For any two-party process matrix $W$, we have

$\displaystyle\mathcal{R}_{s}(W)\leq d_{\bar{O}}^{2}-1\,,$ (86)

where $d_{\bar{O}}=\max(d_{A_{O}},d_{B_{O}})$.

This bound can be achieved by a causally ordered comb if certain dimensional
conditions are satisfied. For example, without loss of generality, let
$d_{A_{O}}\geq d_{B_{O}}$. Then, if $d_{A_{O}}=d_{B_{I}}$, the bound
$d_{\bar{O}}^{2}-1=d_{A_{O}}^{2}-1$ is saturated by a process of the form

$\displaystyle W^{A\rightarrow
B}\coloneqq\frac{\mathbbm{1}_{A_{I}}}{d_{A_{I}}}\otimes\Phi_{A_{O}B_{I}}^{+}\otimes\mathbbm{1}_{B_{O}},$
(87)

where
$\Phi_{A_{O}B_{I}}^{+}=\sum_{ij}\ket{ii\vphantom{jj}}\\!\bra{jj\vphantom{ii}}$
is the Choi matrix of the identity channel. Choosing the proper witness
$S=\mathbbm{1}_{A_{I}}\otimes\Phi_{A_{O}B_{I}}^{+}\otimes\frac{\mathbbm{1}_{B_{O}}}{d_{B_{O}}}$,
we see that $\operatorname{tr}(W^{A\rightarrow B}S)-1=d_{A_{O}}^{2}-1$.

Importantly, while this bound holds for general two-party processes, it is
only tight for scenarios, where the dimension of the largest output space
($A_{O}$ above) is at most as large as that of the other party’s input space
($B_{I}$ above). Intuitively, this is due to the fact that there is no
identity channel from $A_{O}$ to $B_{I}$ when $d_{A_{O}}>d_{B_{I}}$, and the
causal connection between the parties is thus bounded by the upper bound we
gave in Prop. 2 but not necessarily tight anymore (the influence of the
different dimensions of the involved spaces on the tightness of the bounds we
provide is discussed in more detail in Sec. 5.4).

We emphasize that the fact that $W$ is a proper process matrix (and not just a
positive semidefinite matrix which satisfies
$\operatorname{tr}(W)=d_{A_{O}}d_{B_{O}}$) is crucial for the derivation of
the above bounds. For positive semidefinite matrices $W$ with
$\operatorname{tr}(W)=d_{A_{O}}d_{B_{O}}$, it is easy to see that the term
$\operatorname{tr}(WS)$ can achieve its algebraic maximum of
$d_{A_{I}}d_{A_{O}}d_{B_{I}}d_{B_{O}}$ and analogously for more parties. This,
then, also implies that Lem. 1 holds for proper process matrices, but not for
general positive semidefinite matrices with
$\operatorname{tr}(W)=d_{A_{O}}d_{B_{O}}$.

###  5.2 Most resourceful process on two parties

Let us, for the moment, assume that all dimensions of $W$ are the same, i.e.,
$d_{A_{I}}=d_{A_{O}}=d_{B_{I}}=d_{B_{O}}$. As we have seen, in this case, a
process with the maximal signalling robustness is given by

$\displaystyle W^{A\rightarrow
B}\coloneqq\frac{\mathbbm{1}_{A_{I}}}{d_{A_{I}}}\otimes\Phi_{A_{O}B_{I}}^{+}\otimes\mathbbm{1}_{B_{O}},$
(88)

which we will call the fully signalling process. The process $W^{B\rightarrow
A}$ is defined analogously.

In principle, $W^{A\rightarrow B}$ could be a good candidate for the most
resourceful process of the resource theory of causal connection, in the sense
that, starting from $W^{A\rightarrow B}$, all other processes may be reachable
by means of free adapters. This, however, is not the case for two distinct
reasons. On the one hand, if $W^{\prime}_{A^{\prime}B^{\prime}}$ is defined on
spaces $\\{A_{I}^{\prime},A_{O}^{\prime},B_{I}^{\prime},B_{O}^{\prime}\\}$
with $d_{X_{I}^{\prime}}>d_{X_{I}}$ and $d_{X_{O}^{\prime}}>d_{X_{O}}$ for
$X\in\\{A,B\\}$, then the signalling robustness of
$W^{\prime}_{A^{\prime}B^{\prime}}$ can exceed that of $W^{A\rightarrow B}$
(since $d_{\bar{O}^{\prime}}>d_{\bar{O}}$), implying that $W^{A\rightarrow B}$
cannot be transformed to all $W^{\prime}_{A^{\prime}B^{\prime}}$ by means of
free transformations.

On the other hand, even when focusing on transformations that do not change
(or only decrease) the respective dimensions, a free adapter cannot change the
causal ordering of the process matrix it acts on, such that $W^{A\rightarrow
B}$ cannot be transformed to, for example, a process that is ordered
$B\vec{\prec}A$, or a process that is causally non-separable. At best, then,
$W^{A\rightarrow B}$ and $W^{B\rightarrow A}$ can be the most resourceful
processes for all process matrices that are of ordering $A\prec B$ and $B\prec
A$, respectively, and that do not exceed them with respect to the dimensions
of the involved spaces (we will henceforth assume this latter property and not
mention it explicitly anymore). It will turn out that this is indeed the case,
both for two- and the three-party case.

Interestingly, the causally separable process

$\displaystyle W^{\text{mix}}_{q}\coloneqq(1-q)\,W^{A\to B}+q\,W^{B\to A},$
(89)

which is a convex mixture of two fully signalling processes, one in each
direction, also satisfies
$\mathcal{R}_{s}(W^{\text{mix}}_{q})=d_{\bar{O}}^{2}-1$, which can be seen by
setting $S=\Phi^{+}_{A_{O}B_{I}}\otimes\Phi^{+}_{B_{O}A_{I}}$ in the dual SDP
(74).

Processes of this form do not have a definite causal order (they are causally
separable, though), but since free adapters cannot create causal non-
separability, they cannot be transformed to all process matrices. More
interestingly, as we will see, there are causally non-separable processes that
have a lower signalling robustness than $W^{\text{mix}}_{q}$, implying that
the signalling robustness does not impose a total order in the set of all
process matrices.

Let us now show that, within the set of two-party ordered processes $W^{A\prec
B}$, the above process $W^{A\rightarrow B}$ is indeed the most valuable in the
sense that all two-party ordered processes $W^{A\prec B}$ can be obtained from
it. This can be easily seen by considering that every $W^{A\prec B}$ can be
written as a concatenation of an initial pure state $\Psi_{A_{I}E}$ on $A_{I}$
and some appropriate ancillary system $E$ and a CPTP map $A_{O}E\rightarrow
B_{I}$ with corresponding Choi matrix $\Gamma_{A_{O}EB_{I}}$, such that
$W^{A\prec
B}_{AB}=\Psi_{A_{I}E}\star\Gamma_{A_{O}EB_{I}}\star\mathbbm{1}_{B_{O}}$ [4].
It is straightforward to see that, by performing a local adapter in her
laboratory, Alice can prepare all states $\Psi_{A_{I}E}$ and implement all
maps $\Gamma_{A_{O}EB_{I}}$ which, when applying such adapters to
$W^{A\rightarrow B}$, allows her to produce all possible two-party ordered
processes $W^{A\prec B}$ (see Fig. 13 for a graphical representation). We thus
have the following proposition:

######  Proposition 3.

Every process matrix $W^{\prime}_{X^{\prime}Y^{\prime}}$ with causal ordering
$X^{\prime}\prec Y^{\prime}$ can be obtained from $W^{X\rightarrow Y}$ via
free adapters if the dimensions of the spaces
$W^{\prime}_{X^{\prime}Y^{\prime}}$ is defined on do not exceed those of the
corresponding spaces of $W^{X\rightarrow Y}$.

The same holds true in the three-party case, where, for example, the process
matrix

$\displaystyle W^{A\rightarrow B\rightarrow
C}=\eta_{A_{I}}\otimes\Phi^{+}_{A_{O}B_{I}}\otimes\Phi^{+}_{B_{O}C_{I}}\otimes\mathbbm{1}_{C_{O}}$
(90)

can be transformed to all processes of ordering $A\prec B\prec C$. We provide
the proof in App. H, where we also present a conjecture for an ordered process
that cannot be reached in the four-party case. Importantly, this latter four-
party conjecture – or any statement of this kind – only make sense if we put
restrictions on the respective dimensions; having identity channels on
arbitrary dimensions at hand, will allow one to implement all causally ordered
processes on (sufficiently) lower dimensions by simply using the additional
dimensions as a memory carrier. This is similar to the analogous case in the
resource theories of entanglement, coherence and purity, where all states on
lower dimensions can be achieved from a single resourceful state of
sufficiently large dimension [11, 12, 32, 16].

Figure 13: Constructing all $A\prec B$ two-party combs from $W^{A\rightarrow
B}$. Any causally ordered comb $W^{A\prec B}$ can be written as a circuit with
an initial system-environment state and a CPTP map that acts on the system and
the environment. By applying a local adapter, Alice can thus transform
$W^{A\rightarrow B}$ to any process $W^{A\prec B}$ (with system dimensions
smaller or equal than those of $W^{A\rightarrow B}$, that is). For
convenience, we have dropped the primes in the system labeling in the bottom
panel.

###  5.3 Transformations of bipartite processes

As we have seen, some process matrices cannot be reached by transforming the
most robust bipartite process with free adapters. The first example of this
kind we discuss is the process $W^{\text{OCB}}_{A^{\prime}B^{\prime}}$ (named
after Oreshkov, Costa and Brukner [5]), a two-party, causally non-separable
process matrix introduced in Ref. [5] and defined as

$\displaystyle\begin{split}&W^{\text{OCB}}\\\
&=\frac{1}{4}\left[\mathbbm{1}_{AB}+\frac{1}{\mathchoice{{\hbox{$\displaystyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\textstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=6.44444pt,depth=-5.15558pt}}}{{\hbox{$\scriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=4.51111pt,depth=-3.6089pt}}}{{\hbox{$\scriptscriptstyle\sqrt{2\,}$}\lower
0.4pt\hbox{\vrule
height=3.22221pt,depth=-2.57779pt}}}}\left(\sigma_{A_{O}}^{z}\sigma_{B_{I}}^{z}+\sigma_{A_{I}}^{z}\sigma_{B_{I}}^{x}\sigma_{B_{O}}^{z}\right)\right]\,,\end{split}$
(91)

where $\sigma^{z}$ and $\sigma^{x}$ are Pauli matrices and we have omitted the
respective identity matrices and tensor products. This causally non-separable
process cannot be reached from $W^{A\rightarrow B}$ or $W^{B\rightarrow A}$,
since free adapters cannot map causally ordered to causally non-separable
processes. By using the SDP (73), we find that the signalling robustness of
$W^{\text{OCB}}$ is $\mathcal{R}_{s}(W^{\text{OCB}})=1$. Consequently, it is
also impossible to transform $W^{\text{OCB}}$ to $W^{A\rightarrow B}$ or
$W^{B\rightarrow A}$ by means of free adapters, since the latter two processes
have – when all systems are qubits – a signalling robustness of $3$.
Expectedly, there is no total order in the set of processes with respect to
the free transformations $\Theta_{\texttt{NS}}$ (this was already clear from
the fact that processes of order $A\vec{\prec}B$ cannot be transformed to
processes of order $B\vec{\prec}A$ and vice versa by means of free adapters).

While $W^{\text{OCB}}$ cannot be transformed to $W^{A\rightarrow B}$ by means
of free transformations, there exist processes $W^{A\vec{\prec}B}$ with causal
order $A\vec{\prec}B$ such that
$W^{A\vec{\prec}B}=W^{\text{OCB}}\star\Upsilon$ for some free adapter
$\Upsilon\in\Theta_{\texttt{NS}}$. More generally, in the two-party case, any
causally non-separable process matrix $W$ can be ‘degraded’ to a process of
causal order $A\vec{\prec}B$ or $B\vec{\prec}A$, i.e., there exists a free
adapter $\Upsilon$ such that $W\star\Upsilon$ is of causal order
$A\vec{\prec}B$ or $B\vec{\prec}A$. This can be seen from the following, more
general Proposition:

######  Proposition 4.

For any process matrix $W$ with $\mathcal{R}_{s}(W)>0$ there exists a free
adapter $\Upsilon$ such that $W\star\Upsilon$ has a causal ordering
$A\vec{\prec}B$ or $B\vec{\prec}A$.

###### Proof.

For the proof, note that $\mathcal{R}_{s}(W)>0$ implies
$W\neq{}_{A_{O}B_{O}}W$. Now, consider the free adapter $\Upsilon^{(X_{O})}$
that traces out the output space $X_{O}$ and replaces it by a maximally mixed
state, i.e., $\Upsilon^{(X_{O})}\star W={}_{X_{O}}W$. The resulting process is
causally ordered $A\prec B$ if $X=B$ and $B\prec A$ if $X=A$. It remains to
show that for at least one of these possibilities we do not have $A||B$. To
see this, imagine that ${}_{X_{O}}W$ satisfies $A||B$ for all $X$. In this
case we have

$\displaystyle{}_{A_{O}}W={}_{A_{O}B_{O}}W\quad\text{and}\quad{}_{B_{O}}W={}_{A_{O}B_{O}}W\,.$
(92)

Inserting this into $L_{V}[W]=W$ yields $W={}_{A_{O}B_{O}}W$ which contradicts
the assumption $\mathcal{R}_{s}(W)>0$. ∎

Naturally, the above proposition is trivial if $W$ already has a definite
causal order, but becomes more insightful for the case where $W$ corresponds
to a mixture of causal orders or a causally non-separable process.

Finally, one might wonder if there are processes that can be transformed into
both definite causal orderings $A\vec{\prec}B$ and $B\vec{\prec}A$ by means of
free adapters. An answer to this question is provided by the following
Proposition:

######  Proposition 5.

Any causally non-separable process matrix on two parties can be transformed
into a process matrix with ordering $A\vec{\prec}B$, as well as into one with
ordering $B\vec{\prec}A$, by means of free adapters.

Before proving the above statement, we emphasize that it is in line with Prop.
4, since we have $\mathcal{R}_{s}(W)>0$ for all causally non-separable process
matrices, but goes beyond it, since now transformation to both definitive
causal orderings is possible.

###### Proof.

For the proof, consider a free trace-and-replace adapter that traces out the
output of Alice’s operation and feeds forward a fixed state
$\ket{\Psi\vphantom{\Psi}}\\!\bra{\Psi\vphantom{\Psi}}^{\mathrm{T}}$ (where
the additional transpose is for notational convenience), i.e.,
$W^{\prime}=\Upsilon\star
W=\braket{\Psi_{A_{O}}}{W}{\Psi_{A_{O}}}\otimes\mathbbm{1}_{A_{O}^{\prime}}$.
In what follows, for ease of notation, without risk of confusion, we identify
$A_{O}$ and $A_{O}^{\prime}$, thus replacing $\mathbbm{1}_{A_{O}^{\prime}}$ by
$\mathbbm{1}_{A_{O}}$. Since the corresponding adapter is a local operation in
Alice’s laboratory, it corresponds to a free adapter, and the resulting
$W^{\prime}$ is causally ordered $B\prec A$ (but potentially $B||A$ holds).
Now, we show that assuming that $W^{\prime}$ is ordered $B||A$ for all
possible $\ket{\Psi}$ contradicts the assumption that the original $W$ was
causally non-separable.

