimes M^{L_{1}A_{1}A_{0}}\right)\\\
\times\left(\mathds{1}^{A_{3}A_{2}}\otimes Id^{L_{2}L_{2}^{*}}\otimes
Id^{L_{1}^{*}L_{1}}\otimes\mathds{1}^{A_{1}A_{0}}\right)\Big{]}\quad.$ (19)

Computing the trace over $L_{2}^{*}L_{1}^{*}$ yields

$M^{A}=\\\ \mathrm{Tr}_{L_{2}L_{1}}\left[\left(M^{A_{3}A_{2}L_{2}}\otimes
M^{L_{1}A_{1}A_{0}}\right)\left(\mathds{1}^{A}\otimes
Id^{L_{2}L_{1}}\right)\right].$ (20)

In this last expression,
$Id^{L_{2}L_{1}}\equiv\sum_{ij}^{d_{L}^{2}}\outerproduct{ii}{jj}^{L_{2}L_{1}}$
is the transpose of the CJ operator of an identity channel from
$\mathcal{H}^{L_{1}^{*}}$ to $\mathcal{H}^{L_{2}}$ (just like a PM is the
transpose of the CJ operator of a channel from the outputs of nodes to the
inputs of nodes).

Plugging equation (20) into the ‘Born rule’ between the 2-comb $M^{A}$ and a
2-node MPM $W$, one has

$\mathrm{Tr}\left[M^{A}W\right]=\\\
\mathrm{Tr}\left[\left(M^{A_{3}A_{2}L_{2}}\otimes
M^{L_{1}A_{1}A_{0}}\right)\left(W\otimes Id^{L_{2}L_{1}}\right)\right],$ (21)

where $M^{A_{3}A_{2}L_{2}},M^{L_{1}A_{1}A_{0}}$ are quantum 1-combs, and
$W\otimes Id^{L_{2}L_{1}}$ can be shown to be a valid (M)PM, hinting the
formula (16) for the general case. Indeed, formula (18) can be applied
repetitively in the case of a party with more than 2 nodes:
$M^{A}=M^{{A}_{2n_{A}-2}{A}_{2n_{A}-1}{L}_{2n_{A}-2}}\ast\ldots\ast
M^{L_{3}A_{3}A_{2}L_{2}}\ast Id^{L_{2}^{*}L_{1}^{*}}\ast
M^{{L}_{1}{A}_{1}{A}_{0}}$, and then the identity
$\mathrm{Tr}_{Y}\left[A^{Y}\cdot\mathrm{Tr}_{X}\left[B^{XY}\right]\right]=\mathrm{Tr}_{XY}\left[\left(\mathds{1}^{X}\otimes
A^{Y}\right)B^{XY}\right]$ can be used to rewrite the whole expression as one
overall partial trace. The tensor product being a special case of link product
over the trivial system [15], this argument also applies to the multipartite
case.

Consequently, the correlations achievable with an MPM are equivalent to those
in a PM extended by side channels and we can appeal to the concept of causal
separability for PMs in order to define causal separability for MPMs as it is
done in Definition 2.

As shown in Ref. [37], for the PM definition of causal separability, the
possibility of extending the local operations of the parties to act on local
ancillary input systems that may be entangled with the ancillary input systems
of other parties also needs to be considered. Neglecting to do so may
mistakenly lead to the conclusion that a PM is causally separable, although it
is able to violate a causal inequality when using such an extension. Likewise,
we will show in the next section that neglecting the side channels leads to
the same kind of problem. We end this section by reformulating Definition 2
into one closer to Ref. [37, 50], so that the consequences of local causal
ordering are made explicit and extension by arbitrary ancillas and side
channels is explicit.

We only need to rewrite (21) as

$\mathrm{Tr}\left[M^{A}W\right]=\mathrm{Tr}_{A_{3}A_{2}L_{2}}\left[M^{A_{3}A_{2}L_{2}}W_{|M^{L_{1}A_{1}A_{0}}}\right]\,,$
(22)

where we have defined the conditional process matrix

$W_{|M^{L_{1}A_{1}A_{0}}}=\\\
c\;\mathrm{Tr}_{L_{1}A_{1}A_{0}}\left[\left(\mathds{1}\otimes
M^{L_{1}A_{1}A_{0}}\right)(W\otimes Id^{L_{2}L_{1}})\right]\,.$ (23)

This can be shown to be a valid process matrix on the remaining node between
$A_{2}L_{2}$ and $A_{3}$ for all the possible choices of
$M^{L_{1}A_{1}A_{0}}$, as long as the normalisation factor is chosen such that
Eq. (12c) is satisfied. On the contrary, the analogously defined
$W_{|M^{A_{3}A_{2}L_{2}}}$ will not always be a valid PM for all
$M^{A_{3}A_{2}L_{2}}$ as it can lead to post-selection. This implies that when
we reformulate the ‘Born rule’ so as to make the unravelling from an $n$-node
MPM to an $(n-1)$-node MPM apparent, starting the unravelling with a node that
is not the first of some party is automatically forbidden. The generalisation
of this unravelling procedure yields the following definition:

######  Definition 3 (Causal separability of the MPM).

Consider an MPM $W$ shared by $|\mathfrak{N}|$ parties. For $|\mathfrak{N}|=1$
this MPM is a deterministic quantum comb and thus causally separable. For
$|\mathfrak{N}|>1$, the MPM is causally separable if and only if, for any
state
$\rho\in\mathcal{L}\left(\bigotimes_{X\in\mathfrak{N}}\bigotimes_{i=0}^{n_{X}-1}\mathcal{H}^{X_{2i}^{\prime}}\right)$
defined on an extension of the parties’ input subsystems
$\mathcal{H}^{X_{2i}}\rightarrow\mathcal{H}^{\tilde{X}_{2i}}=\mathcal{H}^{X_{2i}}\otimes\mathcal{H}^{X_{2i}^{\prime}}$,
the extended MPM can be decomposed as

$W\otimes\rho=\sum_{X\in\mathfrak{N}}q_{X^{(1)}}W^{\rho}_{X^{(1)}}\,,$ (24)

with $q_{X^{(1)}}\geq 0$, $\sum_{X\in\mathfrak{N}}q_{X^{(1)}}=1$, and where
$W^{\rho}_{X^{(1)}}$ is an MPM compatible with party $X$’s first operation
being first in the causal unravelling (i.e. there can be no signalling from
the rest of the nodes to $X^{(1)}$ [37]), so that the conditional MPM after
the first operation of $X$ has been carried out,

$\left(W^{\rho}_{X^{(1)}}\right)_{\left|M^{X_{0}X^{\prime}_{0}X_{1}L_{1}}\right.}=c\;\mathrm{Tr}_{X_{0}X^{\prime}_{0}X_{1}L_{1}}\left[\left(M^{X_{0}X^{\prime}_{0}X_{1}L_{1}}\otimes\mathds{1}\right)\left(W^{X}\otimes\rho^{X^{\prime}}\otimes
Id^{L_{2}L_{1}}\right)\right]\,,$ (25)

is itself causally separable for all possible CP maps between
$\mathcal{L}\left(\mathcal{H}^{X_{0}}\otimes\mathcal{H}^{X^{\prime}_{0}}\right)$
and
$\mathcal{L}\left(\left(\mathcal{H}^{X_{1}}\right)^{*}\otimes\left(\mathcal{H}^{L_{1}}\right)^{*}\right)$
represented by the CJ operator $M^{X_{0}X^{\prime}_{0}X_{1}L_{1}}$. Here
$L_{2}$ is an extension of the input system of $X^{(2)}$ and $c$ is a
normalisation constant.

In the two equivalent definitions of causal separability above, we have left
the dimensions of the side channels unbounded. Yet, it is natural to ask
whether a bounded dimension could suffice. It is clear that no bounded
dimension could reproduce all correlations achievable with an MPM because, for
example, a given party could follow a protocol where she applies a different
instrument at her second node depending on the outcome of the instrument
applied at her first node. By taking the number of outcomes at the first node
sufficiently large, the side channel required to realise this situation could
be made larger than any assumed bound. However, it may still be the case in
principle that some bounded dimension for the side channels (generally
dependent on the dimensions of the nodes of the MPM) is sufficient for the
definition of causal separability because it implies the analogous property
for all dimensions. The question of whether such a bound exists is left open
for future work.

##  6 Activation of causal nonseparability by a side channel

Since a PM is a special case of an MPM with no partial ordering at all assumed
for its nodes, one may think that applying the PM definition to an MPM would
be sufficient to establish causal (non)separability. Remarkably, this is not
true, as we will now demonstrate by an example.

Consider an MPM for two parties, $A$ and $B$, where $A$ has two nodes and $B$
a single node. Let the dimensions of all input and output systems of all nodes
be equal to 2: $d_{X_{i}}=2\quad\forall X\in\mathfrak{N}\,,0\leq i<2n_{X}-1$.
We take the MPM to have the following expression, here written in the Pauli
basis:

$W^{AB}=\\\
\frac{1}{8}\left(\mathds{1}+\frac{1}{\sqrt{2}}\left[\sigma^{A_{0}}_{x}\sigma^{A_{2}}_{z}\sigma^{A_{3}}_{z}\sigma^{B_{0}}_{z}+\sigma^{A_{0}}_{z}\sigma^{A_{2}}_{z}\sigma^{B_{1}}_{z}\right]\right)\,.$
(26)

To keep the above expression concise, the tensor product between matrices
acting on separate systems has been omitted, and unit matrices on subsystems
are implied. $\sigma^{A_{0}}_{z}\sigma^{A_{2}}_{z}\sigma^{B_{1}}_{z}$ is then
the short form of
$\sigma^{A_{0}}_{z}\otimes\mathds{1}^{A_{1}}\otimes\sigma^{A_{2}}_{z}\otimes\mathds{1}^{A_{3}}\otimes\mathds{1}^{B_{0}}\otimes\sigma^{B_{1}}_{z}$
in our notation.

One can demonstrate that (26) is a valid PM as well as a valid MPM by using
Theorem 1, see Appendix D. This is actually almost the same matrix as the one
given in the activation example in Ref. [37], but the isolated node $A^{(1)}$
between subsystems $A_{0}$ and $A_{1}$ is now getting a non-trivial input
instead of output. The matrix (26) actually corresponds to a deterministic
quantum comb with the fixed causal order $A^{(2)}\prec B\prec A^{(1)}$ as one
can verify using Eqs. (8). This proves that $W^{AB}$ is a causally separable
PM, which admits a physical realisation [15]. However, in this causal
realisation node $A^{(2)}$ is before node $A^{(1)}$.