To this end, we choose a basis
$\\{\ket{\Psi^{(i)}_{A_{O}}\vphantom{\Psi^{(i)}_{A_{O}}}}\\!\bra{\Psi^{(i)}_{A_{O}}\vphantom{\Psi^{(i)}_{A_{O}}}}\\}_{i=1}^{d_{A_{O}}^{2}}$
of $\mathcal{B}(\mathcal{H}_{A_{O}})$ consisting of pure states. Assuming that
$W^{\prime}_{i}:=\braket{\Psi^{(i)}_{A_{O}}}{W}{\Psi^{(i)}_{A_{O}}}\otimes\mathbbm{1}_{A_{O}^{\prime}}$
is causally ordered $B||A$ for all $i$ then implies

$\displaystyle
W_{i}^{\prime}=\rho_{A_{I}B_{I}}^{(i)}\otimes\mathbbm{1}_{A_{O}B_{O}}\quad\forall
i\,,$ (93)

where $\\{\rho_{A_{I}B_{I}}^{(i)}\\}$ are quantum states. With this, we can
express $W$ as

$\displaystyle
W=\sum_{i=1}^{d_{A_{O}}^{2}}\rho_{A_{I}B_{I}}^{(i)}\otimes\Delta_{A_{O}}^{(i)}\otimes\mathbbm{1}_{B_{O}}\,,$
(94)

where $\\{\Delta_{A_{O}}^{(i)}\\}_{i=1}^{d_{A_{O}}^{2}}$ is the dual set to
the basis
$\\{\ket{\Psi^{(i)}_{A_{O}}\vphantom{\Psi^{(i)}_{A_{O}}}}\\!\bra{\Psi^{(i)}_{A_{O}}\vphantom{\Psi^{(i)}_{A_{O}}}}\\}_{i=1}^{d_{A_{O}}^{2}}$,
in the sense that
$\operatorname{tr}(\Delta_{A_{O}}^{(j)}\ket{\Psi_{A_{O}}^{(i)}\vphantom{\Psi_{A_{O}}^{(i)}}}\\!\bra{\Psi_{A_{O}}^{(i)}\vphantom{\Psi_{A_{O}}^{(i)}}})=\delta_{ij}$.
The validity of Eq. (94) can be checked by direct insertion. Since the process
matrix of Eq. (94) factorizes into an identity matrix on $B_{O}$ and the rest,
we see that it is causally ordered $A\prec B$, contradicting the assumption
that $W$ is causally non-separable. Consequently, there must exist at least
one $\ket{\Psi_{A_{O}}^{(i)}}$ for which $W_{i}^{\prime}$ is causally ordered
$B\vec{\prec}A$. Running the same argument for a trace-and-replace adapter on
Bob’s side then shows that it must also be possible to transform a causally
non-separable process matrix $W$ to a process of causal order $A\vec{\prec}B$.
∎

We note that, in the above proof, the requirement of causal non-separability
can be relaxed. Indeed, it is sufficient if $W$ is a convex combination of
causal orders but not of the form $A\prec B$, $B\prec A$, or $A||B$.

The two above propositions provide an a posteriori justification for the
inclusion of causally indefinite processes as valid objects in a resource
theory of causal connection; while free adapters do not allow the
transitioning from one fixed causal order to another, causally indefinite
processes in a sense ‘bridge the gap’ between the two sets, since each
causally indefinite process can be transformed to both $A\vec{\prec}B$ as well
as $B\vec{\prec}A$ processes. This, then, both raises the question, whether
there exists a most resourceful causally non-separable process, as well as
whether there exists a causally indefinite process that allows one to reach
all causally ordered ones by means of free adapters. We return to these
questions in Sec. 6.2, where we introduce a second monotone of causal
connection that is tailored to the distinction between causally separable and
causally non-separable processes. First though, we extend some of the results
we found above for the two-party case to the multi-party scenario.

###  5.4 Signalling robustness of multipartite causally ordered processes

Some of the results on signalling robustness translate straightforwardly to
the multi-party case. Here, we provide a couple of relevant cases for which a
bound on the signalling robustness can be found. The somewhat technical proofs
are relegated to App. I.

For arbitrary many parties with causal order $A\prec B\prec C\prec\cdots$ a
generalization of the fully signalling process $W^{A\to B}$ indeed maximizes
$\mathcal{R}_{s}(W^{A\prec B\prec C\prec\cdots})$. To see this, we require the
following proposition:

######  Proposition 6.

Let $\textup{{Proc}}_{1:N}$ be the set of all causally ordered processes on
$N$ parties with causal order $X^{(1)}\prec X^{(2)}\prec\cdots\prec X^{(N)}$.
For any $W\in\textup{{Proc}}_{1:N}$ we have

$\displaystyle\mathcal{R}_{s}(W)\leq\prod_{i=1}^{N-1}d_{X^{(i)}_{O}}^{2}-1:=d_{\bar{O}}^{2}-1,$
(95)

where $d_{X^{(i)}_{O}}$ is the output dimension of party $X^{(i)}_{O}$.

We emphasize that here, unlike in the two-party case, we restrict our
attention to the causally ordered case, such that there is no maximization
over the output dimensions, but rather the bound on the signalling robustness
only depends on all but the last output space, which is intuitively clear,
since the last output space cannot signal to any of the other parties. The set
of processes for which the above bound (or a variant thereof) holds can be
slightly enlarged to the set of all processes with a definite last party and
convex mixtures thereof (see App. I).

For the case where all involved dimensions are equal (the more general case is
discussed in App. I), it is then straightforward to show the following
corollary:

######  Corollary 1.

If all involved dimensions are equal, then the signalling robustness on the
set $\textup{{Proc}}_{1:N}$ is maximized by the process
$W^{X^{(1)}\rightarrow\cdots\rightarrow X^{(N)}}$.

As we show in App. H, for three parties,
$W^{X^{(1)}\rightarrow\cdots\rightarrow X^{(N)}}$ can also be transformed by
means of free adapters to all other tripartite processes that are ordered in
the same way, making it not only the most robust, but also the most
resourceful process.

For the interpretation of the provided result, for simplicity, consider the
case where all dimensions are equal. Then, it appears to be most advantageous
from a causal connection perspective to perfectly pass information in one
direction. Neither memory, nor convex mixture with different causal orders
yield better causal connections. Upon numerical investigation, it appears like
this bound can also not be beaten by causally non-separable processes in the
tripartite case, and we conjecture that this holds for arbitrary numbers of
parties. As mentioned, a more thorough investigation of the multipartite case
can be found in App. I.

After this short discussion of the multi-party case, let us now return to the
two-party scenario and analyse the interconvertibility of processes by means
of free adapters in more detail, particularly within the set of causally non-
separable processes, where it presents itself more layered than in the
causally ordered one.

##  6 Resource theory of causal non-separability

In the discussion of the resource theory of causal connection, causally non-
separable processes formed the natural ‘bridge’ between the sets of processes
with different causal orders, and allowed for a discussion of the signalling
structure of all processes that quantum mechanics (in principle) allows for.
Beyond this role for the investigation of signalling properties, causal non-
separability can be considered a resource in its own right – as evidenced, for
example, by the fact that causal non-separability can allow for the violation
of causal inequalities [5, 78], or outperform causally ordered processes when
it comes to distinguishing non-signalling channels [53]. Somewhat
unsurprisingly, the corresponding resource theory of causal non-separability
is conceptually closely related to the resource theory of causal connection.
Here, we first detail how the resource theory of causal non-separability can
be developed following the same reasoning as above and show that, unlike in
the case of causal connection, there is only one meaningful set of free
adapters when causal non-separability is the resource of interest. In order to
quantify the resourcefulness of causal non-separability, we provide a
corresponding resource monotone, the causal robustness (originally introduced
in [6]), which also turns out to be a resource monotone for the resource
theory of causal connection. Consequently, it will allow us to investigate the
interconvertibility of causally non-separable processes under free adapters
with respect to causal connection.

###  6.1 Uniqueness of the resource theory of causal non-separability

For simplicity, we limit our discussion of resource theories of causal non-
separability to the two-party case. Naturally, the set of free objects in such
a resource theories is the set Sep of causally separable processes. Following
the same path we took for the resource theory of causal connection, the
different sets of possible free transformations could be derived by imposing
qualitatively different requirements on them. Such requirements on free
operations888While mathematically they are the same type of objects, for
better distinction, we use two different symbols, $\Xi$ and $\Upsilon$ for
adapters in the resource theory of causal non-separability and that of causal
connection, respectively. $\Xi$ are:

  * R$1$’.

To map process matrices to process matrices.

  * R$2$’.

To map causally separable to causally separable processes.

  * R$3$’.

To not allow for the creation of causal non-separability.

  * R$4$’.

To be implemented with only causally ordered resources between the parties.

Requirement R$1$’ is the same as R$1$ and simply ensures that free operations
are also admissible. Requirement R$4$’ seems like the natural analogue of
R$4$. As we have seen, there are adapters that are composed of signalling
channels and still yield admissible adapters, one such example being
$\Upsilon^{2SW}$. However, unlike R$4$ for the resource theory of causal
connection, R$4$’ does not always lead to admissible adapters. To see this,
consider an adapter $\Xi$ that is implemented by Alice and Bob by making use
of a resource that allows for signalling from Bob to Alice. In many cases, the
resulting adapter cannot be applied to a process matrix of causal ordering
$A\prec B$ since causal loops would be closed. An example for such an adapter
is $\Upsilon^{1SW}\notin\Theta_{{\texttt{A}}}$. In addition to potentially
yielding ‘improper adapters’, from a conceptual point of view, it is
questionable why (and how), in a situation where Alice and Bob share a comb of
a given causal order (say $A\prec B$), they should be able to signal in, say,
the opposite direction when making an adapter. Intuitively then, the only
types of adapters that Alice and Bob can implement without potentially running
into logical contradiction seem to be those that do not require signalling,
i.e., adapters in $\Theta_{\texttt{LOSE}}$. While this might be too
restrictive – as we mentioned, $\Upsilon^{2SW}\notin\Theta_{\texttt{LOSE}}$
would be an admissible adapter that can be implemented by means of signalling
resources – $\Theta_{\texttt{LOSE}}$ appears like the largest set of adapters
that Alice and Bob can implement themselves that is not plagued with
interpretational issues. Requirement R$4$’ should thus be replaced by R$4$,
implying that, at the operational level, where Alice and Bob are the ones to
implement the free adapters, there is no fundamental distinction between the
resource theory of causal non-separability and that of causal connection.

Adapters that satisfy both requirements R$1$’ and R$2$’ are the largest set of
possible free transformations in a resource theory of causal non-separability,
and we denote the corresponding set of admissible separability preserving
adapters by $\Theta_{\texttt{ASP}}$. It is natural to compare this maximal set
of free operation to $\Theta_{{\texttt{A}\texttt{FP}}}$, the corresponding
maximal set of free operations for the resource theory of causal connection.
As we have seen, there are free adapters
$\Upsilon\in\Theta_{{\texttt{A}\texttt{FP}}}$ that map causally ordered
processes to causally non-separable processes. Consequently, such adapters
would not lie in $\Theta_{\texttt{ASP}}$. On the other hand, it is easy to
construct adapters that satisfy $\Xi\in\Theta_{\texttt{ASP}}$ but
$\Xi\notin\Theta_{\texttt{A}\texttt{FP}}$. An example for such an adapter is
given by

$\displaystyle\xi=\frac{1}{d_{A_{O}}d_{B_{O}}}\mathbbm{1}_{AB}\otimes
W_{A^{\prime}B^{\prime}}\,,$ (96)

where $W_{A^{\prime}B^{\prime}}\in\texttt{Sep}\setminus\texttt{Free}$. This
adapter, when acting on a process matrix simply discards it and replaces it by
a fixed, signalling process matrix (and thus does not lie in
$\Theta_{\texttt{AFP}}$). With this, we see that, while they naturally share
many common adapters, there is no clear hierarchy or inclusion relation
between the two maximal sets of free adapters for the resource theory of
causal connection and that of causal non-separability.

Requirement R$3$’ appears like the natural analogy to the corresponding
requirement R$3$ for the resource theory of causal connection. However, on its
own, it does not even guarantee that the resulting set of adapters preserves
Sep. Basically, R$3$’ demands that, no matter what Alice and Bob do locally,
when they have access to the adapter, they cannot use it to create a process
matrix that is causally non-separable. Locally, the only operations they can
perform to transform an adapter to a process matrix is to feed in a state and
discard the final degrees of freedom, i.e., given an adapter $\Xi$, they can
transform it to

$\displaystyle
W^{\prime}=\rho_{A_{I}}\otimes\eta_{B_{I}}\otimes\mathbbm{1}_{A_{O}B_{O}}\star\Xi.$
(97)

Now, since $\rho_{A_{I}}\otimes\eta_{B_{I}}\otimes\mathbbm{1}_{A_{O}B_{O}}$
corresponds to a non-signalling process, the requirement that
$W^{\prime}\in\texttt{SEP}$ is weaker than demanding that non-signalling
processes are mapped to non-signalling processes, which, besides being
admissible, is the restriction on adapters that lie in
$\Theta_{{\texttt{A}\texttt{FP}}}$. Since adapters in
$\Theta_{{\texttt{A}\texttt{FP}}}$ can create causally non-separable process
matrices when acting on a causally separable one, this means that the set of
free adapters emerging from requirement R$3$’ is too big to be a meaningful
set of free adapters for the resource theory of causal non-separability. On
the other hand, combining requirements R$2$’ and R$3$’ (and R$1$’) to restrict
ourselves to adapters that preserve Sep does not introduce anything new, since
the resulting set of free adapters simply coincides with
$\Theta_{\texttt{ASP}}$ (i.e., the set already obtained from the combination
of R$1$’ and R$2$’): Since
$\rho_{A_{I}}\otimes\eta_{B_{I}}\otimes\mathbbm{1}_{A_{O}B_{O}}$ is a process
matrix in Sep, satisfaction of R$2$’ ensures that it cannot be mapped to a
causally non-separable one, thus already implying satisfaction of requirement
R$3$’. Consequently, while R$3$ yields an interesting and physically
meaningful set of free adapters, its naïve translation R$3$’ for the resource
theory of causal non-separability does not lead to additional restrictions/a
meaningful new theory.

We conclude thus, given that R$4$ coincides with R$4$’, there seems to only be
a single meaningful resource theory of causal non-separability, namely the one
where the free transformations are given by $\Theta_{\texttt{ASP}}$. This
statement notwithstanding, one could, in principle, introduce other sets of
adapters that preserve Sep. For example, $\Theta_{\texttt{NS}}$, the set of
all non-signalling adapters preserves Sep, since all such adapters preserve
causal order. However, there is no additional operational motivation to
restrict oneself to this set of adapters as a natural set of free adapters in
the resource of causal non-separability.

It remains to briefly comment on the structure of adapters
$\Xi\in\Theta_{\texttt{ASP}}$. As we have seen above, for the resource theory
of causal connection, it is possible to phrase the properties of free adapters
in terms of linear constraints (besides positivity that is). Mathematically,
this is due to the fact that the non-signalling requirement on free process
matrices, i.e., that they are of the form
$\rho_{A_{I}B_{I}}\otimes\mathbbm{1}_{A_{O}B_{O}}$ is – besides the positivity
constraint – a linear one. On the other hand, this fails to hold for the
condition of causal separability; linear combinations of causally separable
processes can well be causally non-separable. This, in turn, makes a concrete
characterization of the adapters in the set $\Theta_{\texttt{ASP}}$ more
elusive and the question of whether a given adapter lies in
$\Theta_{\texttt{ASP}}$ a hard one to answer in general.

Nonetheless, the resource theory of causal non-separability can nicely be
phrased within the framework for the resource theory of causal connection that
we set up above. Here, we do not aim to provide a full account of it, but
merely want to emphasize the structural similarities with the resource theory
of causal connection.

Additionally, in the same vein as above, it is natural to introduce a
robustness measure with respect to the set Sep, the causal robustness, as a
monotone for the resource theory of causal non-separability. While this
monotone is not faithful for causal connection, as it turns out, it is
nonetheless monotone under adapters in $\Theta_{\texttt{NS}}$ and
$\Theta_{\texttt{LOSE}}$. Introducing the causal robustness thus kills two
birds with one stone – it completes our brief discussion of the resource
theory of causal non-separability and provides us with a second monotone for
the resource theory of causal connection, allowing, for example, for the
investigation of the interconvertibility of processes in
$\texttt{Proc}\setminus\texttt{Sep}$ under adapters
$\Upsilon\in\Theta_{\texttt{NS}}$.

###  6.2 Robustness of causal non-separability

We have seen in the previous sections that the stratification of the space of
process matrices with respect to free adapters of the resource theory of
causal connection presents itself as somewhat layered. In particular, there is
no total order, i.e.,
$W\overset{\Theta_{\texttt{NS}}}{\nrightarrow}W^{\prime}$ does not generally
imply $W^{\prime}\overset{\Theta_{\texttt{NS}}}{\rightarrow}W$. Concretely, we
have seen this for pairs of processes, where either both processes were
causally ordered, or one was causally ordered, while the other was not.
However, we have not yet considered the case where both processes are causally
non-separable.

Since the previous results were obtained based on the monotonicity of the
robustness of signalling under free adapters, they hold independent of whether
$\Theta_{\texttt{LOSE}}$, $\Theta_{\texttt{NS}}$, or
$\Theta_{\texttt{A}\texttt{FP}}$ is considered as the set of free adapters.
This will fail to hold true for the results of the present section, where we
require preservation of Sep for the derivation of our results.

By providing a second monotone for the resource theory of causal connection,
we show that the lack of total order also holds within the set of causally
non-separable processes. To this end, we consider the (generalized) robustness
of causal non-separability $\mathcal{R}_{c}$, inspired by analogous measures
for the robustness of entanglement [74] and coherence [31], and already
introduced in Ref. [6] for process matrices (following Ref. [6], we will
simply call $\mathcal{R}_{c}$ ‘causal robustness’). We define
$\mathcal{R}_{c}(W)$ of a process matrix $W$ as the maximal robustness under
worst-case general mixing, now with respect to the set of causally separable
process matrices Sep (instead of the set Free), i.e.,

$\displaystyle\mathcal{R}_{c}(W)=\underset{T\in\texttt{Proc}}{\min}\left\\{s\geq
0\left|\frac{W+sT}{1+s}=R\in\texttt{Sep}\right.\right\\}\,,$ (98)

where $T$ is proper process matrix and Sep is the set of causally separable
process matrices.999While the definition of the set Sep is straightforward in
the bipartite case, it becomes more subtle in the multipartite setting [6, 65,
66]. Here, we predominantly focus on the two-party case. Since free processes
in the resource theory of causal non-separability preserve Sep, the causal
robustness is a monotone of this resource theory. More importantly for our
purposes, it is also a monotone for two of the resource theories of causal
connection we introduced above (those based on $\Theta_{\texttt{NS}}$ and
$\Theta_{\texttt{LOSE}}$), and we now use it to investigate how causally non-
separable processes can be converted into one another by means of adapters in
$\Theta_{\texttt{NS}}$.

###  6.3 Conversion of causally non-separable processes under free
transformations

Analogously to the signalling robustness, one can show that the causal
robustness is a convex function and non-increasing under free adapters (i.e.,
adapters in $\Theta_{\texttt{NS}}$). This latter point follows directly from
the fact that free adapters preserve causal ordering (this fails to hold for
adapters in $\Theta_{\texttt{AFP}}$).

However, the causal robustness is not a faithful measure for causal
connection, since $\mathcal{R}_{c}(W)=0$ for all $W\in\texttt{Sep}$, which is
a strict superset of Free. Nonetheless, the fact that it is non-increasing
under free adapters allows one to show that there are pairs
$\\{W,W^{\prime}\\}$ of causally non-separable process matrices for which
$W\overset{\Theta_{\texttt{NS}}}{\nrightarrow}W^{\prime}$ and
$W^{\prime}\overset{\Theta_{\texttt{NS}}}{\nrightarrow}W$ holds, i.e., there
is no total order in the set of causally non-separable processes with respect
to free adapters in $\Theta_{\texttt{NS}}$.

As for the signalling robustness, the causal robustness of a process matrix
can be phrased as an SDP and equivalently be calculated by its primal and dual
formulation. Its primal SDP problem can be derived starting from the
definition of causal robustness in Eq. (98) and following similar steps as for
the signalling robustness in Eqs. (72) and (73), to arrive at

$\displaystyle\begin{split}\textbf{given}\ \ &W\\\ \textbf{min}\ \
&\frac{1}{d_{O}}\operatorname{tr}(\widetilde{R})-1\\\ \textbf{s.t.}\ \
&\widetilde{R}-W\geq 0\\\ &\widetilde{R}\geq 0\\\ &\widetilde{R}\propto
R\in\texttt{Sep}.\end{split}$ (99)

The dual problem associated to the above SDP reads

$\displaystyle\begin{split}\textbf{given}\ \ &W\\\ \textbf{max}\ \
&\operatorname{tr}(WS)-1\\\ \textbf{s.t.}\ \ &S\geq 0\\\
&\frac{1}{d_{O}}\mathbbm{1}+U-L_{A\prec B}(U)\geq S\\\
&\frac{1}{d_{O}}\mathbbm{1}+V-L_{B\prec A}(V)\geq S,\end{split}$ (100)

where $L_{X\prec Y}(W)\coloneqq{}_{Y_{O}}W-_{Y_{I}Y_{O}}W+_{X_{O}Y_{I}Y_{O}}W$
is the projector onto the subspace of processes with causal order $X\prec Y$.
Analogously as for the robustness of signalling, $S$ can be interpreted as a
witness of causal non-separability.

Evaluating either this SDP or its dual, the causal robustness of
$W^{\text{OCB}}$ [see Eq. (91)] can be computed to be
$\mathcal{R}_{c}(W^{\text{OCB}})=0.1716$, while the causal robustness of
$W^{A\to B}$ is $\mathcal{R}_{c}(W^{A\to B})=0$. Hence, this pair of process
matrices provides us with an interesting example where

$\displaystyle\mathcal{R}_{s}(W^{\text{OCB}})$
$\displaystyle<\mathcal{R}_{s}(W^{A\to B})$ (101) and
$\displaystyle\mathcal{R}_{c}(W^{\text{OCB}})$
$\displaystyle>\mathcal{R}_{c}(W^{A\to B})=0,$ (102)

reinforcing that neither process matrix can be converted into the other with
free adapters.

There are also examples of pairs of process matrices that are both causally
non-separable but that, nevertheless, cannot be converted into each other. In
order to find such examples, we numerically sample process matrices and check
their properties. For bipartite process matrices, this works as follows:

  1. 1.

Fix input dimensions $d_{A_{I}},d_{B_{I}}$ and output dimensions
$d_{A_{O}},d_{B_{O}}$.

  2. 2.

Uniformly sample a positive semidefinite matrix $Q$ of size
($d_{A_{I}}d_{A_{O}}d_{B_{I}}d_{B_{O}}$)$\times$($d_{A_{I}}d_{A_{O}}d_{B_{I}}d_{B_{O}}$).

  3. 3.

Define $\widetilde{W}:=L_{V}(Q)$ to be the projection of $Q$ on the subspace
of valid process matrices, where $L_{V}$ is the projection operator in Eq.
(6).

  4. 4.

Define
$W=d_{A_{O}}d_{B_{O}}\frac{\widetilde{W}}{\operatorname{tr}(\widetilde{W})}$.

  5. 5.

Check whether $W$ is a positive semidefinite matrix. If not, discard $W$ and
repeat the process. If yes, than $W$ is a proper process matrix.

This method can be straightforwardly extended to process matrices of multiple
parties.

Sampling random processes in this way, we find an example of a process matrix
$W^{\\#}$, whose signalling robustness is $\mathcal{R}_{s}(W^{\\#})=1.0581$,
while its causal robustness is $\mathcal{R}_{c}(W^{\\#})=0.0245$. Just like
$W^{\text{OCB}}$, the process matrix $W^{\\#}$ is causally non-separable.
However, there is no clear hierarchy of robustness between them, since they
satisfy the relations

$\displaystyle\mathcal{R}_{s}(W^{\text{OCB}})$
$\displaystyle<\mathcal{R}_{s}(W^{\\#})$ (103) and
$\displaystyle\mathcal{R}_{c}(W^{\text{OCB}})$
$\displaystyle>\mathcal{R}_{c}(W^{\\#})>0.$ (104)

Notice that, even though both process matrices in this example are causally
non-separable, which means that, in principle, there could exist a free
adapter that could convert one into the other, the impossibility of such
conversion is nevertheless guaranteed by the lack of hierarchy between them,
as shown by the different ordering in their signalling and causal robustness
values. A data file containing the matrices that correspond to the processes
$W^{\text{OCB}}$ and $W^{\\#}$, along with the values of their robustness, can
be found in the online repository [79].

It would then be interesting to find out whether in the set of causally non-
separable processes there exist one process matrix that could be transformed
into all others with free adapters (in principle, this is possible despite the
lack of total order). This process matrix would necessarily have higher or
equal causal and signalling robustness then all other causally non-separable
processes.

In order to investigate the potentially most valuable resource with respect to
causal robustness, we applied a seesaw algorithm, alternatingly optimising the
witness $S$ for a given process matrix $W$ and vice versa. Concretely, the
algorithm iterates the SDP (100) and the SDP given by

$\displaystyle\begin{split}\textbf{given}\ \ &S\\\ \textbf{max}\ \
&\operatorname{tr}(WS)\\\ \textbf{s.t.}\ \ &W\in\texttt{Proc}.\end{split}$
(105)

which takes the optimal witness from the solution of SDP (100) as input and
outputs the process matrix that maximally violates it. The output process
matrix then becomes the input of SDP (100) in the next round of iteration, and
so on. As this problem is non-convex, one is not guaranteed to obtain the
global optimum for $\operatorname{tr}(WS)$ by means of this algorithm.
However, running the seesaw for a large number of different initial process
matrices provides a good guess for the process matrices which maximize
$\mathcal{R}_{c}(W)$.

The highest value of causal robustness that we have found for process matrices
in which all subspaces are qubits with our seesaw is
$\mathcal{R}_{c}(W^{*})=0.2104$. This value exceeds
$\mathcal{R}_{c}(W^{\text{OCB}})$ showing that $W^{\text{OCB}}$ cannot be
transformed into $W^{*}$ with free adapters. This same process matrix $W^{*}$
has signalling robustness of $\mathcal{R}_{s}(W^{*})=1.4805$, which is also
greater than $\mathcal{R}_{s}(W^{\text{OCB}})=1$. Hence, these two causally
non-separable process matrices satisfy the relation

$\displaystyle\mathcal{R}_{s}(W^{\text{OCB}})$
$\displaystyle<\mathcal{R}_{s}(W^{*})$ (106) and
$\displaystyle\mathcal{R}_{c}(W^{\text{OCB}})$
$\displaystyle<\mathcal{R}_{c}(W^{*}),$ (107)

making $W^{*}$ a good candidate for a process matrix that can be transformed
into $W^{\text{OCB}}$ with free adapters. A data file containing the matrices
that correspond to the processes $W^{\text{OCB}}$ and $W^{*}$, along with the
values of their robustness, can be found in the online repository [79].

In principle, the question of whether one process can be transformed to
another by means of free adapters can be decided by the SDP

$\displaystyle\begin{split}\textbf{given}\ \ &W,W^{\prime}\\\ \textbf{find}\ \
&\Upsilon\\\ \textbf{s.t.}\ \ &\Upsilon\star W=W^{\prime}\\\
&\Upsilon\in\Theta_{\texttt{NS}}\,.\end{split}$ (108)

Using this SDP, we have verified that the conversion of $W^{*}$ into
$W^{\text{OCB}}$ with free adapters is not possible. Although not conclusive,
this result indicates that there might not be a causally non-separable process
which represents the most valuable resource with respect to causal connection.

Considering the matter of a most resourceful process matrix on more general
grounds, we may derive some insight from the analysis of a hypothetical
process.

Consider the matrix
$Z\in\mathcal{B}(\mathcal{H}_{A_{I}}\otimes\mathcal{H}_{A_{O}}\otimes\mathcal{H}_{B_{I}}\otimes\mathcal{H}_{B_{O}})$
consisting of an identity channel from $A_{O}$ to $B_{I}$ and another identity
channel from $B_{O}$ to $A_{I}$, given by

$\displaystyle Z\coloneqq\Phi^{+}_{A_{O}B_{I}}\otimes\Phi^{+}_{A_{I}B_{O}}.$
(109)

$Z$ is not a proper process matrix, as evidenced by the fact that $Z\neq
L_{V}(Z)$, since by allowing for perfect communication in both directions it
could give rise to causal loops. This non-valid process could be considered a
natural extension of the most valuable process with respect to signalling
robustness – the fully-signalling bipartite ordered processes $W^{A\to B}$ or
$W^{B\to A}$ [see Eq. (88)] that can be transformed into any other ordered
processes of the same order and subspace dimensions by means of free adapters
– to the realm of indefinite causal order. Indeed, all ordered process
matrices can be achieved from $Z$ by simply blocking one of the output spaces
to create a one-way fully-signalling ordered process, and using a free adapter
to transform it. It can also be checked that $W^{\text{OCB}}$ and $W^{*}$ can
be generated by applying free adapters to $Z$, as well as all randomly sampled
causally non-separable processes we tried. Hence, it is not unreasonable to
expect that all bipartite processes could be reached via free transformations
on $Z$.

Although such an unrealistic resource may be overkill when it comes to
generating all bipartite process matrices, the evidence presented here
suggests that it could actually be the required resource, in the sense that no
proper process matrix might be able to fulfil such a role.