Figure 5: Activation of causal nonseparability by a side channel: process
matrix $W$ is causally separable in the PM sense. When extended by a side
channel (in yellow), there exists an operation $M^{A^{(1)}}$ (in red) that
makes it lose this property between the 2 remaining nodes. By measuring
$A_{2}$ and changing the basis of $L_{2}$ accordingly (in green), $A$ obtains
the OCB PM between the remaining nodes.

Yet, if we treat $W$ as an MPM with node $A^{(1)}$ before node $A^{(2)}$,
Alice can use a side channel to pass on her first input to act on it during
her second operation. The extension of $W$ by an identity side channel
$Id=\sum_{i,j}\outerproduct{ii}{jj}^{L_{1}L_{2}}$ between the operations of
$A$,

$W^{\prime ALB}\equiv
W^{AB}\otimes\sum_{i,j}\outerproduct{ii}{jj}^{L_{1}L_{2}}\,,$ (27)

is effectively allowed in the MPM formalism; the situation is represented in
yellow in Fig. 5. When Alice’s first action (in red) is to simply forward her
input through the channel,
$M^{A^{(1)}}=\sum_{k,l}\outerproduct{kk}{ll}^{A_{0}L_{1}}\otimes\frac{1}{2}\mathds{1}^{A_{1}}$,
the conditional 2-partite (M)PM on the remaining nodes is

$\begin{split}&W^{A_{2}A_{3}L_{2}B_{0}B_{1}}=\mathrm{Tr}_{A_{0}A_{1}L_{1}}\left[M^{A^{(1)}}\cdot
W^{\prime ALB}\right]\\\
&=\frac{1}{8}\left(\mathds{1}+\frac{1}{\sqrt{2}}\left[\sigma^{A_{2}}_{z}\sigma^{A_{3}}_{z}\sigma^{L_{2}}_{x}\sigma^{B_{0}}_{z}+\sigma^{A_{2}}_{z}\sigma^{L_{2}}_{z}\sigma^{B_{1}}_{z}\right]\right)\,.\end{split}$
(28)

If $A$ performs a measurement in the $\sigma_{z}$ basis on $A_{2}$ and obtains
the outcome “+”, $A$ and $B$ are left with the OCB process matrix
$W^{A_{3}L_{2}B_{0}B_{1}}=\frac{1}{4}\left(\mathds{1}+\frac{1}{\sqrt{2}}\left[\sigma^{A_{3}}_{z}\sigma^{L_{2}}_{x}\sigma^{B_{0}}_{z}+\sigma^{L_{2}}_{z}\sigma^{B_{1}}_{z}\right]\right)$
on the remaining systems, which can be used to violate a causal inequality as
described in Ref. [38]. If she obtains the outcome “$-$”, $A$ and $B$ are left
with the matrix
$W^{A_{3}L_{2}B_{0}B_{1}}=\frac{1}{4}\left(\mathds{1}-\frac{1}{\sqrt{2}}\left[\sigma^{A_{3}}_{z}\sigma^{L_{2}}_{x}\sigma^{B_{0}}_{z}+\sigma^{L_{2}}_{z}\sigma^{B_{1}}_{z}\right]\right)$,
which differs from the OCB matrix merely by a change of basis. Hence the same
noncausal correlations as with the OCB PM can be obtained if $A$’s operations
are modified to account for this change of basis. She can achieve this by
applying a $\sigma_{y}$ transformation on $L_{2}$ controlled by the output of
the $\sigma_{x}$ measurement of $A_{2}$ (in green). Consequently, the ability
for the parties to use side communication with the MPM exhibits non-trivial
differences in terms of achievable correlations compared to the PM.

##  7 Conclusion

To summarise, we defined an extension of the PM formalism that allows multiple
rounds of information exchange for each party, which we named the “multi-round
process matrix” (MPM). We provided a complete characterisation of the set of
valid MPMs in Theorem 1 using the projective formulation of the validity
constraints on deterministic quantum combs. This highlighted a connection
between the set of MPMs and deterministic quantum combs: Theorem 2
demonstrates that MPMs are affine combination of all the quantum combs
compatible with the local ordering of each party. Finally, we motivated a new
notion of causal separability specific to MPMs, Definition 2, that takes into
account the possibility of using side channels. A non-trivial consequence of
this possibility was that the notion of causal nonseparability for PMs and
MPMs are not equivalent when applied to the same operator – we showed an
example of an operator that is causally separable when considered as a PM, and
causally nonseparable when considered as an MPM.

Several paths can be considered for the continuation of this work. First, the
question of finding whether there exists a bound on the dimensions of the side
channels needed to certify causal non-separability in the MPM sense has been
left open in this work.

Then, a natural path is to analyse the connection of this work with the
axiomatic theory of higher-order quantum computation by Bisio and Perinotti
[39, 11], which captures the MPM as a specific type of process within the
infinite hierarchy of higher-order processes; this will be explored in an
upcoming article. Another related work is Jia’s correlational approach to
quantum theory [30]. Several results derived here were partially found or
hinted in these works.

Finding new protocols that provide an advantage over regular quantum
communications [19, 20] constitutes another research direction. We expect that
the tool provided by the multi-round process matrix would be useful to this
purpose, especially to formulate communication protocols using indefinite
causal order as a resource [48, 32].

Finally, one may expect that the projective methods used here to characterise
MPMs could prove useful in the context of various other problems in the field
of (M)PMs, as well as in the broader field of higher-order processes, such as
for deriving convenient characterisations of more general processes in the
hierarchy. Understanding how the notion of causal nonseparability generalises
to higher orders is another question of great interest, which could
potentially unveil new phenomena specific to these orders.

This work is an adaptation of T. H.’s master’s thesis at the École
polytechnique de Bruxelles (ULB) that can be found on QuIC website [28].
Illustrations were drawn using draw.io. This publication was made possible
through the support of the ID# 61466 grant from the John Templeton Foundation,
as part of the “The Quantum Information Structure of Spacetime (QISS)” Project
(qiss.fr). The opinions expressed in this publication are those of the authors
and do not necessarily reflect the views of the John Templeton Foundation.
This work was supported by the Program of Concerted Research Actions (ARC) of
the Université Libre de Bruxelles. O. O. is a Research Associate of the Fonds
de la Recherche Scientifique (F.R.S.–FNRS).

## References

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##  Appendix A Projective formulation

###  A.1 The projective superoperator

This section aims to introduce the reader to the mapping
$\prescript{}{X}{}:\mathcal{L}\left(\mathcal{H}\right)\rightarrow\mathcal{L}\left(\mathcal{H}\right)$,
defined as

$\prescript{}{X}{\left(\cdot\right)}=\frac{\mathds{1}^{X}}{d_{X}}\otimes\Tr_{X}\left[\cdot\right]\,,$
(29)

where $X$ is a tensor factor of $\mathcal{H}$, and $d_{X}$ its dimension. It
was introduced in Ref. [6] alongside its shorthand prescript notation. To
illustrate its properties, we consider an operator $M$ defined on some space
$\mathcal{L}\left(\mathcal{H}^{A}\otimes\mathcal{H}^{B}\otimes\mathcal{H}^{C}\right)$
of dimension $d^{2}=(d_{A}d_{B}d_{C})^{2}$.

We define a multiplication within the prescript that corresponds formally to a
composition of several maps as

$\frac{\mathds{1}^{A}}{d_{A}}\otimes\Tr_{A}\left[\frac{\mathds{1}^{B}}{d_{B}}\otimes\Tr_{B}\left[M\right]\right]\equiv\prescript{}{AB}{M}\,.$
(30)

This “multiplication” inherits the properties of partial tracing: it possesses
an identity element (see below), it is associative as well as commutative as
partial tracings on different subsystems are associative and commute, but it
does not possess an inverse. Next, note that this mapping is CPTP, as well as
idempotent:

$\prescript{}{AA}{M}=\prescript{}{A^{2}}{M}=\prescript{}{A}{M}\,.$ (31)

Using the Hilbert-Schmidt product as the inner product on the space of linear
operators,

$\displaystyle\left\langle\cdot\left.,\,\cdot\right.\right\rangle\,:\,\mathcal{L}\left(\mathcal{H}\right)\times\mathcal{L}\left(\mathcal{H}\right)\rightarrow\mathbb{C}\quad,$
$\displaystyle M,N\mapsto\left\langle
M\left.,\,N\right.\right\rangle\equiv\mathrm{Tr}\left[M^{\dagger}\cdot
N\right]\quad,$ (32)

one can verify that the mapping is self-adjoint with respect to it. Let
$N\in\mathcal{L}\left(\mathcal{H}^{A}\otimes\mathcal{H}^{B}\otimes\mathcal{H}^{C}\right)$
be another arbitrary operator defined on the same space as $M$, then

$\displaystyle\left\langle M\left.,\,\prescript{}{A}{N}\right.\right\rangle$
$\displaystyle=\mathrm{Tr}_{ABC}\left[M^{\dagger}\cdot\left(\frac{\mathds{1}^{A}}{d_{A}}\otimes\Tr_{A}\left[N\right]\right)\right]$
$\displaystyle=\mathrm{Tr}_{BC}\left[\mathrm{Tr}_{A}\left[M^{\dagger}\right]\cdot\frac{1}{d_{A}}\,\mathrm{Tr}_{A}\left[N\right]\right]$
$\displaystyle=\mathrm{Tr}_{ABC}\left[\left(\frac{\mathds{1}^{A}}{d_{A}}\otimes\Tr_{A}\left[M^{\dagger}\right]\right)\cdot
N\right]\,,$ (33)

where the identity
$\mathrm{Tr}_{ABC}\left[U^{ABC}\cdot\left(\mathds{1}^{A}\otimes
V^{BC}\right)\right]=\mathrm{Tr}_{BC}\left[\mathrm{Tr}_{A}\left[U^{ABC}\right]\cdot
V^{BC}\right]$ have been used to go to the second and third lines. Thus,

$\left\langle
M\left.,\,\prescript{}{A}{N}\right.\right\rangle=\left\langle\prescript{}{A}{M}\left.,\,N\right.\right\rangle\quad.$
(34)

This means that the mapping is idempotent and self-adjoint, making it an
orthogonal projector onto a subspace of
$\mathcal{L}\left(\mathcal{H}^{A}\otimes\mathcal{H}^{B}\otimes\mathcal{H}^{C}\right)$.