In order to better understand the non-valid process $Z$, we investigate how
far it is from the set Proc of valid process matrices of the same dimension.
One potential measure for this distance is the robustness of $Z$ with respect
to the set of valid process matrices, i.e.,

$\displaystyle\mathcal{R}_{\texttt{Proc}}(Z)=\underset{T}{\min}\left\\{s\geq
0\left|\frac{Z+sT}{1+s}=W\in\texttt{Proc}\right.\right\\}.$ (110)

Here, we must carefully choose the characteristics of the noise $T$ against
which we will be measuring robustness. If $T$ is set to be any valid process
matrix, then $\mathcal{R}_{\texttt{Proc}}(Z)$ will diverge, since $Z$ is not
contained in the image of the projector $L_{V}$. A more viable option,
although with an arguably more far-fetched interpretation, would be to set the
noise to any $T\geq 0$ of fixed trace $\operatorname{tr}T=d_{A_{O}}d_{B_{O}}$.
The solution of such a problem for an all-qubit $Z$ is then
$\mathcal{R}_{\texttt{Proc}}(Z)=3$ and $W=W^{\text{mix}}$ [see Eq. (89)].

Perhaps surprinsingly, according to this notion of distance, the valid process
matrix closest to $Z$ is not a causally non-separable process such as $W^{*}$
or $W_{\text{OCB}}$, but instead, an equally-weighted convex mixture of two
one-way fully-signalling processes ordered in opposite directions, precisely
the most robust process matrix with respect to signalling robustness.

While not a physical process, the discussion of the (non-valid) process $Z$
sheds some light on the properties a most resourceful two-party process matrix
$W^{\mathrm{max}}$ would have to satisfy. On the one hand, it would have to
contain perfect channels both from Alice to Bob, as well as from Bob to Alice,
that can be ‘addressed’ independently, such that one can degrade
$W^{\mathrm{\max}}$ to both $W^{A\rightarrow B}$ and $W^{B\rightarrow A}$.
This, already, seems to exclude the existence of $W^{\mathrm{max}}$, since
such a process matrix would most likely contain closed loops and thus lead to
logical paradoxa. On the other hand, $W^{\mathrm{max}}$ would have to be
causally non-separable, since causal separability is preserved under free
adapters. Consequently, we conjecture that $Z$ (or variants thereof) is the
only positive matrix of the correct trace that can be transformed to all
processes by means of free adapters, but, as already mentioned, $Z$ is not a
valid process itself.

##  7 Conclusion

In this paper, we have constructed a resource theory of causal connection,
making a first step towards a systematic resource-theoretic understanding of
signalling in general (non-)causal structures. We derived the fundamental
building blocks of this theory – the sets of free objects and free
transformations – both on operational grounds as well as more axiomatic
considerations. In turn, the resulting different levels of control of the
involved parties over the adapter yield a strict hierarchy of sets of free
transformations.

The axiomatic perspective leads to the largest set of possible free
transformations, which, despite being free, still allows for internal
signalling as well as for the creation of causal non-separability. The guiding
principle in the more operationally motivated settings were generalized non-
signalling conditions between the involved parties via free transformations.
Like in the case of many other resource theories – most prominently that of
entanglement – all of these approaches lead to the same set of free objects,
but differ with respect to the free transformations, thus giving rise to
distinct resource theories of causal connection.

In order to quantify the causal connection of general processes, we introduced
the signalling robustness as a faithful monotone of the resource theory of
causal connection and provided a general upper bound for this monotone for the
case of two parties, as well as tight upper bounds for many relevant multi-
party scenarios. Somewhat surprisingly, the process that maximizes the
signalling robustness for two parties is causally ordered, implying that
causal non-separability does not increase ‘the amount of communication’ that
is possible between parties. Based on numerical evidence, we conjecture that
this also holds true for an arbitrary numbers of parties, suggesting that the
communication advantages displayed by causally indefinite processes are not
rooted in an overall greater ability to communicate when the assumption of
global causal order is dropped. For two and three parties, in addition, we
provided the most resourceful causally ordered process, respectively, i.e.,
the process that can be transformed to all other processes of the same causal
order by means of free transformations.

Besides quantifying causal connection in general processes, the signalling
robustness has a direct interpretation in terms of a witness of causal
connection. Concurrently, this robustness can be understood as a measure of
how many causal loops can be closed in a given process.

Our results allow one to investigate and quantify – under one common umbrella
– the ability to signal, both in causally ordered as well as causally
disordered processes. Naturally, analogous considerations for the set of
causally separable processes as the set of free objects lead to a resource
theory of causal non-separability, which we introduced, showed uniqueness of,
and compared to the resource theory of causal connection. In the same vein,
using the framework we provided, one could also consider other sets of
resources in process matrices, like, for example, the possibility of two-way
signalling. Such considerations will be subject to future work.

As we showed by means of causal robustness, a second monotone of the resource
theory of causal connection, causally non-separable processes display a
layered structure with respect to free transformations, and there exist pairs
of causally non-separable processes where no process can freely be transformed
to the other. On the other hand, while causally ordered processes can only be
transformed to processes of the same causal order by means of free adapters,
for two parties, every causally indefinite process can be transformed to
processes of either causal order by means of free transformations.

While quantifying causal connection, it is currently unclear what operational
task is naturally related to the signalling robustness. Using techniques
provided in Refs. [80, 75, 76], for every process $W$ that satisfies
$\mathcal{R}_{s}(W)>0$, one can tailor a particular information-theoretic game
for which the process $W$ will outperform every process with vanishing
signalling robustness (and this outperformance is quantified by
$\mathcal{R}_{s}(W)$). However, no single operational task is known, for which
the signalling robustness faithfully quantifies performance. In particular,
such a task would have to involve bidirectional signalling, since it would
have to highlight both signalling in the direction $A\rightarrow B$ as well as
$B\rightarrow A$ (for example, a task that only requires signalling from Alice
to Bob would not faithfully represent the resourcefulness of a process that
allows for signalling from Bob to Alice). Besides this operational question,
it is of interest to both find a general tight bound for the signalling
robustness for any number of parties, and to investigate how the signalling
robustness ties in with the complete set of monotones for the resource theory
of causal connection that we provide in App. J.

Additionally, there are still many natural open questions with respect to the
interconvertibility of processes under free transformations, both single shot
as well as asymptotically, approximately, and catalytically [38]. In
particular these latter points require detailed analysis since – unlike in
other resource theories like those involving states and channels as resources
– the tensor product of process matrices in general does not yield a proper
process matrix [81].

Finally, as mentioned, in the causally ordered case, there are processes from
which all others can be reached by means of free transformations for the two-
and three-party scenario. This suggests that these processes play a similar
role as that played by the ebit in the resource theory of entanglement, giving
rise to the question of distillability [82, 83, 12] of causal connection for
general causally ordered processes.

All code developed for this work is openly available in the online repository
[79].

We thank Tristan Kraft, Kavan Modi, Felix A. Pollock, Benjamin Yadin and Marco
Túlio Quintino for valuable discussions, and Giuseppe Vitagliano for providing
the idea for the proof of Prop. 2. The authors are also grateful to Princess
Bubblegum for performing very demanding calculations. S. M. acknowledges
funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska Curie grant agreement No 801110, and the
Austrian Federal Ministry of Education, Science and Research (BMBWF). The
opinions expressed in this publication are those of the authors, the EU Agency
is not responsible for any use that may be made of the information it
contains. JB acknowledges funding from the Austrian Science Fund (FWF) through
the Zukunftskolleg ZK03 and the START project Y879-N27. GC acknowledges
financial support from the Hong Kong Research Grant Council through grant
17300918 and through the Senior Research Fellowship Scheme SRFS2021-7S02, by
the Croucher Foundation, by the John Templeton Foundation through grant 61466,
The Quantum Information Structure of Spacetime (qiss.fr). Research at the
Perimeter Institute is supported by the Government of Canada through the
Department of Innovation, Science and Economic Development Canada and by the
Province of Ontario through the Ministry of Research, Innovation and Science.
The opinions expressed in this publication are those of the authors and do not
necessarily reflect the views of the John Templeton Foundation.

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## APPENDIX

##  Appendix A Characterisation of admissible adapters

An admissible adapter
$\Upsilon^{\texttt{A}}_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}$
should – besides being positive – map process matrices
$W\in\mathcal{B}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$ onto process matrices
$W^{\prime}\in\mathcal{B}(\mathcal{H}_{A^{\prime}}\otimes\mathcal{H}_{B^{\prime}})$.
Equivalently, as discussed, it can be understood as a map that transforms non-
signalling maps
$M^{\prime}_{A^{\prime}B^{\prime}}\in\mathcal{B}(\mathcal{H}_{A^{\prime}}\otimes\mathcal{H}_{B^{\prime}})$
to non-signalling maps
$M_{AB}\in\mathcal{B}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$. Let us take the
latter standpoint to derive the linear constraints on the set of adapters
using techniques from [84]. First, we see that for any non-signalling map
$N^{\prime}\in\mathcal{B}(\mathcal{H}_{A^{\prime}}\otimes\mathcal{H}_{B^{\prime}})$
(whenever there is no risk of confusion, we will drop the subscripts from now
on) we have

$\displaystyle
N^{\prime}=N^{\prime}-{}_{A_{O}^{\prime}}N^{\prime}+{}_{A_{I}^{\prime}A_{O}^{\prime}}N^{\prime}\quad\text{and}\quad
N^{\prime}=N^{\prime}-{}_{B_{O}^{\prime}}N^{\prime}+{}_{B_{O}^{\prime}B_{I}^{\prime}}N^{\prime}\,.$
(111)

Combining these two conditions, we see that

$\displaystyle\begin{split}N^{\prime}&=N^{\prime}-{}_{A_{O}^{\prime}}N^{\prime}+{}_{A_{I}^{\prime}A_{O}^{\prime}}N^{\prime}-{}_{B_{O}^{\prime}}N^{\prime}+{}_{A_{O}^{\prime}B_{O}^{\prime}}N^{\prime}-{}_{A_{I}^{\prime}A_{O}^{\prime}B_{O}^{\prime}}N^{\prime}+{}_{B_{I}^{\prime}B_{O}^{\prime}}N^{\prime}-{}_{A_{O}^{\prime}B_{I}^{\prime}B_{O}^{\prime}}N^{\prime}+{}_{A_{I}^{\prime}A_{O}^{\prime}B_{I}^{\prime}B_{O}^{\prime}}N^{\prime}\\\
&=:L_{ns}^{\prime}[N^{\prime}]\end{split}$ (112)

holds for all non-signalling maps. Naturally, this equation is stronger than
the requirement for general trace preserving maps, since it implies
${}_{A_{O}^{\prime}B_{O}^{\prime}}N^{\prime}={}_{A_{I}^{\prime}A_{O}^{\prime}B_{I}^{\prime}B_{O}^{\prime}}N^{\prime}$,
which is the TP requirement on the Choi matrices of maps
$\mathcal{B}(\mathcal{H}_{A_{I}^{\prime}}\otimes\mathcal{H}_{B_{I}^{\prime}})\rightarrow\mathcal{B}(\mathcal{H}_{A_{O}^{\prime}}\otimes\mathcal{H}_{B_{O}^{\prime}})$.
The operator $L^{\prime}_{NS}$ defined above is a trace-preserving self-dual
projector. Now, demanding that $\Upsilon^{\texttt{A}}$ maps non-signalling
maps to non-signalling maps implies

$\displaystyle L_{ns}[\Upsilon^{\texttt{A}}\star
N^{\prime}]=\Upsilon^{\texttt{A}}\star N^{\prime}.$ (113)

for all non-signalling maps $N^{\prime}$. Since the span of the set of non-
signalling maps is not the full space of Hermitian matrices, the above
equation does not yet fully determine the linear constraints on
$\Upsilon^{\texttt{A}}$. However, since $L_{ns}^{\prime}$ is a projector, we
see that every matrix $N^{\prime}=L_{ns}^{\prime}[R^{\prime}]$ satisfies
$N^{\prime}=L_{ns}^{\prime}[N^{\prime}]$ for arbitrary
$R^{\prime}\in\mathcal{B}(\mathcal{H}_{A^{\prime}}\otimes\mathcal{H}_{B^{\prime}})$,
implying that the span of the set of non-signalling maps is given by
$L^{\prime}_{ns}[\mathcal{B}(\mathcal{H}_{A^{\prime}}\otimes\mathcal{H}_{B^{\prime}})]$.
With this, using the self-duality of $L_{ns}^{\prime}$, we can transform the
above equation to a full linear constraint on $\Upsilon^{\texttt{A}}$ such
that it reads

$\displaystyle(L_{ns}\otimes L_{ns}^{\prime})[\Upsilon^{\texttt{A}}]\star
R^{\prime}=(\mathcal{I}_{AB}\otimes
L_{ns}^{\prime})[\Upsilon^{\texttt{A}}]\star R^{\prime}$ (114)

for all
$R^{\prime}\in\mathcal{B}(\mathcal{H}_{A^{\prime}}\otimes\mathcal{H}_{B^{\prime}})$.
This, then, implies that

$\displaystyle\Upsilon^{\texttt{A}}=\Upsilon^{\texttt{A}}-(\mathcal{I}_{AB}\otimes
L_{ns}^{\prime})[\Upsilon^{\texttt{A}}]+(L_{ns}\otimes
L_{ns}^{\prime})[\Upsilon^{\texttt{A}}]\,.$ (115)

Additionally, $\Upsilon^{\texttt{A}}$ is trace-rescaling on the set of non-
signalling maps (since it maps channels to channels), i.e.,

$\displaystyle\operatorname{tr}(\Upsilon^{\texttt{A}}\star
N^{\prime})=\frac{d_{A_{I}}d_{B_{I}}}{d_{A^{\prime}_{I}}d_{B^{\prime}_{I}}}\operatorname{tr}(N^{\prime})$
(116)

for all non-signalling maps $N^{\prime}$. Using the same arguments as above,
as well as the trace preservation of $L_{ns}^{\prime}$, this translates to

$\displaystyle\operatorname{tr}[(\mathcal{I}_{AB}\otimes
L_{ns}^{\prime}[\Upsilon^{\texttt{A}}])(R^{\prime\mathrm{T}}\otimes\mathbbm{1}_{AB})]=\frac{d_{A_{I}}d_{B_{I}}}{d_{A^{\prime}_{I}}d_{B^{\prime}_{I}}}\operatorname{tr}(R^{\prime})\,,$
(117)

which implies

$\displaystyle\operatorname{tr}_{AB}\\{\mathcal{I}_{AB}\otimes
L_{ns}^{\prime}[\Upsilon^{\texttt{A}}]\\}=\frac{d_{A_{I}}d_{B_{I}}}{d_{A^{\prime}_{I}}d_{B^{\prime}_{I}}}\mathbbm{1}_{A^{\prime}B^{\prime}}\,.$
(118)

With this, we obtain

$\displaystyle\operatorname{tr}(\Upsilon^{\texttt{A}})=d_{A_{I}}d_{B_{I}}d_{A_{O}^{\prime}}d_{B_{O}^{\prime}}.$
(119)

Using this, we see that

$\displaystyle{}_{AB}(L_{ns}^{\prime}[\Upsilon^{\texttt{A}}])={}_{ABA^{\prime}B^{\prime}}\Upsilon^{\texttt{A}}\,,$
(120)

where we omitted the identity operator $\mathcal{I}_{AB}$. This gives us the
conditions on an admissible adapter presented in Def. 1.

##  Appendix B Free-preserving adapters

In the main text, we show that free-preserving adapters, besides being
positive, have to satisfy the following constraints:

$\displaystyle\begin{split}{}_{A_{O}^{\prime}B_{O}^{\prime}}(\Upsilon^{\texttt{FP}}\star
W^{A||B})&=(\Upsilon^{\texttt{FP}}\star W^{A||B})\\\
\text{and}\quad\operatorname{tr}(\Upsilon^{\texttt{FP}}\star
W^{A||B})&=d_{A^{\prime}_{O}}d_{B_{O}^{\prime}}\,.\end{split}$ (121)

Following the same arguments as in the previous section, any free process
matrix $W^{A||B}$ can be understood as $W^{A||B}={}_{A_{O}B_{O}}R$ for some
$R$, so that the first of the two above equations can be rewritten as

$\displaystyle{}_{A_{O}^{\prime}B_{O}^{\prime}}(\Upsilon^{\texttt{FP}}\star{}_{A_{O}B_{O}}R)=(\Upsilon^{\texttt{FP}}\star{}_{A_{O}B_{O}}R)\,.$
(122)

Using the self-duality of
${}_{A_{O}B_{O}}\,\begin{picture}(-1.0,1.0)(-1.0,-3.0)\circle*{2.5}\end{picture}\
$ and the fact that the above has to hold for all
$R\in\mathcal{B}(\mathcal{H}_{A}\otimes\mathcal{B}_{B})$, we obtain

$\displaystyle{}_{A_{O}^{\prime}B_{O}^{\prime}A_{O}B_{O}}\Upsilon^{\texttt{FP}}={}_{A_{O}B_{O}}\Upsilon^{\texttt{FP}}$
(123)

as desired. Again, analogously to the proof in the previous section, the trace
rescaling property of $\Upsilon^{\texttt{FP}}$ on the set of free process
matrices can be rewritten as

$\displaystyle\operatorname{tr}(\Upsilon^{\texttt{FP}}\star{}_{A_{O}B_{O}}R)=\frac{d_{A^{\prime}_{O}}d_{B_{O}^{\prime}}}{d_{A_{O}}d_{B_{O}}}\operatorname{tr}R\,$
(124)

for all $R\in\mathcal{B}(\mathcal{H}_{A}\otimes\mathcal{B}_{B})$, where we
used the fact that
${}_{A_{O}B_{O}}\,\begin{picture}(-1.0,1.0)(-1.0,-3.0)\circle*{2.5}\end{picture}\
$ is a trace preserving operator. This implies
$\operatorname{tr}_{A^{\prime}B^{\prime}}({}_{A_{O}B_{O}}\Upsilon^{\texttt{FP}})=\frac{d_{A^{\prime}_{O}}d_{B_{O}^{\prime}}}{d_{A_{O}}d_{B_{O}}}\mathbbm{1}_{AB}$,
which, in turn, yields

$\displaystyle\operatorname{tr}(\Upsilon^{\texttt{FP}})=d_{A_{I}}d_{B_{I}}d_{A_{O}^{\prime}}d_{B_{O}^{\prime}}\quad\text{and}\quad{}_{A^{\prime}B^{\prime}A_{O}B_{O}}\Upsilon^{\textup{{FP}}}={}_{A^{\prime}B^{\prime}AB}\Upsilon^{\textup{{FP}}}\,,$
(125)

as claimed in the main text.

##  Appendix C Admissible and free-preserving adapters can create causal non-
separability

As we have seen in the main text, admissible and free-preserving adapters
$\Upsilon^{AFP}$ can change the causal order of a non-free process and even
create mixtures of causal orders. Now we are left to verify that admissible
and free-preserving adapters can also map non-free causally separable
processes to causally non-separable processes. We tackle this question
numerically and we make use of the witnesses for causal non-separability that
where introduced in [6]. These witnesses are such that, if for a witness $G$
and a process matrix $W$ we have $\operatorname{tr}(GW)<0$, then the process
matrix $W$ is causally non-separable. With this, one can search for an adapter
$\Upsilon^{\texttt{A}\texttt{FP}}\in\Theta_{{\texttt{A}\texttt{FP}}}$ that
satisfies $\Upsilon^{{\texttt{A}\texttt{FP}}}\star W\notin\texttt{Sep}$ for
some $W\in\texttt{Sep}$ via a see-saw SDP. In particular, if there is a
causally separable $W$ such that $\Upsilon^{{\texttt{A}\texttt{FP}}}\star
W\notin\texttt{Sep}$, then there must also be a causally ordered process
matrix $W^{A\prec B}$ such that $\Upsilon^{{\texttt{A}\texttt{FP}}}\star
W^{A\prec B}\notin\texttt{Sep}$. As we see in Sec. 5.1, for two parties, the
most valuable process (in the sense that it maximizes the causal robustness)
with ordering $A\prec B$ is of the form

$\displaystyle W^{A\rightarrow
B}=\Psi_{A_{I}}\otimes\Phi^{+}_{A_{O}B_{I}}\otimes\mathbbm{1}_{B_{O}}\,,$
(126)

where $\Psi_{A_{I}}$ is an arbitrary quantum state.

Intuitively, this process matrix provides a good starting point when searching
for causally non-separable processes resulting from the action of an adapter
$\Upsilon^{{\texttt{A}\texttt{FP}}}\in\Theta_{{\texttt{A}\texttt{FP}}}$ on a
causally ordered process. Employing this guess, we run the following program
to find such an admissible and free-preserving adapter; first, we sample a
random witness $G$ for causal non-separability. For this given witness, we run
the SDP

$\displaystyle\begin{split}\textbf{given}\ \ &\text{Witness of causal non-
separability}\ G\\\ \textbf{minimize}\ \ &\operatorname{tr}[(\Upsilon\star
W^{A\rightarrow B})G]\\\ \textbf{subject to}\ \
&\Upsilon^{{\texttt{A}\texttt{FP}}}\in\Theta_{\texttt{A}\texttt{FP}}\end{split}$
(127)

If the resulting $\Upsilon^{{\texttt{A}\texttt{FP}}}$ is such that
$\operatorname{tr}[(\Upsilon^{{\texttt{A}\texttt{FP}}}\star W^{A\rightarrow
B})G]<0$, then the resulting process matrix
$\Upsilon^{{\texttt{A}\texttt{FP}}}\star W^{A\rightarrow B}$ is causally non-
separable and we have indeed found an admissible and free-preserving adapter
that maps causally separable to causally non-separable process matrices. To
obtain better and more robust results, the resulting adapter of the above SDP
can be fed into a second SDP to optimize the respective witness $G$, i.e.,

$\displaystyle\begin{split}\textbf{given}\ \ &\text{Adapter}\
\Upsilon^{{\texttt{A}\texttt{FP}}}\in\Theta_{\texttt{A}\texttt{FP}}\\\
\textbf{minimize}\ \
&\operatorname{tr}[(\Upsilon^{{\texttt{A}\texttt{FP}}}\star W^{A\rightarrow
B})G]\\\ \textbf{subject to}\ \ &G\ \text{is proper witness of causal non-
separability.}\end{split}$ (128)

While running these two SDPs is not guaranteed to yield an adapter
$\Upsilon^{{\texttt{A}\texttt{FP}}}\in\Theta_{\texttt{A}\texttt{FP}}$ with the
desired properties, it actually provides adapters for which
$\operatorname{tr}[(\Upsilon^{{\texttt{A}\texttt{FP}}}\star W^{A\rightarrow
B})G]\approx-0.1661$ (as a comparison, for the best witness of
$W^{\text{OCB}}$, one obtains
$\operatorname{tr}(W^{\text{OCB}}G)\approx-0.1716$). While not necessarily the
best achievable value, $-0.1661$ is well-beyond any potential numerical
imprecision thresholds, implying that we indeed found an adapter with the
desired properties. Since it does not appear to possess a particularly nice
structure, we do not report it here explicitly, however, the code used to find
this adapter and a data file containing the matrices that correspond to the
adapter $\Upsilon^{{\texttt{A}}\texttt{FP}}$, the causally ordered, fully
signalling process matrix $W^{A\rightarrow B}$, and the causally non-separable
process matrix $\Upsilon^{{\texttt{A}\texttt{FP}}}\star W^{A\rightarrow B}$
can be found in the online repository [79].

##  Appendix D Definition of non-signalling adapters

In Sec. 3.2.4 we defined the set $\Theta_{\texttt{NS}}$ of non-signalling
adapters
$\Upsilon^{\texttt{NS}}_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}\in\mathcal{B}(\mathcal{H}_{\mathbbm{A}_{I}}\otimes\mathcal{H}_{\mathbbm{A}_{O}}\otimes\mathcal{H}_{\mathbbm{B}_{I}}\otimes\mathcal{H}_{\mathbbm{B}_{O}})$
as the set of positive semidefinite matrices that satisfy

$\displaystyle\begin{split}&\Upsilon^{\texttt{NS}}=L_{A}[\Upsilon^{\texttt{NS}}],\quad\Upsilon^{\texttt{NS}}=L_{B}[\Upsilon^{\texttt{NS}}],\quad\\\
\text{and}\
&\operatorname{tr}[\Upsilon^{\texttt{NS}}]=d_{A_{O}^{\prime}}d_{A_{I}}d_{A_{O}^{\prime}}d_{A_{I}}\,,\end{split}$
(129)

where
$L_{X}[\Upsilon^{\texttt{NS}}]=\Upsilon^{\texttt{NS}}-{}_{X_{O}}\Upsilon^{\texttt{NS}}+{}_{X_{O}X_{O}^{\prime}}\Upsilon^{\texttt{NS}}-{}_{X_{I}^{\prime}X_{O}X_{O}^{\prime}}\Upsilon^{\texttt{NS}}+{}_{X_{I}X_{I}^{\prime}X_{O}X_{O}^{\prime}}\Upsilon^{\texttt{NS}}$.
While positivity guarantees that adapters will lead to positive probabilities
even when applied to process matrices with additional degrees of freedom
(i.e., degrees of freedom the adapter does not act on), the above trace
conditions and the normalization of $\Upsilon^{\texttt{NS}}$ ensure that
$\Upsilon^{\texttt{NS}}$ does not allow for communication between Alice and
Bob, and that the resulting process matrix is properly normalized, i.e.,
$\operatorname{tr}[\Upsilon^{\texttt{NS}}\star
W]=d_{A_{O}^{\prime}}d_{B_{O}^{\prime}}$ for all $W\in\texttt{Proc}$ and all
$\Upsilon^{\texttt{NS}}\in\Theta_{\texttt{NS}}$.

Let us first show the former statement, namely that a non-signalling adapter
that satisfies Eq. (129) does not allow for communication between Alice and
Bob. To see this, consider a deterministic operation
$\Omega_{\mathbbm{A}_{I}A_{O}^{\prime}}\otimes\mathbbm{1}_{A_{O}}$ that Alice
can perform. Due to its causal ordering we have
${}_{A_{O}^{\prime}}\Omega_{\mathbbm{A}_{I}A_{O}^{\prime}}={}_{A_{O}^{\prime}A_{I}^{\prime}}\Omega_{\mathbbm{A}_{I}A_{O}^{\prime}}$
and
$\operatorname{tr}[\Omega_{\mathbbm{A}_{I}A_{O}^{\prime}}]=d_{A_{I}^{\prime}}$.
It is easy to see that with this, we have
$L_{A}[\Omega\otimes\mathbbm{1}_{A_{O}}]={}_{A_{I}A_{I}^{\prime}A_{O}A_{O}^{\prime}}[\Omega\otimes\mathbbm{1}_{A_{O}}]$.
We thus obtain

$\displaystyle\begin{split}\Upsilon^{\texttt{NS}}\star(\Omega\otimes\mathbbm{1}_{A_{O}})&=(L_{A}\otimes
L_{B})[\Upsilon^{\texttt{NS}}]\star(\Omega\otimes\mathbbm{1}_{A_{O}})=(L_{A}\otimes
L_{B})[\Upsilon^{\texttt{NS}}]\star L_{A}[\Omega]=\\\
&=\Upsilon^{\texttt{NS}}\star{}_{A_{I}A_{I}^{\prime}A_{O}A_{O}^{\prime}}\Omega=\frac{1}{d_{A_{I}}d_{A_{O}^{\prime}}}\operatorname{tr}_{\mathbbm{A}_{I}\mathbbm{A}_{O}}\Upsilon^{\texttt{NS}},\end{split}$
(130)

where we have dropped the labels for concise notation and we have used the
fact that the operators $L_{X}$ are self-dual projections. Since the last line
of the above equation is independent of the specific comb $\Omega$ Alice
employs, Bob’s local adapter is independent of Alice’s operations, implying
that she cannot signal to him via the non-signalling adapter
$\Upsilon^{\texttt{NS}}$. Running the same argument for operations on Bob’s
side then shows that a non-signalling adapter indeed does not allow for
signalling between the involved parties. This, then, implies that adapters
that satisfy the properties laid out in Def. 4 automatically satisfy the
requirements of Eqs. (49) and (50).

Finally, it would remain to show that non-signalling adapters indeed satisfy
$\operatorname{tr}(\Upsilon^{\texttt{NS}}_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}\star
W_{AB})=d_{A_{O}^{\prime}}d_{B_{O}^{\prime}}$, i.e., they transform process
matrices to correctly-normalized objects. This has already been done in the
main text. As we saw, the free adapters form a proper subset of all valid
adapters, i.e., $\Theta_{\texttt{NS}}\subset\Theta_{\texttt{A}}$, and all
admissible adapters map proper process matrices to proper process matrices,
i.e., they also preserve the correct normalization.

##  Appendix E Shared entanglement and local operations

In the main text, we claimed that
$\Theta_{\texttt{LOSE}}\subset\Theta_{\texttt{NS}}$, implying that adapters
from local operations and shared entanglement satisfy not only Requirement
R$3$, i.e., they do not allow for signalling between Alice and Bob, but also
Requirements R$1$ and R$2$, by mapping proper processes to proper processes
and free processes to free processes. We state this here in more detail as the
following Lemma:

######  Lemma 2.

An adapter of the form

$\displaystyle\Upsilon^{\textup{{LOSE}}}=\rho_{\widetilde{A}_{I}\widetilde{A}^{\prime}_{O}\widetilde{B}_{I}\widetilde{B}^{\prime}_{O}}\star\Lambda_{\widetilde{\mathbbm{A}}_{I}\mathbbm{A}_{I}}\star\Gamma_{\widetilde{A}^{\prime}\mathbbm{A}_{O}}\star\Lambda_{\widetilde{\mathbbm{B}}_{I}\mathbbm{B}_{I}}\star\Gamma_{\widetilde{B}^{\prime}\mathbbm{B}_{O}}\,,$
(131)

stemming from shared entanglement and local operations, maps proper process
matrices to proper processes and does not allow for signalling between Alice
and Bob.