We define a linear addition operation within the prescript as

$\prescript{}{A}{M}+\prescript{}{B}{M}\equiv\prescript{}{A+B}{M}\,,$ (35)

and for each element $\prescript{}{A}{\left(\cdot\right)}$ we define the
inverse element
$\prescript{}{-A}{\left(\cdot\right)}\equiv-\frac{\mathds{1}^{A}}{d_{A}}\otimes\Tr_{A}\left[\cdot\right]$,
such that $\prescript{}{A+(-A)}{M}\equiv\prescript{}{0}{M}=0$, where 0 is the
additive identity element, which corresponds to the zero mapping. We will
introduce the minus sign as a shorthand notation
$\prescript{}{A+(-A)}{M}\equiv\prescript{}{A-A}{M}$ for addition of inverse
elements. With this addition and multiplication, the prescripts have the
algebraic structure of a ring [26]. One could also promote the structure to an
algebra by defining a scalar multiplication the same way the inverse additive
element have been defined, i.e. $\prescript{}{\lambda
A}{\left(\cdot\right)}\equiv\lambda\frac{\mathds{1}^{A}}{d_{A}}\otimes\Tr_{A}\left[\cdot\right]$,
for all $\lambda\in\mathbb{C}$. However, note that the idempotency property is
not in general conserved under scalar multiplication nor under addition of
prescripts. Hence, for the purposes of this article, we restrict the algebra
to a ring and we only consider combinations under addition that result in
valid projectors, i.e. idempotent elements.

Finally, we define 2 particular elements of the ring. One is the identity
mapping $\mathcal{I}$, such that $\mathcal{I}[M]=M$ for all operators, which
corresponds to the multiplicative identity element in the ring. In prescript
notation it is then denoted as a 1:

$\mathcal{I}[M]\equiv\prescript{}{1}{M}\,.$ (36)

The other is the multiplicative absorbing element
$\mathcal{D}:\mathcal{D}\mathcal{P}=\mathcal{P}\mathcal{D}=\mathcal{D}$, for
every $\mathcal{P}$ in the ring, which is no other than the projector to the
subspace spanned by the unit matrix $\frac{1}{d}\mathds{1}$, or equivalently
the mapping applied on the full system
$\mathcal{D}(M)=\frac{\mathds{1}}{d}\otimes\Tr\left[M\right]$. In our example,
$\mathcal{D}$ is written in prescript notation as

$\mathcal{D}[M]\equiv\prescript{}{ABC}{M}\,.$ (37)

We will also use superscripts when those projectors are applied on parts of
composite subsystems, e.g.
$\left(\mathcal{I}^{A}\otimes\mathcal{D}^{BC}\right)[M]\equiv\prescript{}{1BC}{M}=\prescript{}{BC}{M}$.

###  A.2 Comb conditions in projective formulation

In this section, we derive an alternative formulation of the set of necessary
and sufficient constraints an operator has to satisfy to be a valid
deterministic quantum comb (Eqs. (4), see [15, Theorems 3 and 5]). This
formulation is in the spirit of what was done in Ref. [6] for the PM validity
conditions. We will find a recursive way of building the projector onto the
subspace of valid quantum combs for an increasing number of teeth. To do so,
we use the shorthand notation $M^{(j)}\equiv M^{X_{0}X_{1}\cdots X_{2j-1}}$,
so that $M^{X}\equiv M^{(n_{X})}$, to refer to a deterministic quantum comb
with $j$ teeth.

The set of conditions to be proven equivalent to (4) are equations (8) and
(9), rewritten here for convenience:

$\displaystyle M^{X}\geq 0\,,$ (38a)
$\displaystyle\mathcal{P}_{n_{X}}^{X}\left[M^{X}\right]=M^{X}\,,$ (38b)
$\displaystyle\mathrm{Tr}\left[M^{X}\right]=\prod_{j=0}^{n_{X}-1}d_{X_{2j}}\,,$
(38c)

where
$\mathcal{P}_{n_{X}}^{X}\equiv\mathcal{P}_{n_{X}}^{X^{(1)}\prec\ldots\prec
X^{(n_{X})}}$ is the projector to the validity subspace of $n_{X}$-combs,
which is given by the recursive relation (9)

$\begin{split}&\mathcal{P}_{0}=\prescript{}{1}{\left(\cdot\right)}\,,\\\
&\mathcal{P}_{n_{X}}^{X^{(1)}\prec\ldots\prec X^{(n_{X}-1)}\prec
X^{(n_{X})}}=\\\
&\,\prescript{}{\left(1-X_{2n_{X}-1}\right)}{\left(\cdot\right)}+\prescript{}{X_{2n_{X}-2}X_{2n_{X}-1}}{\left(\mathcal{P}^{X^{(1)}\prec\ldots\prec
X^{(n_{X}-1)}}_{n_{X}-1}\right)}.\end{split}$ (39)

Keep in mind that we are using the shorthand notation
$\mathcal{P}^{X^{(1)}\prec\ldots\prec
X^{(n_{X}-1)}}_{n_{X}-1}\equiv\mathcal{P}^{X_{0}\ldots
X_{2n_{X}-3}}_{n_{X}-1}\otimes\mathcal{I}^{X_{2n_{X}-2}X_{2n_{X}-1}}$ to
indicate that we are projecting the subsystems $X_{0}\ldots X_{2n_{X}-3}$ on a
subspace of quantum ($n_{X}$-1)-combs with causal ordering
$X^{(1)}\prec\ldots\prec X^{(n_{X}-1)}$.

Note that condition (38a) is just condition (4a)—the condition that a quantum
comb is PSD, which reflects the fact that it is the CJ operator of a CP map.
Therefore, equivalence between the set of conditions (4) and the set of
conditions (38) with (39) would follow if we show that (4b) is equivalent to
(38b) and (38c), with the projector in (38b) given by (39). This is what we
show next.

The normalisation condition (38c) for deterministic combs is obtained directly
by taking the total trace of the comb. We can indeed use the identity
$\mathrm{Tr}_{AB}\left[M^{A}\otimes
N^{B}\right]=\mathrm{Tr}_{A}\left[M^{A}\right]\mathrm{Tr}_{B}\left[N^{B}\right]$,
which allows to nest the set of linear constraints (4b) within one another. As
it yields
$\mathrm{Tr}_{X_{2i-1}X_{2i-2}}\left[M^{(i)}\right]=d_{X_{2i-2}}\;M^{(i-1)}$,
one can see that the total trace of a valid comb is equal to the product of
its input dimensions (reflecting the fact that a deterministic quantum comb
correspond to a channel from all inputs to all outputs). Hence, Eq. (38c) is a
necessary condition.

We next show that for a 1-comb, Eq. (38b) with the projector
$\mathcal{P}^{X}_{1}$ defined as in (39) is necessary. Indeed, the constraint
for a 1-comb is equivalent to

$\displaystyle\mathrm{Tr}_{X_{1}}\left[M^{(1)}\right]\,$
$\displaystyle=\,\mathds{1}^{X_{0}}$ $\displaystyle\iff$
$\displaystyle\prescript{}{X_{1}}{M^{(1)}}$
$\displaystyle=\frac{\mathds{1}^{X_{0}X_{1}}}{d_{X_{0}}d_{X_{1}}}\;d_{X_{0}}$
$\displaystyle\overset{\eqref{eq:app:QuComb_normalisation}}{\Longrightarrow}$
$\displaystyle\prescript{}{X_{1}}{M^{(1)}}$
$\displaystyle=\prescript{}{X_{0}X_{1}}{M^{(1)}}$ , (40)

where we multiplied by $\frac{\mathds{1}^{X_{1}}}{d_{X_{1}}}\otimes$ on both
sides between the first and second lines, and we used the definition (29) on
the left-hand side, then we used Eq. (38c), i.e.
$\mathrm{Tr}\left[M^{(1)}\right]=d_{X_{0}}$, together with (29) on the right-
hand side to go to from the second line to the last. Hence, (40) is necessary.

The relation (40) is rephrased as a projector as
$M^{(1)}-\prescript{}{X_{1}}{M^{(1)}}+\prescript{}{X_{0}X_{1}}{M^{(1)}}=M^{(1)}$
[13], or

$\mathcal{P}^{X}_{1}\left[M^{(1)}\right]\equiv\prescript{}{1-X_{1}+X_{0}X_{1}}{M^{(1)}}=M^{(1)}\,,$
(41)

where we have defined the 1-comb projector
$\mathcal{P}^{X}_{1}\equiv\mathcal{P}^{X^{(1)}}_{1}$ from the space of
operators
$\mathcal{L}\left(\mathcal{H}^{X_{0}}\otimes\left(\mathcal{H}^{X_{1}}\right)^{*}\right)$
to the subspace of (nonnormalised) deterministic quantum 1-combs. It is a
projector as the relation
$\left(1-X_{1}+X_{0}X_{1}\right)^{2}=1-X_{1}+X_{0}X_{1}$ follows from the
algebraic rules of the ring defined in Sec. A.1. This last equation, (41), is
exactly (38b) with (39) in the case of 1-combs, proving their necessity in
this case.

To prove that Eqs. (38b) and (38c) with (39) are sufficient to enforce Eqs.
(4b) in the 1-comb case, we start from the projective condition,

$\prescript{}{X_{1}}{M^{(1)}}-\prescript{}{X_{0}X_{1}}{M^{(1)}}=0\\\ \iff\\\
\frac{\mathds{1}^{X_{1}}}{d_{X_{1}}}\otimes\left(\mathrm{Tr}_{X_{1}}\left[M^{(1)}\right]-\mathrm{Tr}_{X_{1}}\left[\prescript{}{X_{0}}{M^{(1)}}\right]\right)=0\\\
\iff\\\
\mathrm{Tr}_{X_{1}}\left[M^{(1)}\right]-\frac{\mathds{1}^{X_{0}}}{d_{X_{0}}}\mathrm{Tr}_{X_{0}X_{1}}\left[{M^{(1)}}\right]=0\\\
\overset{\eqref{eq:app:QuComb_normalisation}}{\Longrightarrow}\\\
\mathrm{Tr}_{X_{1}}\left[M^{(1)}\right]-\mathds{1}^{X_{0}}=0\,,$ (42)

where we successively used the fact that prescripts commute and the
distributive property of the tensor product to go to the second line, and (29)
to make the full trace appear in the third line. Injecting (38c) we reach (4b)
in the last line. Therefore, (38b) and (38c) with (39) imply (4b) for the case
of 1-combs. This completes the proof of equivalence in this case.