###### Proof.

We start by proving the former. Since the link product of positive objects is
again a positive object, we see that for an adapter of the form of Eq. (131)
we have $\Upsilon^{\texttt{LOSE}}\geq 0$ and thus
$W^{\prime}_{A^{\prime}B^{\prime}}=\Upsilon^{\texttt{LOSE}}\star W_{AB}\geq 0$
for all proper process matrices $W_{AB}$. It remains to show that
$W^{\prime}_{A^{\prime}B^{\prime}}$ is a proper process matrix. This can be
done by checking its action on products $M_{A^{\prime}}\otimes M_{B^{\prime}}$
of CPTP maps. We have

$\displaystyle\begin{split}&W^{\prime}_{A^{\prime}B^{\prime}}\star
M_{A^{\prime}}\star
M_{B^{\prime}}=\rho_{\widetilde{A}_{I}\widetilde{A}^{\prime}_{O}\widetilde{B}_{I}\widetilde{B}^{\prime}_{O}}\star\Lambda_{\widetilde{\mathbbm{A}}_{I}\mathbbm{A}_{I}}\star\Gamma_{\widetilde{A}^{\prime}\mathbbm{A}_{O}}\star\Lambda_{\widetilde{\mathbbm{B}}_{I}\mathbbm{B}_{I}}\star\Gamma_{\widetilde{B}^{\prime}\mathbbm{B}_{O}}\star
W_{AB}\star M_{A^{\prime}}\star M_{B^{\prime}}\\\ &=:N_{AB}\star
W_{AB}\end{split}$ (132)

It is easy to see that the map $N_{AB}$ satisfies
$\operatorname{tr}_{A_{O}}N_{AB}=\mathbbm{1}_{A_{I}}\otimes N_{B}$ and
$\operatorname{tr}_{B_{O}}N_{AB}=\mathbbm{1}_{B_{I}}\otimes N_{A}$ with CPTP
maps $N_{A}$ and $N_{B}$, implying that $N_{AB}$ is non-signalling between
Alice and Bob.

Figure 14: Free adapters from shared entanglement and local operations. For
better orientation, here, we reproduce Fig. 10 from the main text.

Figure 15: No signalling for adapters. Given the adapter
$\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}$, the
most general thing Alice could do to send information to Bob would be to
perform a causally ordered comb $\Omega_{\mathbbm{A}_{I}\mathbbm{A}_{O}}$. If
the adapter is non-signalling, then the resulting comb on Bobs side (defined
on $B_{I},B_{I}^{\prime},B_{O}^{\prime}$, and $B_{O}$) must be independent of
$\Omega_{\mathbbm{A}_{I}\mathbbm{A}_{O}}$.

Concretely, we have (see Fig. 15 for better orientation):

$\displaystyle\begin{split}&\operatorname{tr}_{A_{O}}N_{AB}=\rho_{\widetilde{A}_{I}\widetilde{A}^{\prime}_{O}\widetilde{B}_{I}\widetilde{B}^{\prime}_{O}}\star\Lambda_{\widetilde{\mathbbm{A}}_{I}\mathbbm{A}_{I}}\star\operatorname{tr}_{A_{O}}\Gamma_{\widetilde{A}^{\prime}\mathbbm{A}_{O}}\star\Lambda_{\widetilde{\mathbbm{B}}_{I}\mathbbm{B}_{I}}\star\Gamma_{\widetilde{B}^{\prime}\mathbbm{B}_{O}}\star
M_{A^{\prime}}\star M_{B^{\prime}}\\\
&=\operatorname{tr}_{\widetilde{A}^{\prime}_{O}}\rho_{\widetilde{A}_{I}\widetilde{A}^{\prime}_{O}\widetilde{B}_{I}\widetilde{B}^{\prime}_{O}}\star\operatorname{tr}_{\widetilde{A}_{I}^{\prime}}\Lambda_{\widetilde{\mathbbm{A}}_{I}\mathbbm{A}_{I}}\star\Lambda_{\widetilde{\mathbbm{B}}_{I}\mathbbm{B}_{I}}\star\Gamma_{\widetilde{B}^{\prime}\mathbbm{B}_{O}}\star\operatorname{tr}_{A_{O}^{\prime}}M_{A^{\prime}}\star
M_{B^{\prime}}\\\
&=\rho_{\widetilde{A}_{I}\widetilde{B}_{I}\widetilde{B}^{\prime}_{O}}\star\operatorname{tr}_{\widetilde{A}_{I}^{\prime}A_{I}^{\prime}}\Lambda_{\widetilde{\mathbbm{A}}_{I}\mathbbm{A}_{I}}\star\Lambda_{\widetilde{\mathbbm{B}}_{I}\mathbbm{B}_{I}}\star\Gamma_{\widetilde{B}^{\prime}\mathbbm{B}_{O}}\star
M_{B^{\prime}}\\\
&=\mathbbm{1}_{A_{I}}\star\rho_{\widetilde{B}_{I}\widetilde{B}_{O}^{\prime}}\star\Lambda_{\widetilde{\mathbbm{B}}_{I}\mathbbm{B}_{I}}\star\Gamma_{\widetilde{B}^{\prime}\mathbbm{B}_{O}}\star
M_{B^{\prime}}\\\ &=:\mathbbm{1}_{A_{I}}\otimes N_{B}\end{split}$ (133)

where we have used
$\operatorname{tr}_{A_{O}}\Gamma_{\widetilde{A}^{\prime}\mathbbm{A}_{O}}=\mathbbm{1}_{\widetilde{A}^{\prime}A_{O}^{\prime}}$,
$\operatorname{tr}_{A_{O}^{\prime}}M_{A^{\prime}}=\mathbbm{1}_{A_{I}^{\prime}}$,
$\operatorname{tr}_{\widetilde{A}_{I}^{\prime}A_{I}^{\prime}}\Lambda_{\widetilde{\mathbbm{A}}_{I}\mathbbm{A}_{I}}$,
and the fact that $\mathbbm{1}_{X}$ is the Choi matrix of
$\operatorname{tr}_{X}$. In a similar vein, it is easy to see that $N_{B}$
satisfies $\operatorname{tr}_{B_{O}}N_{B}=\mathbbm{1}_{B_{I}}$, i.e., it is
CPTP. The corresponding relations for $\operatorname{tr}_{B_{O}}N_{AB}$ follow
analogously. $N_{AB}$ is thus a non-signalling map. Invoking the decomposition
(47) of general non-signalling maps in terms of products of CPTP maps implies
that $\Upsilon^{\texttt{LOSE}}$ maps non-signalling maps on ‘the primed
degrees of freedom’ to non-signalling maps ‘on the unprimed degrees of
freedom’. As we have seen in the main text, this implies that it maps proper
process matrices to proper process matrices, which concludes the first part of
the proof.

To show that the above $\Upsilon^{\texttt{LOSE}}$ does not allow for
signalling between Alice and Bob, we first note that, in the above reasoning,
we have already shown that $\Upsilon^{\texttt{LOSE}}$ maps all non-signalling
maps to non-signalling maps (i.e.,
$\Upsilon^{\texttt{LOSE}}\in\Theta_{\texttt{A}\texttt{FP}}$). However, as
discussed in detail in Sec. 3.2.4, this is not sufficient to guarantee that
the adapter does not enable any signalling between Alice and Bob. In addition,
we have to show that, given the adapter $\Upsilon^{\texttt{LOSE}}$, no matter
what Alice ‘inserts’ into it, Bob locally ‘sees’ the same comb (and vice
versa). In the language of Def. 4, where we defined non-signalling adapters,
this means that $\Upsilon^{\texttt{LOSE}}$ satisfies
$L_{A}[\Upsilon^{\texttt{LOSE}}]=L_{B}[\Upsilon^{\texttt{LOSE}}]=\Upsilon^{\texttt{LOSE}}$.
Here, for concreteness, we will not use the projectors $L_{A}$ and $L_{B}$ to
prove the non-signalling conditions, but rather show them directly (yet
equivalently) by proving that Alice cannot send signals to Bob by performing
deterministic operations on her side of the adapter.

In detail, this means that
$\Upsilon^{\texttt{LOSE}}\star\Omega_{\mathbbm{A}_{I}\mathbbm{A}_{O}}$ is
independent of any comb $\Omega_{\mathbbm{A}_{I}\mathbbm{A}_{O}}$ with
ordering $A_{I}\prec A_{I}^{\prime}\prec A_{O}^{\prime}\prec A_{O}$ that Alice
could ‘insert’ into the adapter (see Fig. 15).

This can be checked by direct insertion into Eq. (131) and using the causality
constraints on
$\Omega_{\mathbbm{A}_{I}\mathbbm{A}_{O}}=\mathbbm{1}_{A_{O}}\otimes\Omega_{\mathbbm{A}_{I}A_{O}^{\prime}}$:

$\displaystyle\begin{split}&\Upsilon^{\texttt{LOSE}}\star\Omega_{\mathbbm{A}_{I}\mathbbm{A}_{O}}=\mathbbm{1}_{A_{O}}\star\Gamma_{\widetilde{A}^{\prime}\mathbbm{A}_{O}}\star\Omega_{\mathbbm{A}_{I}A_{O}^{\prime}}\star\rho_{\widetilde{A}_{I}\widetilde{A}^{\prime}_{O}\widetilde{B}_{I}\widetilde{B}^{\prime}_{O}}\star\Lambda_{\widetilde{\mathbbm{A}}_{I}\mathbbm{A}_{I}}\star\Lambda_{\widetilde{\mathbbm{B}}_{I}\mathbbm{B}_{I}}\star\Gamma_{\widetilde{B}^{\prime}\mathbbm{B}_{O}}\\\
&=\mathbbm{1}_{\widetilde{A}_{O}^{\prime}A_{O}^{\prime}\widetilde{A}_{I}^{\prime}}\star\Omega_{\mathbbm{A}_{I}A_{O}^{\prime}}\star\rho_{\widetilde{A}_{I}\widetilde{A}^{\prime}_{O}\widetilde{B}_{I}\widetilde{B}^{\prime}_{O}}\star\Lambda_{\widetilde{\mathbbm{A}}_{I}\mathbbm{A}_{I}}\star\Lambda_{\widetilde{\mathbbm{B}}_{I}\mathbbm{B}_{I}}\star\Gamma_{\widetilde{B}^{\prime}\mathbbm{B}_{O}}\\\
&=\rho_{\widetilde{B}_{I}\widetilde{B}^{\prime}_{O}}\star\Lambda_{\widetilde{\mathbbm{B}}_{I}\mathbbm{B}_{I}}\star\Gamma_{\widetilde{B}^{\prime}\mathbbm{B}_{O}}\,,\end{split}$
(134)

which is independent of Alice’s operations. As for the previous derivation,
here, we have alternatingly used the properties of CPTP maps and causally
ordered combs, in particular
$\operatorname{tr}_{A_{O}}\Gamma_{\widetilde{A}^{\prime}\mathbbm{A}_{O}}=\mathbbm{1}_{\widetilde{A}_{O}^{\prime}A_{O}^{\prime}\widetilde{A}_{I}^{\prime}}$,
$\operatorname{tr}_{A_{I}^{\prime}\widetilde{A}_{I}^{\prime}}\Lambda_{\widetilde{\mathbbm{A}}_{I}\mathbbm{A}_{I}}=\mathbbm{1}_{\widetilde{A}_{I}A_{I}}$,
and
$\operatorname{tr}_{A_{O}^{\prime}}\Omega_{\mathbbm{A}_{I}A_{O}^{\prime}}=\mathbbm{1}_{A_{I}^{\prime}}\otimes\Omega_{A_{I}}$.
Naturally, the same independence can be shown with respect to Bob’s
operations, implying that
$\Upsilon^{\texttt{LOSE}}_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}$
does not enable any signalling between Alice and Bob. Consequently, we have
$\Theta_{\texttt{LOSE}}\subseteq\Theta_{\texttt{NS}}$. The fact that the
inclusion is strict is proven in the main text. ∎

##  Appendix F Equivalence of $\Theta_{\texttt{CA}}$ and
$\Theta_{\texttt{NS}}$

Here, we provide the prove of Prop. 1, where we stated that the set
$\Theta_{\texttt{CA}}$ of completely admissible adapters coincides with the
set $\Theta_{\texttt{NS}}$ of non-signalling adapters. As laid out in Sec.
3.3, a completely admissible adapter
$\Upsilon^{\texttt{CA}}_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}$
must – besides being a positive semidefinite matrix – map any non-signalling
map $\bar{M}_{A^{\prime}\bar{A}B^{\prime}\bar{B}}$ with
$A_{I}^{\prime}\bar{A}_{I}\not\rightarrow B_{O}^{\prime}\bar{B}_{O}$ and
$B_{I}^{\prime}\bar{B}_{I}\not\rightarrow A_{O}^{\prime}\bar{A}_{O}$ to a non-
signalling map
$\bar{N}_{A\bar{A}B\bar{B}}=\Upsilon^{\texttt{CA}}_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}\star\bar{M}_{A^{\prime}\bar{A}B^{\prime}\bar{B}}$,
where $\bar{N}_{A\bar{A}B\bar{B}}$ is non-signalling
$A_{I}\bar{A}_{I}\not\rightarrow B_{O}\bar{B}_{O}$ and
$B_{I}\bar{B}_{I}\not\rightarrow A_{O}\bar{A}_{O}$.

As was the case for admissible adapters, complete admissibility is equivalent
to mapping any non-signalling map of the form $M_{A^{\prime}\bar{A}}\otimes
M_{B^{\prime}\bar{B}}$ – where $M_{X^{\prime}\bar{X}}$ is an arbitrary CPTP
map $X_{I}^{\prime}\bar{X}_{I}\rightarrow X_{O}^{\prime}\bar{X}_{O}$ – to a
non-signalling map $N_{A\bar{A}B\bar{B}}$ (see Fig. 16).

Figure 16: Completely admissible adapters. An adapter $\Upsilon^{\texttt{CA}}$
is completely admissible if it maps any non-signalling map to a non-signalling
map, even when only acting on parts of the map. As for the case of admissible
adapters, this is equivalent to mapping arbitrary tensor products of CPTP maps
to a non-signalling map, i.e., if the adapter in the figure is completely
admissible, then $N_{A\bar{A}B\bar{B}}$ is non-signalling
$A_{I}\bar{A}_{I}\not\rightarrow B_{O}\bar{B}_{O}$ and
$B_{I}\bar{B}_{I}\not\rightarrow A_{O}\bar{A}_{O}$ for all CPTP maps
$M_{A^{\prime}\bar{A}}$ and $M_{B^{\prime}\bar{B}}$.

With this, one could derive the linear constraints on the set
$\Theta_{\texttt{CA}}$ of completely admissible adapters in terms of a
projector $L_{CA}$ and show that its properties correspond to those of the
projectors that define $\Theta_{\texttt{NS}}$ [given in Eqs. (54) and (55)].
However, here, we take a more explicit route and directly show that both
$\Theta_{\texttt{NS}}\subseteq\Theta_{\texttt{CA}}$ and
$\Theta_{\texttt{CA}}\subseteq\Theta_{\texttt{NS}}$ hold.