Equivalence in the general case as well as a rule for building the
$n_{X}$-comb projector can be proven by induction. Suppose that the
reformulation holds up to $n_{X}-1$ teeth such that, for $2\leq j\leq
n_{X}-1$,

$\begin{gathered}\mathrm{Tr}_{X_{2j-1}}\left[M^{(j)}\right]=\mathds{1}^{X_{2j-2}}\otimes
M^{(j-1)},\\\ \mathrm{Tr}_{X_{1}}\left[M^{(1)}\right]=\mathds{1}^{X_{0}}\,,\\\
\iff\\\
\mathcal{P}^{X}_{n_{X}-1}\left[M^{(n_{X}-1)}\right]=M^{(n_{X}-1)}\,,\\\
\mathrm{Tr}\left[M^{(n_{X}-1)}\right]=\prod_{j=0}^{n_{X}-2}d_{X_{2j}}\,,\end{gathered}$
(43)

where $\mathcal{P}^{X}_{n_{X}-1}\equiv\mathcal{P}_{n_{X}}^{X^{(1)}\prec
X^{(2)}\prec\ldots\prec X^{(n_{X}-1)}}$ is the $(n_{X}-1)$-comb projector. To
prove that the $n_{X}$ case of (38) follows from (4), we start with
$\mathrm{Tr}_{X_{2n_{X}-1}}\left[M^{(n_{X})}\right]=\mathds{1}^{X_{2n_{X}-2}}\otimes
M^{(n_{X}-1)}$. We already know that the trace condition will be satisfied, so
to find the projective condition we write

$\mathrm{Tr}_{X_{2n_{X}-1}}\left[M^{(n_{X})}\right]=\mathds{1}^{X_{2n_{X}-2}}\otimes
M^{(n_{X}-1)}\,,\\\
\prescript{}{X_{2n_{X}-1}}{M^{(n_{X})}}=\frac{\mathds{1}^{X_{2n_{X}-2}X_{2n_{X}-1}}}{d_{X_{2n_{X}-1}}}\otimes
M^{(n_{X}-1)}\,,\\\
\prescript{}{X_{2n_{X}-1}X_{2n_{X}-2}}{M^{(n_{X})}}=\frac{\mathds{1}^{X_{2n_{X}-2}X_{2n_{X}-1}}}{d_{X_{2n_{X}-1}}}\otimes
M^{(n_{X}-1)}\,,\\\
\prescript{}{X_{2n_{X}-1}X_{2n_{X}-2}}{\left(\mathcal{P}^{X}_{n_{X}-1}\left[M^{(n_{X})}\right]\right)}=\\\
\frac{\mathds{1}^{X_{2n_{X}-2}X_{2n_{X}-1}}}{d_{X_{2n_{X}-1}}}\otimes
M^{(n_{X}-1)}\,,$ (44)

where we applied successively
$\frac{\mathds{1}^{X_{2n_{X}-1}}}{d_{X_{2n_{X}-1}}}\otimes$,
$\prescript{}{X_{2n_{X}-2}}{\left(\cdot\right)}$, and
$\mathcal{P}^{X}_{n_{X}-1}\left[\cdot\right]$ on both sides to show that
equivalence in the $n_{X}-1$ case implies the following relation in the
$n_{X}$ case:

$\mathrm{Tr}_{X_{2n_{X}-1}}\left[M^{(n_{X})}\right]=\mathds{1}^{X_{2n_{X}-2}}\otimes
M^{(n_{X}-1)}\Longrightarrow\\\
\prescript{}{X_{2n_{X}-1}}{M^{(n_{X})}}=\prescript{}{X_{2n_{X}-1}X_{2n_{X}-2}}{\left(\mathcal{P}^{X}_{n_{X}-1}\left[M^{(n_{X})}\right]\right)}.$
(45)

This last relation is actually the recursive definition $\eqref{eq:PC}$ of the
projector as it can be rephrased as

$\begin{split}&M^{(n_{X})}-\prescript{}{X_{2n_{X}-1}}{M^{(n_{X})}}\\\
&+\prescript{}{X_{2n_{X}-1}X_{2n_{X}-2}}{\left(\mathcal{P}^{X}_{n_{X}-1}\left[M^{(n_{X})}\right]\right)}=M^{(n_{X})}\,,\\\
&\equiv\mathcal{P}^{X^{(1)}\prec\ldots\prec X^{(n_{X}-1)}\prec
X^{(n_{X})}}_{n_{X}}\left[M^{(n_{X})}\right]=M^{(n_{X})}\,.\end{split}$ (46)

This yields the definition (39).

To prove that this relation indeed defines a projector, notice the pattern in
the prescripts of the recursive relation: let
$\mathcal{P}_{n_{X}}^{X^{(1)}\prec\ldots\prec X^{(n_{X}-1)}\prec
X^{(n_{X})}}=1-B+ABC$, where $A\equiv\prescript{}{X_{2n_{X}-2}}{}$ and
$B\equiv\prescript{}{X_{2n_{X}-1}}{}$ are idempotent elements, while
$C\equiv\mathcal{P}^{X^{(1)}\prec\ldots\prec X^{(n_{X}-1)}}_{n_{X}-1}$. In the
case $n_{X}=1$, $\prescript{}{C}{}=1$, which is idempotent, in the case
$n_{X}=2$, $\prescript{}{C}{}=\mathcal{P}^{X^{(1)}}_{1}$, which has been
proven idempotent as well. If we assume $C$ to be idempotent up to $n_{X}-1$,
then for $n_{X}$:

$\displaystyle\left(1-B+ABC\right)^{2}=$
$\displaystyle\left(1-B\right)^{2}+2\left(1-B\right)ABC$ (47)
$\displaystyle\,+\left(ABC\right)^{2}$ $\displaystyle=$ $\displaystyle
1-B+0+ABC\,,$

where we have used the distributive property of multiplication as well as
idempotency of all elements to show that $1-B+ABC$ is idempotent as well. This
proves that the objects built in equation (39) are idempotent for all $n_{X}$,
thus they define projectors.

Hence, by induction we have that conditions (38) with (39) are necessary for
(4). Sufficiency is also proven by induction. Suppose (43) holds. For the
$n_{X}$ case we have the implication

$\mathcal{P}^{X^{(1)}\prec\ldots\prec X^{(n_{X}-1)}\prec
X^{(n_{X})}}_{n_{X}}\left[M^{(n_{X})}\right]=M^{(n_{X})}\\\ \iff\\\
\prescript{}{X_{2n_{X}-1}}{M^{(n_{X})}}-\prescript{}{X_{2n_{X}-1}X_{2n_{X}-2}}{\left(\mathcal{P}^{X}_{n_{X}-1}\left[M^{(n_{X})}\right]\right)}=0\\\
\Longrightarrow\\\
\mathrm{Tr}_{X_{2n_{X}-1}}\left[M^{(n_{X})}-\prescript{}{X_{2n_{X}-2}}{\left(\mathcal{P}^{X}_{n_{X}-1}\left[M^{(n_{X})}\right]\right)}\right]=0,$
(48)

yielding

$\mathrm{Tr}_{X_{2n_{X}-1}}\left[M^{(n_{X})}\right]=\\\
\frac{\mathds{1}^{X_{2n_{X}-2}}}{d_{X_{2n_{X}-2}}}\otimes\mathrm{Tr}_{X_{2n_{X}-1}X_{2n_{X}-2}}\left[\mathcal{P}^{X}_{n_{X}-1}\left[M^{(n_{X})}\right]\right]\,.$
(49)

Because of equations (43), we must have that

$\mathrm{Tr}_{X_{2n_{X}-1}X_{2n_{X}-2}}\left[\mathcal{P}^{X}_{n_{X}-1}\left[M^{(n_{X})}\right]\right]\propto
M^{(n_{X}-1)}.$ (50)

Tracing out both sides, we find that the proportionality constant has to be
equal to $d_{X_{2n_{X}-2}}$ in order for condition (38c) to hold. Thus,
$\mathrm{Tr}_{X_{2n_{X}-1}}\left[M^{(n_{X})}\right]=\mathds{1}^{X_{2n_{X}-2}}\otimes
M^{(n_{X}-1)}$ follows from (38). Adding this condition to the $n_{X}-1$ other
conditions
$\mathrm{Tr}_{X_{2j-1}}\left[M^{(j)}\right]=\mathds{1}^{X_{2j-2}}\otimes
M^{(j-1)}$, $2\leq j\leq n_{X}-1$ and
$\mathrm{Tr}_{X_{1}}\left[M^{(1)}\right]=\mathds{1}^{X_{0}}$, that were
already assumed in Eqs. (43), gives the $n_{X}$ conditions (4b) for an
deterministic quantum $n_{X}$-comb. Hence (38) with (39) implies (4) by
induction, proving sufficiency.

Note that Eq. (39) admits an “unravelled” formulation,

$\mathcal{P}^{X}_{n_{X}}=\prescript{}{\left(1-X_{2n_{X}-1}\left(1-X_{2n_{X}-2}\big{(}\ldots\big{(}1-X_{1}\big{(}1-X_{0}\big{)}\big{)}\ldots\big{)}\right)\right)}{\left(\cdot\right)}\,,$
(51)

shortened in

$\mathcal{P}^{X}_{n_{X}}=\mathcal{I}-\prescript{}{X_{2n_{X}-1}}{\left(\mathcal{I}-\prescript{}{X_{2n_{X}-2}}{\left(\mathcal{P}^{X}_{n_{X}-1}\right)}\right)}\,.$
(52)

This will be useful in the proof of Theorem 1 presented in the next section.

##  Appendix B Proof of Theorem 1

As (6) is the same condition as (12a), the proof consists in showing that
condition (7) is equivalent to conditions (12b) and (12c).

The first step of the proof is to notice that since
$\frac{\bigotimes_{X\in\mathfrak{N}}\mathds{1}^{X}}{\prod_{X\in\mathfrak{N}}\prod_{i=0}^{n_{X}-1}d_{X_{2i+1}}}\equiv\frac{\mathds{1}}{d_{out}}$
is a valid tensor product of deterministic quantum combs, the normalisation
(7) implies that the set $\\{W\\}$ have fixed trace norm,

$\mathrm{Tr}\left[W\cdot\frac{\mathds{1}}{d_{out}}\right]=1\iff\mathrm{Tr}\left[W\right]=d_{out}\,.$
(53)

This is condition (12c), which is thus necessary. It also implies that
$W=\frac{\bigotimes_{X\in\mathfrak{N}}\mathds{1}^{X}}{\prod_{X\in\mathfrak{N}}\prod_{i=0}^{n_{X}-1}d_{X_{2i}}}\equiv\frac{\mathds{1}}{d_{in}}$
is an element of the set of valid MPMs since
$\mathrm{Tr}\left[\frac{\mathds{1}}{d_{in}}\cdot\bigotimes_{X\in\mathfrak{N}}M^{X}\right]=1$
for all $M^{X}$. This actually corresponds to enforcing condition (8c) on each
comb in the tensor product.

For proving the necessity of (12b) (and subsequently its sufficiency together
with (12c)), we will work in a convenient Hilbert-Schmidt (HS) basis
$\\{\sigma_{i}\\}$ for an operator space of dimension $d^{2}$, whose elements
satisfy

$\begin{gathered}\sigma_{i}^{\dagger}=\sigma_{i}\,,\forall i\,,\\\
\sigma_{0}=\mathds{1}\,,\\\ \mathrm{Tr}\left[\sigma_{i}\right]=0\,,\forall
i\neq 0\,,\\\
\mathrm{Tr}\left[\sigma_{i}\cdot\sigma_{j}\right]=d\,\delta_{i,j}\,,\,\forall
i,j\,,\end{gathered}$ (54)

where $\delta_{i,j}$ is the Kronecker delta.

An arbitrary operator
$O\in\mathcal{L}\left(\mathcal{H}^{X_{0}}\otimes\mathcal{H}^{X_{1}}\otimes\ldots\right)$
can always be expressed in a product basis of the kind described above:
$O=\sum_{i_{0},i_{1}\ldots}o_{i_{0},i_{1}\ldots}\;\sigma^{X_{0}}_{i_{0}}\otimes\sigma^{X_{1}}_{i_{1}}\otimes\cdots$,
where $\sigma_{j}^{Y}$ is the $j$-th element of a basis of subsystem $Y$ that
respects (54), while the $o_{i_{0}i_{1}\ldots}\in\mathbb{C}\,,\forall
i_{0},\forall i_{1},\ldots$ are the associated coefficients of the basis
expansion, and the sum runs over all indices:
$\sum_{i_{0},i_{1}\ldots}\equiv\sum_{i_{0}=0}^{d_{X_{0}}^{2}-1}\sum_{i_{1}=0}^{d_{X_{1}}^{2}-1}\cdots$.
Let $\mathbf{i}=(i_{0},i_{1},\ldots)$ be a multi-index, ranging from 0 to
$d_{X}^{2}-1$, with first element $\mathbf{0}\equiv(0,0,\ldots)$. Each sub-
index corresponds to a subsystem of X, i.e. $i_{0}$ refers to the index of the
$X_{0}$ component, $i_{1}$ to the one of $X_{1}$, etc… We use it to shorten
the basis expansion:

$O=\sum_{i_{0}i_{1}\ldots}o_{i_{0}i_{1}\ldots}\;\sigma^{X_{0}}_{i_{0}}\otimes\sigma^{X_{1}}_{i_{1}}\otimes\ldots\equiv\sum_{\mathbf{i}}o_{\mathbf{i}}\;\sigma^{X}_{\mathbf{i}}\,.$
(55)

The action of the mapping (29) defined in section A.1 on an operator is to
remove certain terms in the basis expansion, as it is projecting on a subset
of the basis elements of this type of basis. One can see it as if the
superoperator was setting its corresponding sub-index in the HS basis to 0:
$\prescript{}{X_{0}}{O}=\sum_{\mathbf{i}=(i_{0}=0,i_{1},i_{2},\ldots)}o_{\mathbf{i}}\;\sigma^{X}_{\mathbf{i}}$;
the notation under the sum is to be understood as “summing only over all
$\mathbf{i}$ for which $i_{0}=0$”. For example,
$\prescript{}{X_{0}}{\left(\sum_{i_{0}}o_{i_{0}}\;\sigma^{X_{0}}_{i_{0}}\right)}=o_{0}\;\sigma^{X_{0}}_{0}$,
$\prescript{}{X_{0}}{\left(\sum_{i_{0}i_{1}}o_{i_{0}i_{1}}\;\sigma^{X_{0}}_{i_{0}}\otimes\sigma^{X_{1}}_{i_{1}}\right)}=\sum_{i_{1}}o_{0i_{1}}\;\sigma^{X_{0}}_{0}\otimes\sigma^{X_{1}}_{i_{1}}$,
etc…

Actually, the projector on the comb validity subspace also shares this
property, and so does any tensor product of several projectors of this kind
(i.e. the projector on $\mathrm{Span}\left\\{\bigotimes M\right\\}$). This is
the content of the following lemma.

######  Lemma 1.

The (tensor product of) mapping(s) $\mathcal{P}^{X}_{n_{X}}$ defined as in Eq.
(39) is a superoperator projector whose action is to remove certain basis
elements in the Hilbert-Schmidt expansion of an arbitrary CJ operator.

###### Proof.

The proof is based on the following two observations: (a) Let $\mathcal{P}$ be
a projector whose action in the HS basis expansion amounts to removing certain
types of terms while leaving the others intact. Then,
$\mathcal{I}-\mathcal{P}$ is a projector that also has this property. Indeed,
it will leave the terms that are removed by $\mathcal{P}$ and will remove
those that are left by $\mathcal{P}$. (b) Let $\mathcal{P}$ and
$\mathcal{P}^{{}^{\prime}}$ be two projectors whose action in the basis
expansion amounts to removing certain types of terms. Then the product
$\mathcal{P}^{{}^{\prime}}\mathcal{P}$ is also a projector of this kind.
Indeed, $\mathcal{P}$ would first remove the terms it removes, and then
$\mathcal{P}^{{}^{\prime}}$ would remove those that it removes from what
remains after the action of $\mathcal{P}$. Note that this observation implies
in particular that if the claimed property holds for $\mathcal{P}^{X}$ and
$\mathcal{P}^{{}^{\prime}Y}$ that act on separate systems, it also holds for
their tensor product
$\left(\mathcal{P}^{X}\otimes\mathcal{I}^{Y}\right)\left(\mathcal{I}^{X}\otimes\mathcal{P}^{{}^{\prime}Y}\right)$,
since $\left(\mathcal{P}^{X}\otimes\mathcal{I}^{Y}\right)$ and
$\left(\mathcal{I}^{X}\otimes\mathcal{P}^{{}^{\prime}Y}\right)$ are obviously
also projectors with this property.

Consider the “unravelled” version of Eq. (39), Eq. (51). We have seen that
$\prescript{}{X_{0}}{}$ is such a projector, hence $\prescript{}{1-X_{0}}{}$
is also a projector of this kind by (a), so is
$\prescript{}{X_{1}\left(1-X_{0}\right)}{}$ by (b), and so is
$\prescript{}{1-X_{1}\left(1-X_{0}\right)}{}$ by (a) again. This proves the
lemma in the case $n_{X}=1$ (that is, for Eq. (41)). The general case follows
by induction: if it is true for $\mathcal{P}^{X}_{n_{X}-1}$, then it is true
for $\prescript{}{X_{2n_{X}-2}}{\left(\mathcal{P}^{X}_{n_{X}-1}\right)}$ by
(b), which is true for
$\mathcal{I}-\prescript{}{X_{2n_{X}-2}}{\left(\mathcal{P}^{X}_{n_{X}-1}\right)}$
by (a), which is again true for
$\prescript{}{X_{2n_{X}-1}}{\left(\mathcal{I}-\prescript{}{X_{2n_{X}-2}}{\left(\mathcal{P}^{X}_{n_{X}-1}\right)}\right)}$
by (b), and which is true for
$\mathcal{I}-\prescript{}{X_{2n_{X}-1}}{\left(\mathcal{I}-\prescript{}{X_{2n_{X}-2}}{\left(\mathcal{P}^{X}_{n_{X}-1}\right)}\right)}$
by (a). This is Eq. (51) for $n_{X}$, hereby proving the induction. ∎

For example, in the case of a single party with one operation
$O=\sum_{i_{0}i_{1}}o_{i_{0}i_{1}}\;\sigma^{X_{0}}_{i_{0}}\otimes\sigma^{X_{1}}_{i_{1}}$,
one has
$\mathcal{P}^{X}_{1}\left[O\right]=\prescript{}{1-X_{1}+X_{0}X_{1}}{O}=o_{00}\;\sigma^{X_{0}}_{0}\otimes\sigma^{X_{1}}_{0}+\sum_{\mathbf{i}=(i_{0},i_{1}>0)}o_{i_{0}i_{1}}\;\sigma^{X_{0}}_{i_{0}}\otimes\sigma^{X_{1}}_{i_{1}}$;
the removed terms have indices $\mathbf{i}=(i_{0}>0,i_{1}=0)$.

In the general case for a single party, we will write the action of a
projector on the basis expansion as
$\mathcal{P}^{X}_{n_{X}}\left[\sum_{\mathbf{i}}o_{\mathbf{i}}\;\sigma_{\mathbf{i}}^{X}\right]=\sum_{\mathbf{i}\in\\{\\{M^{X}\\}\\}}o_{\mathbf{i}}\;\sigma_{\mathbf{i}}^{X}$,
where $\sum_{\mathbf{i}\in\\{\\{M^{X}\\}\\}}$ is to be understood as “summing
only over the basis elements that belong to the subspace in which the set of
$n_{X}$-combs $\\{M^{X}\\}$ is defined”. In the 1-node example of above,
$\mathbf{i}\in\\{\\{M^{X}\\}\\}$ is thus equivalent to
$\mathbf{i}\in\\{(0,0)\\}\cup\\{(i_{0},i_{1})\\}_{i_{0}=0,i_{1}>0}^{d^{2}_{X_{0}}-1,d^{2}_{X_{1}}-1}$.

We next observe that the subspace spanned by the operators that are a tensor
product of valid deterministic combs for the separate parties is the subspace
on which the projector
$\mathcal{P}\equiv\bigotimes_{X\in\mathfrak{N}}\mathcal{P}^{X}_{n_{X}}$
projects. Indeed, the action of $\mathcal{P}$ on an operator is to remove all
terms in its Hilbert-Schmidt expansion that are not tensor products of terms
allowed (left intact) by the local projectors $\mathcal{P}^{X}_{n_{X}}$. In
other words, $\mathcal{P}$ projects on the subspace spanned by the tensor
products of all locally allowed basis elements. Since each locally allowed
basis element can be expressed as a linear combination of local deterministic
combs, every operator in the subspace on which $\mathcal{P}$ projects is of
the form $M=\sum_{k}q_{k}\left(\bigotimes_{X}M^{X}\right)_{k}$, where $M^{X}$
is a valid deterministic comb associated with party $X$, and $q_{k}$ are real
coefficients. Conversely, every operator of this form is obviously left
invariant by $\mathcal{P}$. The trace of this last expression gives
$\mathrm{Tr}\left[M\right]=\sum_{k}q_{k}\left(\prod_{X\in\mathfrak{N}}d_{X_{in}}\right)_{k}$,
where for each $k$ the terms in parenthesis are exactly the input dimension of
the Hilbert space, so they can be factored out of the sum:
$\mathrm{Tr}\left[M\right]=d_{in}\sum_{k}q_{k}$. Hence, requiring that the
trace is equal to $d_{in}\equiv\prod_{X\in\mathfrak{N}}d_{X_{in}}$ is
equivalent to requiring that $\sum_{k}q_{k}=1$. Therefore, conditions (11) are
equivalent to requiring that $M$ is an affine sum of tensor products of
deterministic quantum combs.