We start by showing $\Theta_{\texttt{NS}}\subseteq\Theta_{\texttt{CA}}$.
Recall that for any
$\Upsilon^{\texttt{NS}}_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}\in\Theta_{\texttt{NS}}$
we have
${}_{B_{O}}\Upsilon^{\texttt{NS}}={}_{B_{O}B_{O}^{\prime}}\Upsilon^{\texttt{NS}}$
and
${}_{B_{O}B_{O}^{\prime}B_{I}^{\prime}}\Upsilon^{\texttt{NS}}={}_{B_{O}B_{O}^{\prime}B_{I}^{\prime}B_{I}}\Upsilon^{\texttt{NS}}$.
With this, we can show that
$N_{A\bar{A}B\bar{B}}:=\Upsilon^{\texttt{NS}}\star(M_{A^{\prime}\bar{A}}\otimes
M_{B^{\prime}\bar{B}})$ is non-signalling $B_{I}\bar{B}_{I}\not\rightarrow
A_{O}\bar{A}_{O}$, i.e.,
${}_{B_{O}\bar{B}_{O}}N_{A\bar{A}B\bar{B}}={}_{B_{O}\bar{B}_{O}B_{I}\bar{B}_{I}}N_{A\bar{A}B\bar{B}}$.
Concretely, we have

$\displaystyle\begin{split}&{}_{B_{O}\bar{B}_{O}}N_{A\bar{A}B\bar{B}}\\\
&={}_{B_{O}\bar{B}_{O}}(\Upsilon^{\texttt{NS}}_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}\star
M_{A^{\prime}\bar{A}}\otimes
M_{B^{\prime}\bar{B}})={}_{B_{O}B_{O}^{\prime}}\Upsilon^{\texttt{NS}}_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}\star
M_{A^{\prime}\bar{A}}\star{}_{\bar{B}_{O}}M_{B^{\prime}\bar{B}}\\\
&={}_{B_{O}}\Upsilon^{\texttt{NS}}_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}\star
M_{A^{\prime}\bar{A}}\star{}_{\bar{B}_{O}B_{O}^{\prime}B_{I}^{\prime}\bar{B}_{I}}M_{B^{\prime}\bar{B}}={}_{B_{O}B_{O}^{\prime}B_{I}^{\prime}B_{I}}\Upsilon^{\texttt{NS}}_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}\star
M_{A^{\prime}\bar{A}}\star{}_{\bar{B}_{O}\bar{B}_{I}}M_{B^{\prime}\bar{B}}\\\
&={}_{B_{O}\bar{B}_{O}B_{I}\bar{B}_{I}}(\Upsilon^{\texttt{NS}}_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}\star
M_{A^{\prime}\bar{A}}\star{}_{\bar{B}_{O}B_{O}^{\prime}}M_{B^{\prime}\bar{B}})\,\end{split}$
(135)

where we have both used the fact that the operators
${}_{X}\,\begin{picture}(-1.0,1.0)(-1.0,-3.0)\circle*{2.5}\end{picture}\ $ can
be moved around freely in the link product and the property
${}_{\bar{B}_{O}B^{\prime}_{O}}M_{B^{\prime}\bar{B}}={}_{\bar{B}_{O}B^{\prime}_{O}\bar{B}_{I}B_{I}^{\prime}}M_{B^{\prime}\bar{B}}$
of CPTP maps. In the same vein (and using
${}_{X}({}_{X}\,\begin{picture}(-1.0,1.0)(-1.0,-3.0)\circle*{2.5}\end{picture}\
)={}_{X}\,\begin{picture}(-1.0,1.0)(-1.0,-3.0)\circle*{2.5}\end{picture}\ $),
we see that

$\displaystyle{}_{B_{O}\bar{B}_{O}B_{I}\bar{B}_{I}}N_{A\bar{A}B\bar{B}}={}_{B_{O}\bar{B}_{O}B_{I}\bar{B}_{I}}(\Upsilon^{\texttt{NS}}_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}\star
M_{A^{\prime}\bar{A}}\star{}_{\bar{B}_{O}B_{O}^{\prime}}M_{B^{\prime}\bar{B}})\,.$
(136)

Comparison of Eqs. (135) and (136) implies
${}_{B_{O}\bar{B}_{O}}N_{A\bar{A}B\bar{B}}={}_{B_{O}\bar{B}_{O}B_{I}\bar{B}_{I}}N_{A\bar{A}B\bar{B}}$.
The fact that $M_{A\bar{A}B\bar{B}}$ is non-signalling from $A_{I}\bar{A}_{I}$
to $B_{O}\bar{B}_{O}$ follows in the same way. Non-signalling adapters thus
map non-signalling maps to non-signalling maps, even when only acting on a
part of them, implying $\Theta^{\texttt{NS}}\subseteq\Theta^{\texttt{CA}}$.

To prove $\Theta^{\texttt{NS}}\supseteq\Theta^{\texttt{CA}}$, let us consider
two particular CPTP maps on Alice’s and Bob’s side, respectively, namely a
swap $\bar{A}_{I}\leftrightarrow
A_{O}^{\prime},A^{\prime}_{I}\leftrightarrow\bar{A}_{O}$ on Alice’s side, and
a CPTP map
$M_{B^{\prime}\bar{B}}=L_{B^{\prime}}\otimes\Phi^{+}_{\bar{B}_{I}\bar{B}_{O}}$
(where $L_{B^{\prime}}$ is CPTP and $\Phi^{+}_{\bar{B}_{I}\bar{B}_{O}}$
corresponds to an identity channel from $\bar{B}_{I}$ to $\bar{B}_{O}$) on
Bob’s side (see Fig. 17).

Figure 17: Complete admissibility and non-signalling. Taking the swap and a
product map as the two maps on Alice’s and Bob’s side, respectively, if the
adapter $\Upsilon^{\texttt{CA}}$ is completely admissible, then the resulting
channel should be non-signalling (here, this requirement is shown for non-
signalling from Alice to Bob).

Since the swap on Alice’s side is merely a relabelling, we drop the usage of
bars on the labels for Alice from here on. Demanding that, for this particular
choice of CPTP maps, the resulting channel is non-signalling
$B_{I}\bar{B}_{I}\not\rightarrow A_{O}A_{I}^{\prime}$ is equivalent to the
requirement (see Fig. 17)

$\displaystyle\operatorname{tr}_{B_{O}\bar{B}_{O}}(\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}^{\texttt{CA}}\star
L_{B^{\prime}})=\mathbbm{1}_{B_{I}\bar{B}_{I}}\otimes
C_{\mathbbm{A}_{I}\mathbbm{A}_{O}}\,,$ (137)

where $C_{\mathbbm{A}_{I}\mathbbm{A}_{O}}$ is a causally ordered comb (since
$\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}^{\texttt{CA}}$
is an admissible adapter), but its concrete form is not important for the
following argument. We now show that $C_{\mathbbm{A}_{I}\mathbbm{A}_{O}}$ must
be independent of $L_{B}^{\prime}$ if
$\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}^{\texttt{CA}}$
is completely admissible. To see this, let us assume the contrary, that there
exist two different CPTP maps $L_{B^{\prime}}$ and $K_{B^{\prime}}$ such that

$\displaystyle\operatorname{tr}_{B_{O}\bar{B}_{O}}(\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}^{\texttt{CA}}\star
L_{B^{\prime}}):=\mathbbm{1}_{B_{I}\bar{B}_{I}}\otimes
C_{\mathbbm{A}_{I}\mathbbm{A}_{O}}\neq\mathbbm{1}_{B_{I}\bar{B}_{I}}\otimes
C^{\\#}_{\mathbbm{A}_{I}\mathbbm{A}_{O}}:=\operatorname{tr}_{B_{O}\bar{B}_{O}}(\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}^{\texttt{CA}}\star
K_{B^{\prime}})\,.$ (138)

Then, Bob can perform a CPTP map $H_{B^{\prime}\bar{B}}$ that classically
controls the CPTP maps $L_{B^{\prime}}$ (for measurement outcome $0$ on
$\bar{B}_{I}$) and $K_{B^{\prime}}$ (for measurement outcome 1 on
$\bar{B}_{I}$), i.e.,

$\displaystyle
H_{B^{\prime}\bar{B}^{\prime}}=\ket{00\vphantom{00}}\\!\bra{00\vphantom{00}}_{\bar{B}_{I}\bar{B}_{O}}\otimes
L_{B^{\prime}}+\ket{11\vphantom{11}}\\!\bra{11\vphantom{11}}_{\bar{B}_{I}\bar{B}_{O}}\otimes
K_{B^{\prime}}\,,$ (139)

and we assume that the degrees of freedom labeled with bars correspond to
qubits. Now, contracting this map with the adapter
$\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}^{\texttt{CA}}$
(and tracing out the final degrees of freedom $B_{O}$ and $\bar{B}_{O}$)
yields [cf. (137)]

$\displaystyle\operatorname{tr}_{B_{O}\bar{B}_{O}}(\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}^{\texttt{CA}})\star
H_{B^{\prime}\bar{B}}=\ket{0\vphantom{0}}\\!\bra{0\vphantom{0}}_{\bar{B}_{I}}\otimes\mathbbm{1}_{B_{I}}\otimes
C_{\mathbbm{A}_{I}\mathbbm{A}_{O}}+\ket{1\vphantom{1}}\\!\bra{1\vphantom{1}}_{\bar{B}_{I}}\otimes\mathbbm{1}_{B_{I}}\otimes
C_{\mathbbm{A}_{I}\mathbbm{A}_{O}}^{\\#}\,.$ (140)

However, since $H_{B^{\prime}\bar{B}}$ is itself a CPTP map, the above must –
due to complete admissibility of
$\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}^{\texttt{CA}}$
– also factorize as

$\displaystyle\operatorname{tr}_{B_{O}\bar{B}_{O}}(\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}^{\texttt{CA}})\star
H_{B^{\prime}\bar{B}}=\mathbbm{1}_{B_{I}\bar{B}_{I}}\otimes
D_{\mathbbm{A}_{I}\mathbbm{A}_{O}}\,,$ (141)

for some causally ordered comb $D_{\mathbbm{A}_{I}\mathbbm{A}_{O}}$. This, in
turn, shows that
$C_{\mathbbm{A}_{I}\mathbbm{A}_{O}}=C_{\mathbbm{A}_{I}\mathbbm{A}_{O}}^{\\#}$
holds, i.e., Alice’s part of the adapter is independent of the CPTP map that
Bob performs (if Bob’s final degree of freedom is discarded). The same
argument can be employed to show that Bob’s part of the adapter
$\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}^{\texttt{CA}}$
is independent of the CPTP map that Alice performs (if Alice’s final degrees
of freedom are discarded). In fact, this is already sufficient to conclude
that $\Theta^{\texttt{NS}}\supseteq\Theta^{\texttt{CA}}$, since it is exactly
this property that we used to derive the non-signalling conditions of
$\Theta_{\texttt{NS}}$. Combining this with the fact that
$\Theta^{\texttt{NS}}\subseteq\Theta^{\texttt{CA}}$ (shown above), we see that
the requirements on completely admissible adapters are equivalent to those on
non-signalling adapters and we have
$\Theta_{\texttt{NS}}=\Theta_{\texttt{CA}}$.

##  Appendix G Convexity of $\mathcal{R}_{s}(W)$

In order to prove the convexity of $\mathcal{R}_{s}(W)$, assume that for a
given $W=pW_{1}+(1-p)W_{2}$, we have found the process matrices
$T^{*}\in\texttt{Proc}$ and $C^{*}\in\texttt{Free}$ that lead to the minimal
value in Eq. (69). Then,

$\displaystyle W=(1+\mathcal{R}_{s}(W))C^{*}-\mathcal{R}_{s}(W)T^{*}.$ (142)

Importantly, any other such pseudo-mixture representation
$W=(1+\mathcal{R}^{\prime})C^{\prime}-\mathcal{R}^{\prime}T^{\prime}$, where
$C^{\prime}\in\texttt{Free}$ and $T^{\prime}\in\texttt{Proc}$ satisfies (by
definition) $\mathcal{R}^{\prime}\geq\mathcal{R}_{s}(W)$.

Let $C_{1}^{*},C_{2}^{*},T_{1}^{*},$ and $T_{2}^{*}$ be the corresponding
process matrices for $W_{1}$ and $W_{2}$, i.e.,
$W_{1}=(1+\mathcal{R}_{s}(W_{1}))C_{1}^{*}-\mathcal{R}_{s}(W_{1})T_{1}^{*}$
and
$W_{2}=(1+\mathcal{R}_{s}(W_{2}))C_{2}^{*}-\mathcal{R}_{s}(W_{2})T_{2}^{*}$.
For convenience, we will set $\mathcal{R}_{i}:=\mathcal{R}_{s}(W_{i})$.
Consequently, we have

$\displaystyle
W=p(1+\mathcal{R}_{1})C_{1}^{*}+(1-p)(1+\mathcal{R}_{2})C_{2}^{*}-p\mathcal{R}_{1}T_{1}^{*}-(1-p)\mathcal{R}_{2}T_{2}^{*}.$
(143)

Up to normalization, the first two terms together form the convex combination
of two free processes, while the second two terms form a convex combination of
proper process matrices. Thus, we can express $W$ as

$\displaystyle
W=[p(1+\mathcal{R}_{1})+(1-p)(1+\mathcal{R}_{2})]C^{\prime}-[p\mathcal{R}_{1}+(1-p)\mathcal{R}_{2}]T^{\prime},$
(144)

where $T^{\prime}\in\texttt{Proc}$ and $C^{\prime}\in\texttt{Free}$. This
implies

$\displaystyle\mathcal{R}_{s}(pW_{1}+(1-p)W_{2})\leq
p\mathcal{R}_{s}(W_{1})+(1-p)\mathcal{R}_{s}(W_{2}),$ (145)

i.e., $\mathcal{R}_{s}(W)$ is a convex function.

##  Appendix H Three-party processes from $W^{A\rightarrow B\rightarrow C}$

In the main text we showed that all processes $W^{A\prec B}$ that satisfied
the dimensional constraints we laid out can be obtained from the fully
signalling process matrix $W^{A\rightarrow B}$ by means of free adapters.
Here, we show that the analogous statement holds for processes of the form
$W^{A\prec B\prec C}$, which can all be obtained from the fully signalling
three-party process matrix

$\displaystyle W^{A\rightarrow B\rightarrow
C}=\rho_{A_{I}}\otimes\Phi^{+}_{A_{O}B_{I}}\otimes\Phi^{+}_{B_{O}C_{I}}\otimes\mathbbm{1}_{C_{I}}$
(146)

by means of free adapters.

Figure 18: General $W^{A\prec B\prec C}$ from $W^{A\rightarrow B\rightarrow
C}$. By performing the correct local adapters and sharing an entangled state,
Alice and Bob can implement the representation of general three-step combs.
For convenience, we have relabelled the spaces in the resulting comb (bottom
panel).

Figure 19: Conjectured unreachable comb. Starting from $W^{A\rightarrow
B\rightarrow C\rightarrow D}$, we conjecture that the above comb, which
basically corresponds to an identity channel from $A_{O}$ to $C_{I}$ and an
identity channel from $B_{O}$ to $C_{I}$, cannot be reached by free adapters.