We will now prove the necessity of (12b) by using this observation to compute
the quasiorthogonal projector explicitly. Let $M$ be an operator on
$\mathcal{L}\left(\mathcal{H}\right)$ that satisfies conditions (11). It is
written as

$M=\bigotimes_{X\in\mathfrak{N}}\left(\sum_{\mathbf{i}_{X}\in\\{\\{M^{X}\\}\\}}m_{\mathbf{i}_{X}}\;\sigma^{X}_{\mathbf{i}_{X}}\right)\equiv\sum_{\mathbf{i}\in\\{\\{M\\}\\}}m_{\mathbf{i}}\;\sigma_{\mathbf{i}}\,,$
(56)

where we have introduced a shortened formulation by making the tensor product
implicit. Let there be an arbitrary operator
$W=\sum_{\mathbf{j}}w_{\mathbf{j}}\sigma_{\mathbf{j}}$, where $\mathbf{j}$ is
a multi-index defined analogously to $\mathbf{i}$. Applying the normalisation
condition (7) on $W$ is equivalent to normalising it on all affine sums of
tensor product of deterministic combs (see main text), hence Eq. (10) gives

$\mathrm{Tr}\left[W\cdot
M\right]=\sum_{\mathbf{j}}\sum_{\mathbf{i}\in\\{\\{M\\}\\}}w_{\mathbf{j}}m_{\mathbf{i}}\;d\;\delta_{{\mathbf{j}},\mathbf{i}}=1\,.$
(57)

Since $\mathrm{Tr}\left[W\right]=w_{\mathbf{0}}d$ and
$\mathrm{Tr}\left[M\right]=m_{\mathbf{0}}d$, the values of these two
coefficients are fixed using Eqs. (38c) and (53) obtained above. This allows
us to write

$\displaystyle\sum_{\mathbf{j}}\sum_{\mathbf{i}\in\\{\\{M\\}\\}}w_{\mathbf{j}}m_{\mathbf{i}}d\delta_{\mathbf{j},\mathbf{i}}$
$\displaystyle=\sum_{\mathbf{i}\in\\{\\{M\\}\\}}w_{\mathbf{i}}m_{\mathbf{i}}d$
(58)
$\displaystyle=w_{\mathbf{0}}m_{\mathbf{0}}d+\sum_{\mathbf{i}\neq\mathbf{0}\in\\{\\{M\\}\\}}w_{\mathbf{i}}m_{\mathbf{i}}d$
$\displaystyle=1+\sum_{\mathbf{i}\neq\mathbf{0}\in\\{\\{M\\}\\}}w_{\mathbf{i}}m_{\mathbf{i}}d\,,$

turning the normalisation condition into

$\sum_{\mathbf{i}\neq\mathbf{0}\in\\{\\{M\\}\\}}w_{\mathbf{i}}m_{\mathbf{i}}d=0\,.$
(59)

As $w_{\mathbf{i}},m_{\mathbf{i}}\in\mathbb{R}\;\forall\mathbf{i}$ (since $W$
and $M$ are Hermitian), as $d$ is a non-negative integer, and as in general
$m_{\mathbf{i}}\neq 0$ for at least one given value of
$\mathbf{i}\neq\mathbf{0}$ (otherwise we are back to the case
$M=\frac{\mathds{1}}{\prod d_{X_{out}}}$), we are left with two possibilities:
either
$\sum_{\mathbf{i}\neq\mathbf{0}\in\\{\\{M\\}\\}}w_{\mathbf{i}}m_{\mathbf{i}}=0$
for some nonvanishing $w_{\mathbf{i}}$, or
$w_{\mathbf{i}}=0\,,\forall\mathbf{i}\neq\mathbf{0}\in\\{\\{M\\}\\}$. We will
show that only the second possibility is viable.

Following Ref. [37], one can always construct a valid deterministic quantum
comb of the form
$M_{\mathbf{a}}=m_{\mathbf{0}}\sigma_{{\mathbf{0}}}+m_{\mathbf{a}}\sigma_{\mathbf{a}}$
for $\mathbf{a}\neq{\mathbf{0}}\in\\{\\{M\\}\\}$ with a small enough
$m_{\mathbf{a}}$ coefficient so that Eqs. (8) are satisfied (it is sufficient
to set $m_{\mathbf{0}}$ to $\left(\prod d_{X_{out}}\right)^{-1}$ and
$m_{\mathbf{a}}^{2}\leq m_{\mathbf{0}}^{2}$). Then, equation (59) yields
$w_{\mathbf{a}}=0$. Doing it again with a different valid comb
$M_{\mathbf{b}}=m_{\mathbf{0}}\sigma_{{\mathbf{0}}}+m_{\mathbf{b}}\sigma_{\mathbf{b}}$
such that $\mathbf{b}\neq\mathbf{a}$ yields a second condition,
$w_{\mathbf{b}}=0$. Repeating this argument for all possible choice of basis
element,
$\left\\{M_{\mathbf{k}}=m_{\mathbf{0}}\sigma_{{\mathbf{0}}}+m_{\mathbf{k}}\sigma_{\mathbf{k}}\right\\}_{\mathbf{k}\neq{\mathbf{0}}\in\\{\\{M\\}\\}}$,
proves the latter possibility:
$w_{\mathbf{i}}=0\,,\forall\;{\mathbf{i}}\neq{\mathbf{0}}\;\in\\{\\{M\\}\\}$.

Note that the last condition defines a subspace in the space of operators to
which $W$ must belong. This subspace is quasiorthogonal to the subspace
$\mathrm{Span}\left\\{\bigotimes M\right\\}$, meaning that the intersection of
the two subspaces is the span of the unit matrix. Noticing that
$w_{\mathbf{0}}\sigma_{\mathbf{0}}=\prescript{}{X}{W}\equiv\mathcal{D}[W]$,
one expresses this last statement in a basis-independent formulation:

$\\{W\\}:\sum_{\mathbf{i}\in\\{\\{M\\}\\}}w_{\mathbf{i}}\sigma_{\mathbf{i}}=w_{\mathbf{0}}\sigma_{{\mathbf{0}}}\\\
\iff\left(\mathcal{P}-\mathcal{D}\right)[W]=0\,.$ (60)

One can check that
$\left(\mathcal{P}-\mathcal{D}\right)^{2}=\mathcal{P}-\mathcal{D}$, so
$\mathcal{P}-\mathcal{D}$ is a projector. Taking the orthogonal complement,
one gets

$W=\left(\mathcal{I}-\mathcal{P}+\mathcal{D}\right)[W]\,,$ (61)

which is the sought formula.

The fact that Eqs. (12b) and (12c) are sufficient for (10) (and therefore (7))
is straightforward to check. Indeed, plugging (60), which is equivalent to
(12b), together with (12c) in Eq. (57), one verifies that
$m_{\mathbf{0}}\frac{d}{d_{in}}=1$ because Eq. (11b) implies that
$m_{\mathbf{0}}=1/d_{out}$. This completes the proof of the theorem.

##  Appendix C Proof of Theorem 2

The proof will be split among several lemmas. Given a finite set of orthogonal
projectors $\mathcal{P}^{1},\mathcal{P}^{2},\ldots$, we call their
intersection the orthogonal projector
$\mathcal{P}_{\mathcal{P}^{1}\cap\mathcal{P}^{2}\cap\ldots}$ that projects on
the intersection of the subspaces on which
$\mathcal{P}^{1},\mathcal{P}^{2},\ldots$ project. As shown in [41], if the
projectors commute, this intersection is given by the “product” of the
projectors (formally, their composition):

$\mathcal{P}_{\mathcal{P}^{1}\cap\mathcal{P}^{2}\cap\ldots}\equiv\mathcal{P}^{1}\cap\mathcal{P}^{2}\cap\ldots=\mathcal{P}^{1}\mathcal{P}^{2}\ldots\quad.$
(62)

Let $\mathcal{P}^{A\prec B}_{n_{A}+n_{B}}$ be the projector on the space of
valid $(n_{A}+n_{B})$-combs formed by composing $n_{A}$ operations (or
“teeth”) of a party $A$ with the $n_{B}$ operations of another party $B$ into
an overall deterministic $(n_{A}+n_{B})$-comb where the operations of Alice
are all before those of Bob (see the second case from the left in Fig. 6 for a
graphical example). Let $\mathcal{P}^{B\prec A}_{n_{A}+n_{B}}$ be the same
kind of comb projector but where the operations of $B$ are put before those of
$A$ (rightmost case in Fig. 6). Then the intersection of these two projectors
is

$\mathcal{P}^{A\prec B}_{n_{A}+n_{B}}\cap\mathcal{P}^{B\prec
A}_{n_{A}+n_{B}}=\mathcal{P}^{A\prec B}_{n_{A}+n_{B}}\mathcal{P}^{B\prec
A}_{n_{A}+n_{B}}\,.$ (63)

Figure 6: Examples of ways to append $n_{A}=1$ operations of Alice with
$n_{B}=2$ operations of Bob into valid 3-combs, with the associated projectors
on a particular subspace of valid 3-comb written below its graphical
representation.

Any comb in the intersection should therefore be compatible with either all
the operations of Alice being first or those of Bob. Intuitively, one can
conceive that the only kind of combs valid within this requirement are those
where the part of Alice is forbidden to communicate with the part of Bob and
vice versa. This is the content of the first lemma.

######  Lemma 2.

Consider a quantum comb in which some of the teeth are associated with a party
$A$ and the others with $B$. Then, the intersection of the subspace of
deterministic quantum combs in which the teeth of $A$ are all before those of
$B$ (e.g. Fig. 6, second case from the left) with the one in which the teeth
of $B$ are all before those of $A$ (Fig. 6, rightmost case) is equivalent to
$\mathrm{Span}\left\\{M^{A}\otimes M^{B}\right\\}$, which is the subspace
spanned by the tensor product of a smaller deterministic comb acting on the
nodes of $A$ only together with a smaller deterministic comb acting on the
nodes of $B$ only (Fig. 6, leftmost case). This is because their projectors
are equivalent:

$\mathcal{P}^{A\prec B}_{n_{A}+n_{B}}\mathcal{P}^{B\prec
A}_{n_{A}+n_{B}}=\mathcal{P}^{A}_{n_{A}}\otimes\mathcal{P}^{B}_{n_{B}}\,.$
(64)

###### Proof.

The proof relies upon the recursive formulation of a quantum comb projector
(39). When inspecting formula (51) for $\mathcal{P}^{A\prec B}_{n_{A}+n_{B}}$,
we see that it is possible to express the projector as:

$\mathcal{P}^{A\prec
B}_{n_{A}+n_{B}}=\mathcal{I}^{A}\otimes\mathcal{P}^{B}_{n_{B}}-\left(\mathcal{I}^{A}-\mathcal{P}^{A}_{n_{A}}\right)\otimes\mathcal{D}^{B}\,.$
(65)

Indeed, it holds for the case $n_{B}=1$:

$\displaystyle\mathcal{P}^{A\prec B}_{n_{A}+1}$
$\displaystyle=\prescript{}{1-B_{1}}{\left(\cdot\right)}+\prescript{}{B_{0}B_{1}}{\left(\mathcal{P}^{A}_{n_{A}}\right)}$
(66)
$\displaystyle=\prescript{}{1-B_{1}+B_{0}B_{1}}{\left(\cdot\right)}-\prescript{}{B_{0}B_{1}}{\left(1-\mathcal{P}^{A}_{n_{A}}\right)}$
$\displaystyle=\mathcal{I}^{A}\otimes\mathcal{P}^{B}_{1}-\prescript{}{B_{0}B_{1}}{\left(\mathcal{I}^{A}-\mathcal{P}^{A}_{n_{A}}\right)}$
$\displaystyle=\mathcal{I}^{A}\otimes\mathcal{P}^{B}_{1}-\left(\mathcal{I}^{A}-\mathcal{P}^{A}_{n_{A}}\right)\otimes\mathcal{D}^{B}\,.$

Suppose the decomposition (65) is true for $n_{A}+n_{B}-1$ operations. We
define the projector for such a case as $\mathcal{P}^{A\prec
B}_{n_{A}+n_{B}-1}\equiv\mathcal{P}^{A\prec B^{(1)}\prec\ldots\prec
B^{(n_{B}-1)}}_{n_{A}+n_{B}-1}$. Then, for $n_{A}+n_{B}$:

$\mathcal{P}^{A\prec
B}_{n_{A}+n_{B}}=\prescript{}{1-B_{2n_{B}-1}}{\left(\cdot\right)}+\prescript{}{B_{2n_{B}-2}B_{2n_{B}-1}}{\left(\mathcal{P}^{A\prec
B}_{n_{A}+n_{B}-1}\right)}\\\
=\prescript{}{1-B_{2n_{B}-1}}{\left(\cdot\right)}+\prescript{}{B_{2n_{B}-2}B_{2n_{B}-1}}{\left(\mathcal{I}^{A}\otimes\mathcal{P}^{B}_{n_{B}-1}\right)}\\\
-\prescript{}{B_{2n_{B}-2}B_{2n_{B}-1}}{\left(\left(\mathcal{I}^{A}-\mathcal{P}^{A}_{n_{A}}\right)\otimes\mathcal{D}^{B}_{n_{B}-1}\right)}\\\
=\mathcal{I}^{A}\otimes\mathcal{P}^{B}_{n_{B}}-\left(\mathcal{I}^{A}-\mathcal{P}^{A}_{n_{A}}\right)\otimes\mathcal{D}^{B}_{n_{B}}\,,$
(67)

where we injected $\mathcal{P}^{A\prec
B}_{n_{A}+n_{B}-1}=\mathcal{I}^{A}\otimes\mathcal{P}^{B}_{n_{B}-1}-\left(\mathcal{I}^{A}-\mathcal{P}^{A}_{n_{A}}\right)\otimes\mathcal{D}^{B}_{n_{B}-1}$
to go from the first equality to the second, with
$\mathcal{D}^{B}_{n_{B}-1}\equiv\prescript{}{B_{0}\ldots B_{2n_{B}-3}}{}$.
Next, we used Eq. (39) together with
$\prescript{}{B_{2n_{B}-2}B_{2n_{B}-1}}{\left(\mathcal{D}^{B}_{n_{B}-1}\right)}=\mathcal{D}^{B}$
to go to the last equality. This proves decomposition (65) by induction.

Now, applying an analogous decomposition to $\mathcal{P}^{B\prec
A}_{n_{A}+n_{B}}$, the left-hand side of Eq. (64) becomes

$\mathcal{P}^{A\prec B}_{n_{A}+n_{B}}\mathcal{P}^{B\prec A}_{n_{A}+n_{B}}=\\\
\left(\mathcal{I}^{A}\otimes\mathcal{P}^{B}_{n_{B}}-\left(\mathcal{I}^{A}-\mathcal{P}^{A}_{n_{A}}\right)\otimes\mathcal{D}^{B}\right)\\\
\times\left(\mathcal{P}^{A}_{n_{A}}\otimes\mathcal{I}^{B}-\mathcal{D}^{A}\otimes\left(\mathcal{I}^{B}-\mathcal{P}^{B}_{n_{B}}\right)\right)\\\
=\mathcal{P}^{A}_{n_{A}}\otimes\mathcal{P}^{B}_{n_{B}}\,.$ (68)

To go to the last line, we used the fact that $\mathcal{D}$ is the absorbing
(i.e. “zero”) element of the multiplication in the ring (see Sec. A.1), making
all the terms where it appears vanish, since
$\left(\mathcal{I}^{X}-\mathcal{P}^{X}\right)\mathcal{D}^{X}=\left(\mathcal{D}^{X}-\mathcal{D}^{X}\right)=0$.
∎

The projector of this first lemma is simply the projector on a subspace
containing all the $(n_{A}+n_{B})-$combs obtained by taking the tensor product
of an $n_{A}$-comb with an $n_{B}$-comb (e.g. leftmost case in Fig. 6). It
should also be compatible with permutations of Bob’s and Alice’s teeth in the
full comb that respect the local causal ordering for each party. We will need
the $\pi_{i}(A,B)$ notation introduced in Sec. 4 to refer to the $i$-th valid
permutation in the causal ordering of the teeth of parties $A$ and $B$ in
order to express this as a corollary.

######  Corollary 1.

The subspace made by the intersection of the $(n_{A}+n_{B})$-combs such that
all the teeth of $A$ are before those of $B$ with the $(n_{A}+n_{B})$-combs
such that all the teeth of $B$ are before those of $A$ is inside the subspace
of all $(n_{A}+n_{B})$-combs whose teeth orderings are permutations of the
teeth of $A$ and $B$ that respect the local order assumed for $A$ and $B$:

$\mathcal{P}^{A\prec B}_{n_{A}+n_{B}}\mathcal{P}^{B\prec
A}_{n_{A}+n_{B}}\mathcal{P}^{\pi_{i}(A,B)}_{n_{A}+n_{B}}=\mathcal{P}^{A\prec
B}_{n_{A}+n_{B}}\mathcal{P}^{B\prec A}_{n_{A}+n_{B}}\,\forall i\,.$ (69)

###### Proof.

The subset of deterministic quantum combs on
$\mathcal{L}\left(\mathcal{H}^{A}\otimes\mathcal{H}^{B}\right)$ formed by the
tensor product of combs on $A$ and $B$ is automatically a valid comb, no
matter how one defines the relative ordering between the teeth of $A$ and $B$
[15]. ∎

In other words, this corollary says that when an operator is a tensor product
of a deterministic $n_{A}$-comb with a deterministic $n_{B}$-comb, then it is
a valid deterministic $(n_{A}+n_{B})$-comb for any ordering of the teeth
compatible with the partial ordering of the individual combs.

The second lemma needed is just a simplification of the formula for the union
of projectors when applied to the case of quasiorthogonal projectors for
different teeth orderings.

######  Lemma 3.

The union of two arbitrary quasiorthogonal projectors
$\prescript{Q}{}{\mathcal{P}}^{\pi_{i}(A,B)}_{n_{A}+n_{B}}$ and
$\prescript{Q}{}{\mathcal{P}}^{\pi_{j}(A,B)}_{n_{A}+n_{B}}$ is given by

$\prescript{Q}{}{\mathcal{P}}^{\pi_{i}(A,B)}_{n_{A}+n_{B}}\cup\prescript{Q}{}{\mathcal{P}}^{\pi_{j}(A,B)}_{n_{A}+n_{B}}=\mathcal{I}-\mathcal{P}^{\pi_{i}(A,B)}_{n_{A}+n_{B}}\mathcal{P}^{\pi_{j}(A,B)}_{n_{A}+n_{B}}+\mathcal{D}\,.$
(70)

###### Proof.

It is proven by doing the computation explicitly, then simplifying by using
again the fact that the projector $\mathcal{D}$ on the span of unity is the
zero element. Hence,

$\displaystyle\prescript{Q}{}{\mathcal{P}}^{\pi_{i}}\prescript{Q}{}{\mathcal{P}}^{\pi_{j}}$
$\displaystyle=\left(\mathcal{I}-\mathcal{P}^{\pi_{i}}+\mathcal{D}\right)\left(\mathcal{I}-\mathcal{P}^{\pi_{j}}+\mathcal{D}\right)$
(71)
$\displaystyle=\mathcal{I}-\mathcal{P}^{\pi_{i}}-\mathcal{P}^{\pi_{j}}+\mathcal{P}^{\pi_{i}}\mathcal{P}^{\pi_{j}}+\mathcal{D}\,,$

where we have dropped the $n_{A}+n_{B}$ subscript and simplified the
superscript (i.e. $\pi_{i}(A,B)\equiv\pi_{i}$ ) in order to shorten the
notation. This gives, when plugged into (13),

$\displaystyle\prescript{Q}{}{\mathcal{P}}^{\pi_{i}}\cup\prescript{Q}{}{\mathcal{P}}^{\pi_{j}}$
$\displaystyle=\mathcal{I}-\mathcal{P}^{\pi_{i}}+\mathcal{D}+\mathcal{I}-\mathcal{P}^{\pi_{j}}+\mathcal{D}$
(72)
$\displaystyle\quad-\left(\mathcal{I}-\mathcal{P}^{\pi_{i}}-\mathcal{P}^{\pi_{j}}+\mathcal{P}^{\pi_{i}}\mathcal{P}^{\pi_{j}}+\mathcal{D}\right)$
$\displaystyle=\mathcal{I}-\mathcal{P}^{\pi_{i}}\mathcal{P}^{\pi_{j}}+\mathcal{D}\,.$

∎

All the elements needed for the main proof are in place. Note that the lemmas
have been derived assuming only 2 parties. Actually, these hold analogously
for any number of parties, since every formula that was used is associative.
In the case where $|\mathfrak{N}|>2$, one just has to group the parties
together to get back to the 2-partite scenario.

For the main proof, we start from the right-hand side of equation (14) to
which we apply the result of Lemma 3, equation (70):

$\bigcup_{\pi_{i}}\prescript{Q}{}{\mathcal{P}}^{\pi_{i}(A,B,\ldots)}_{n_{A}+n_{B}+\ldots}=\mathcal{I}-\prod_{\pi_{i}}\mathcal{P}^{\pi_{i}(A,B,\ldots)}_{n_{A}+n_{B}+\ldots}+\mathcal{D}\,.$
(73)

We then apply Corollary 1 so the permutations $\pi_{i}$ of all the nodes are
restricted to permutations $\pi_{i}^{\prime}$ of whole parties only:

$\displaystyle\mathcal{I}-\prod_{\pi_{i}^{\prime}}\mathcal{P}^{\pi_{i}^{\prime}(A,B,\ldots)}_{n_{A}+n_{B}+\ldots}$
$\displaystyle+\mathcal{D}=$ (74) $\displaystyle\mathcal{I}-$
$\displaystyle\mathcal{P}^{A\prec
B\prec\ldots}_{n_{A}+n_{B}+\ldots}\mathcal{P}^{B\prec
A\prec\ldots}_{n_{A}+n_{B}+\ldots}...+\mathcal{D}\,.$

Remember that here a superscript $X$ is referring to $n_{X}$ nodes like
$X\equiv X^{(1)}\prec X^{(2)}\prec\ldots\prec X^{(n_{X})}$. The final step is
to apply Lemma 2, equation (64), yielding

$\mathcal{I}-\mathcal{P}^{A\prec
B\prec\ldots}_{n_{A}+n_{B}+\ldots}\mathcal{P}^{B\prec
A\prec\ldots}_{n_{A}+n_{B}+\ldots}\ldots+\mathcal{D}=\\\
\mathcal{I}-\left(\mathcal{P}^{A}_{n_{A}}\otimes\mathcal{P}^{B}_{n_{B}}\otimes\ldots\right)+\mathcal{D}\,.$
(75)

We can now invoke the definition of the quasiorthogonal projector (12b) on the
right-hand side, and we have proven that

$\bigcup_{\pi_{i}}\prescript{Q}{}{\mathcal{P}}^{\pi_{i}(A,B,\ldots)}_{n_{A}+n_{B}+\ldots}=\prescript{Q}{}{\left(\mathcal{P}^{A}_{n_{A}}\otimes\mathcal{P}^{B}_{n_{B}}\otimes\ldots\right)}\quad.$
(76)

##  Appendix D Details for the activation example

As an illustration of Theorem 1, we will now sketch the proof that the
operator (26), rewritten here for convenience

$W^{AB}=\\\
\frac{1}{8}\left(\mathds{1}+\frac{1}{\sqrt{2}}\left[\sigma^{A_{0}}_{x}\sigma^{A_{2}}_{z}\sigma^{A_{3}}_{z}\sigma^{B_{0}}_{z}+\sigma^{A_{0}}_{z}\sigma^{A_{2}}_{z}\sigma^{B_{1}}_{z}\right]\right)\,,$
(77)

is a valid MPM with 3 nodes, where $A^{(1)}\prec A^{(2)}$.

This is a PSD matrix with trace
$\mathrm{Tr}\left[W^{AB}\right]=8=d_{A_{1}}d_{A_{3}}d_{B_{1}}=d_{out}$, so
conditions (12a) and (12c) are directly verified. The projective condition
(12b) is given by

$\displaystyle W^{AB}=\mathcal{I}^{AB}\left[W^{AB}\right]-$
$\displaystyle\prescript{}{\left(1-A_{3}+A_{2}A_{3}-A_{1}A_{2}A_{3}+A_{0}A_{1}A_{2}A_{3}\right)\left(1-B_{1}+B_{0}B_{1}\right)}{\left(W^{AB}\right)}$
$\displaystyle+\mathcal{D}^{AB}\left[W^{AB}\right]\,,$ (78)

where (9) have been used to find the 2-comb projector on $A$,
$\mathcal{P}^{A^{(1)}\prec
A^{(2)}}_{2}=\prescript{}{\left(1-A_{3}+A_{2}A_{3}-A_{1}A_{2}A_{3}+A_{0}A_{1}A_{2}A_{3}\right)}{}$,
as well as the 1-comb projector on $B$,
$\mathcal{P}^{B}_{1}=\prescript{}{\left(1-B_{1}+B_{0}B_{1}\right)}{}$. Notice
that
$\prescript{}{A_{3}}{\left(W^{AB}\right)}=\frac{1}{8}\left(\mathds{1}+\frac{1}{\sqrt{2}}\sigma^{A_{0}}_{z}\sigma^{A_{2}}_{z}\sigma^{B_{1}}_{z}\right)$,
and
$\prescript{}{B_{1}}{\left(W^{AB}\right)}=\frac{1}{8}\left(\mathds{1}+\frac{1}{\sqrt{2}}\sigma^{A_{0}}_{x}\sigma^{A_{2}}_{z}\sigma^{A_{3}}_{z}\sigma^{B_{0}}_{z}\right)$,
while all the other combinations of prescripts are equivalent to the action of
$\mathcal{D}^{AB}$ on the matrix, i.e.
$\prescript{}{A_{3}B_{1}}{\left(W^{AB}\right)}=\prescript{}{B_{0}B_{1}}{\left(W^{AB}\right)}=\prescript{}{A_{2}A_{3}}{\left(W^{AB}\right)}=\ldots=\frac{\mathds{1}}{8}=\mathcal{D}^{AB}\left[W^{AB}\right]$.
This allows us to simplify the projective condition (78) into

$W^{AB}=W^{AB}-\prescript{}{1-A_{3}-B_{1}}{\left(W^{AB}\right)}-\mathcal{D}^{AB}\left[W^{AB}\right]\,,$
(79)

which is effectively verified:

$W^{AB}=\prescript{}{A_{3}}{\left(W^{AB}\right)}+\prescript{}{B_{1}}{\left(W^{AB}\right)}-\mathcal{D}^{AB}\left[W^{AB}\right]\\\
=\frac{\mathds{1}}{4}+\frac{1}{8\sqrt{2}}\left[\sigma^{A_{0}}_{x}\sigma^{A_{2}}_{z}\sigma^{A_{3}}_{z}\sigma^{B_{0}}_{z}+\sigma^{A_{0}}_{z}\sigma^{A_{2}}_{z}\sigma^{B_{1}}_{z}\right]-\frac{\mathds{1}}{8}\\\
=\frac{1}{8}\left(\mathds{1}+\frac{1}{\sqrt{2}}\left[\sigma^{A_{0}}_{x}\sigma^{A_{2}}_{z}\sigma^{A_{3}}_{z}\sigma^{B_{0}}_{z}+\sigma^{A_{0}}_{z}\sigma^{A_{2}}_{z}\sigma^{B_{1}}_{z}\right]\right).$
(80)

The same way, operator (28),

$W=\\\
\frac{1}{8}\left(\mathds{1}+\frac{1}{\sqrt{2}}\left[\sigma^{A_{2}}_{z}\sigma^{A_{3}}_{z}\sigma^{L_{2}}_{x}\sigma^{B_{0}}_{z}+\sigma^{A_{2}}_{z}\sigma^{L_{2}}_{z}\sigma^{B_{1}}_{z}\right]\right)\,,$
(81)

written here without superscript, will now be proven to be a valid (M)PM on 2
nodes $A$ and $B$, where $L_{2}$ is extending the input system of party $A$.
It is straightforward to check that it is a PSD matrix with
$\mathrm{Tr}\left[W\right]=4=d_{A_{3}}d_{B_{1}}$, hence verifying (12a) and
(12c). The projective condition (12b) is expressed as

$W=\\\
\mathcal{I}\left[W\right]-\prescript{}{\left(1-A_{3}+L_{2}A_{2}A_{3}\right)\left(1-B_{1}+B_{0}B_{1}\right)}{\left(W\right)}+\mathcal{D}\left[W\right]=\\\
\prescript{}{A_{3}+B_{1}-A_{3}B_{1}+A_{3}B_{0}B_{1}-B_{0}B_{1}+L_{2}A_{2}A_{3}B_{1}-L_{2}A_{2}A_{3}}{\left(W\right)}\,.$
(82)

Computing each term on the right hand side, we find

$\displaystyle\prescript{}{A_{3}}{W}=\frac{1}{8}\left(\mathds{1}+\frac{1}{\sqrt{2}}\sigma^{A_{2}}_{z}\sigma^{L_{2}}_{z}\sigma^{B_{1}}_{z}\right),$
(83)
$\displaystyle\prescript{}{B_{1}}{W}=\frac{1}{8}\left(\mathds{1}+\frac{1}{\sqrt{2}}\sigma^{A_{2}}_{z}\sigma^{A_{3}}_{z}\sigma^{L_{2}}_{x}\sigma^{B_{0}}_{z}\right),$
$\displaystyle\prescript{}{A_{3}B_{1}}{W}=\prescript{}{A_{3}B_{0}B_{1}}{W}=\prescript{}{L_{2}A_{2}A_{3}B_{1}}{W}=\prescript{}{L_{2}A_{2}A_{3}}{W}=$
$\displaystyle\quad\prescript{}{B_{0}B_{1}}{W}=\frac{\mathds{1}}{8}\,.$

Therefore, (82) becomes

$W=\prescript{}{A_{3}}{W}+\prescript{}{B_{1}}{W}-\prescript{}{A_{3}B_{1}}{W}\\\
=\frac{\mathds{1}}{4}+\frac{1}{8\sqrt{2}}\left[\sigma^{A_{2}}_{z}\sigma^{L_{2}}_{z}\sigma^{B_{1}}_{z}+\sigma^{A_{2}}_{z}\sigma^{A_{3}}_{z}\sigma^{L_{2}}_{x}\sigma^{B_{0}}_{z}\right]-\frac{\mathds{1}}{8}\\\
=\frac{1}{8}\left(\mathds{1}+\frac{1}{\sqrt{2}}\left[\sigma^{A_{2}}_{z}\sigma^{A_{3}}_{z}\sigma^{L_{2}}_{x}\sigma^{B_{0}}_{z}+\sigma^{A_{2}}_{z}\sigma^{L_{2}}_{z}\sigma^{B_{1}}_{z}\right]\right)\,,$
(84)

proving that the projective condition (12b) is indeed verified for $W$.