As for the two-party case, let us motivate this statement graphically (see
Fig. 19 for a graphical representation). Any causally ordered comb $W^{A\prec
B\prec C}$ can be represented as a concatenation of an initial system-
environment state $\Psi_{A_{I}E_{1}}$ and two CPTP mappings
$A_{O}E_{1}\rightarrow B_{I}E_{2}$ and $B_{O}E_{2}\rightarrow C_{I}$ with
corresponding Choi matrices $\Lambda_{E_{1}A_{O}E_{2}B_{I}}$ and
$\Gamma_{E_{2}B_{O}C_{I}}$. It is then easy to see that, by only using shared
entantglement and local operations, starting from $W^{A\rightarrow
B\rightarrow C}$ Alice and Bob can implement any such $\Psi,\Lambda$ and
$\Gamma$, and thus reach all $W^{A\prec B\prec C}$ (see Fig. 19). In
particular, let $\tau_{E_{O}E_{1}}=\Psi_{E_{O}E_{1}}\cong\Psi_{A_{I}E_{1}}$ be
the initially shared entangled state, and let
$\xi_{A_{I}E_{O}A_{I}^{\prime}}=\mathbbm{1}_{A_{I}}\otimes\Phi^{+}_{E_{O}A_{I}^{\prime}}$
be the Choi matrix of a local map in Alice’s laboratory that traces out the
system on $A_{I}$ and relabels $E_{O}\mapsto A_{I}^{\prime}$. Then
$\rho_{A_{I}}\star\tau_{E_{O}E_{1}}\star\xi_{A_{I}E_{O}A_{I}^{\prime}}=\Psi_{A_{I}^{\prime}E_{1}}\cong\Psi_{A_{I}E_{1}}$.
Since the degree of freedom $E_{1}$ is shared with Bob’s laboratory, it is
then sufficient for Bob to locally perform CPTP maps with corresponding Choi
matrices $\Lambda_{B_{I}E_{1}E_{2}B_{I}^{\prime}}$ and
$\Gamma_{E_{2}B_{O}^{\prime}B_{O}}$. The identity maps in the fully signalling
comb $W^{A\rightarrow B\rightarrow C}$ then lead to a relabelling
$B_{I}\mapsto A_{O}$ and $B_{O}\mapsto C_{I}$. Finally then, identifying
$A_{I}^{\prime}$ with $A_{I}$, $B_{I}^{\prime}$ with $B_{I}$ and
$B_{O}^{\prime}$ with $B_{O}$, we find that

$\displaystyle\begin{split}W^{A\rightarrow B\rightarrow
C}\star\Upsilon&=(\rho_{A_{I}}\otimes\Phi^{+}_{A_{O}B_{I}}\otimes\Phi^{+}_{B_{O}C_{I}}\otimes\mathbbm{1}_{C_{I}})\star(\tau_{E_{O}E_{1}}\star\xi_{A_{I}E_{O}A_{I}^{\prime}}\star\Lambda_{B_{I}E_{1}E_{2}B_{I}^{\prime}}\star\Gamma_{E_{2}B_{O}^{\prime}B_{O}})\\\
&\cong\Psi_{A_{I}E_{1}}\star\Lambda_{E_{1}A_{O}E_{2}B_{I}}\star\Gamma_{E_{2}B_{O}C_{I}}\,,\end{split}$
(147)

and thus all combs $W^{A\prec B\prec C}$ can be reached from $W^{A\rightarrow
B\rightarrow C}$ by means of free adapters.

While the above derivation works for the case of three parties, it cannot be
employed in the four-party scenario, where we conjecture the existence of
ordered processes that cannot be reached from $W^{A\rightarrow B\rightarrow
C\rightarrow D}$ by means of free adapters. In particular, we expect the
following process to not be reachable (see Fig. 19):

$\displaystyle
W_{4\text{parties}}=\Psi_{A_{I}}\otimes\Phi^{+}_{A_{O}D_{I}}\otimes\eta_{B_{I}}\otimes\Phi^{+}_{B_{O}C_{I}}\otimes\mathbbm{1}_{C_{O}D_{O}}$
(148)

This process consists of an identity channel between Alice and Dave, and
identity channel between Bob and Alice, and either states that are fed in, or
degrees that are traced out on the remaining degrees of freedom. The reason
why we conjecture that this process cannot be reached from $W^{A\rightarrow
B\rightarrow C\rightarrow D}$ by means of an adapter that does not enable any
communication between the parties is as follows: If an adapter $\Upsilon$ does
not enable direct communication between the respective parties, all signalling
between Alice and Dave, as well as the signalling between Bob and Charlie must
have come from $W^{A\rightarrow B\rightarrow C\rightarrow D}$. Consequently,
signals that go from Alice to Dave must have gone via Bob (since there is no
direct channel between Alice and Dave in $W^{A\rightarrow B\rightarrow
C\rightarrow D}$). However, Bob has an identity channel to Charlie, but, at
the same time, needs to feed forward signals from Alice to Dave, so that the
resulting $W_{4\text{parties}}$ can contain an identity channel between Alice
and Dave. This, then, should not be possible, since Bob cannot, at the same
time, send his information undisturbed to Charlie, but concurrently also
perfectly feed Alice’s signals forward towards Charlie.

While somewhat intuitive, these arguments are not sufficient to exclude the
possibility to obtain $W_{4\text{parties}}$ from $W^{A\rightarrow B\rightarrow
C\rightarrow D}$ by means of free adapters. As we have seen, there are free
adapters that cannot be obtained from shared entanglement and local
operations. They thus require internal signalling for their implementation,
however, this signalling is such that it cannot be employed by the respective
parties to signal to each other. In principle, though, this signalling might
be activated by concatenation with $W^{A\rightarrow B\rightarrow C\rightarrow
D}$. Nonetheless, we conjecture that $W_{4\text{parties}}$ cannot be obtained
from $W^{A\rightarrow B\rightarrow C\rightarrow D}$ by means of free adapters.

##  Appendix I Signalling robustness of multipartite causally ordered
processes

Here, we provide proofs and generalizations of the results on the bounds on
the signalling robustness in the multi-party scenario mentioned in the main
text. In Sec. 5.4 we claimed the following result:

Proposition 5. Let $\textup{{Proc}}_{1:N}$ be the set of all causally ordered
processes on $N$ parties with causal order $X^{(1)}\prec
X^{(2)}\prec\cdots\prec X^{(N)}$. For any $W\in\textup{{Proc}}_{1:N}$ we have

$\displaystyle\mathcal{R}_{s}(W)\leq\prod_{i=1}^{N-1}d_{X^{(i)}_{O}}^{2}-1:=d_{\bar{O}}^{2}-1,$
(149)

where $d_{X^{(i)}_{O}}$ is the output dimension of party $X^{(i)}_{O}$.

###### Proof.

The proof follows a similar line as that of Thm. B3 in Ref. [51]. First,
consider the quantum channel
$\widetilde{\Lambda}[\rho]=\operatorname{tr}(\rho)\mathbbm{1}/d_{X}$, where
$d$ is the dimension of the space $X$ on which $\rho$ is defined. This channel
can be written as
$\widetilde{\Lambda}[\rho]=\frac{1}{d^{2}}\sum_{j=0}^{d^{2}-1}U_{j}\rho
U_{j}^{\dagger}$, where the $d^{2}$ unitary matrices $\\{U_{j}\\}$ form an
orthogonal basis of the space spanned by unitary matrices and
$U_{0}=\mathbbm{1}$. Using this decomposition of $\widetilde{\Lambda}$, we can
write its action as

$\displaystyle\widetilde{\Lambda}[\rho]=\frac{1}{d^{2}}\rho+\frac{1}{d^{2}}\sum_{j=1}^{d^{2}-1}U_{j}\rho
U_{j}^{\dagger}=:(\frac{1}{d^{2}}\mathcal{I}+\frac{d^{2}-1}{d^{2}}\widetilde{\Gamma})[\rho]\,,$
(150)

where $\widetilde{\Gamma}$ is a CPTP map and $\mathcal{I}$ is the identity
map. The map $\widetilde{\Lambda}$ traces out the state it acts on and
replaces it by a maximally mixed state. Consequently, applying it to all
output spaces $X_{O}^{(i)}$ (except the last one) of
$W\in\textup{{Proc}}_{1:N}$ yields a process of the form
$\rho_{X_{I}}\otimes\mathbbm{1}_{X_{O}}$. In detail, due to its causal
ordering, we have $W=W_{X_{I}^{(1)}X_{O}^{(1)}\cdots
X_{I}^{(N)}}\otimes\mathbbm{1}_{X_{O}^{(N)}}$. Applying $\widetilde{\Lambda}$
to each output space of $W$ except for the last one yields

$\displaystyle{}_{X_{O}^{(1)}\cdots X_{O}^{(N-1)}}W=1/(d_{X^{(1)}_{O}}\cdots
d_{X^{(N-1)}_{O}})\operatorname{tr}_{X^{(1)}_{O}\cdots
X^{(N-1)}_{O}}W\otimes\mathbbm{1}_{X_{O}}=:\rho_{X_{I}}\otimes\mathbbm{1}_{X_{O}}\,,$
(151)

where $X_{I}$ $(X_{O})$ denotes all input (output) spaces. Now, employing Eq.
(150) and setting $d_{\bar{O}}^{2}:=\prod_{i=1}^{N-1}d_{X^{(i)}_{O}}^{2}$, we
obtain

$\displaystyle{}_{X_{O}^{(1)}\cdots
X_{O}^{(N-1)}}W=\prod_{i=1}^{N-1}(\frac{1}{d_{X^{(i)}_{O}}^{2}}\mathcal{I}_{X_{O}^{(i)}}+\frac{d_{X^{(i)}_{O}}^{2}-1}{d_{X^{(i)}_{O}}^{2}}\widetilde{\Gamma}_{X_{O}^{(i)}})[W]=:(\frac{1}{d_{\bar{O}}^{2}}\mathcal{I}_{X_{O}}+(1-\frac{1}{d_{\bar{O}}^{2}})\widetilde{\Omega}_{X_{O}})[W]\,,$
(152)

where $\widetilde{\Omega}_{X_{O}}$ is a CPTP map. Let us assume that
$T:=\widetilde{\Omega}_{X_{O}}[W]$ is a proper process matrix. In this case,
using the above considerations, we have
$1/d_{\bar{O}}^{2}W+(1-1/d_{\bar{O}}^{2})T\in\texttt{Free}$. Recalling the
definition of the signalling robustness

$\displaystyle\mathcal{R}_{s}(W)=\underset{T\in\texttt{Proc}}{\min}\left\\{s\geq
0\left|\frac{W+sT}{1+s}=C\in\texttt{Free}\right.\right\\}\,,$ (153)

we see that $\mathcal{R}_{s}(W)\leq\prod_{i=1}^{N-1}d_{X^{(i)}_{O}}^{2}-1$,
which corresponds to the claim of the Proposition. It remains to show that
$\widetilde{\Omega}_{X_{O}}[W]$ is indeed a proper process matrix. To this
end, we note from Eq. (152) that $\widetilde{\Omega}_{X_{O}}$ corresponds to a
convex combination of local CPTP maps that act on the respective output spaces
of each party. Since each such tensor product of maps simply corresponds to a
post-processing of the local output spaces, it transforms a proper process
matrix to a proper process matrix, and analogously so does a convex
combination of such maps. This concludes the proof. ∎

For the case where all involved dimensions are equal (the more general case
will be discussed below), it is now straightforward to show that a process of
the form

$\displaystyle W^{X^{(1)}\rightarrow\cdots\rightarrow
X^{(N)}}=\frac{\mathbbm{1}_{X^{(1)}_{I}}}{d_{X^{(1)}_{I}}}\otimes\Phi^{+}_{X^{(1)}_{O}X^{(2)}_{I}}\otimes\cdots\otimes\Phi^{+}_{X^{(N-1)}_{O}X^{(N)}_{I}}\otimes\mathbbm{1}_{X^{(N)}_{O}}$
(154)

maximizes the signalling robustness on $\textup{{Proc}}_{1:N}$, as claimed in
the main text. In clear analogy to the two-party case, the matrix $S\geq 0$
used in the dual SDP (74) for the computation of the signalling robustness in
the multi-party case satisfies

$\displaystyle\operatorname{tr}_{X_{O}^{(1)}\cdots
X_{O}^{(N)}}S=\mathbbm{1}_{X_{I}^{(1)}\cdots X_{I}^{(N)}}.$ (155)

Then, choosing

$\displaystyle
S=\mathbbm{1}_{X^{(1)}_{I}}\otimes\Phi^{+}_{X^{(1)}_{O}X^{(2)}_{I}}\otimes\cdots\otimes\Phi^{+}_{X^{(N-1)}_{O}X^{(N)}_{I}}\otimes\mathbbm{1}_{X^{(N)}_{O}}/d_{X^{(N)}_{O}}\,,$
(156)

we see that $\operatorname{tr}(SW^{X^{(1)}\rightarrow\cdots\rightarrow
X^{(N)}})=d_{\bar{O}}^{2}-1$. This proofs Cor. 1 of the main text.

Evidently, the proof of Prop. 6 does not work for general, causally
indefinite, processes, since we explicitly used the fact that $W$ factorizes,
i.e., it satisfies $W={}_{X_{O}^{(N)}}W$. Nonetheless, we can slightly
generalize the above statement without changing the proof. In particular, even
though we focused on causally ordered processes, the only actual assumption we
employ in the above proof is that there is a definite last party ($X^{(N)}$ in
the notation of the proof). Whether or not the remainder of the process is
causally ordered did not factor in. Consequently, we have the following
corollary:

######  Corollary 2.

Let $\textup{{Proc}}_{\prec N}$ be the set of all processes that have
$X^{(N)}$ as the definite last party, i.e., if $W\in\textup{{Proc}}_{\prec
N}$, then ${}_{X^{(N)}_{O}}W=W$ and
${}_{X^{(N)}_{O}X^{(N)}_{I}}W={}_{X^{(N)}_{O}X^{(N)}_{I}X^{(N-1)}_{O}}W$. For
all $W\in\textup{{Proc}}_{\prec N}$ we have
$\mathcal{R}_{s}(W)\leq\prod_{i=1}^{N-1}d_{X^{(i)}_{O}}^{2}-1$, where
$d_{X^{(i)}_{O}}$ is the output dimension of party $X^{(i)}$.

The proof of this statement follows along the same lines as that of Prop. 6
for causally ordered processes.

Having found these upper bounds for the signalling robustness on the sets
$\textup{{Proc}}_{1:N}$ and $\textup{{Proc}}_{\prec N}$ of causally ordered
processes and processes with a definite last party, respectively, then allows
one to find an analogous bound on their convex hulls.

######  Corollary 3.

Let $\textup{{Proc}}_{\pi(N)}^{\textup{{conv}}}$ be the convex hull of all
processes that have a definitive last party. Then, for all
$W\in\textup{{Proc}}_{\pi(N)}^{\textup{{conv}}}$ we have
$\mathcal{R}_{s}(W)\leq\underset{\bar{O}}{\max}\\{d_{\bar{O}}^{2}-1\\}$, where
$d_{\bar{O}}^{2}$ is defined as in Prop. 6, i.e., the dimension of all spaces
except for the last one, and the maximization is over all possible causal
orders – given by the permutation $\pi$ – of $N$ parties.

Since $\textup{{Proc}}_{1:N}\subset\textup{{Proc}}_{\prec N}$, the above
Corollary also applies to the convex hull of $\textup{{Proc}}_{1:N}$.

###### Proof.

If $W\in\textup{{Proc}}_{\pi(N)}^{\textup{{conv}}}$ then it can be written as
$W=\sum_{\pi}p_{\pi}W^{\pi(N)}$, where $\pi$ is a permutation of $N$ elements,
$\\{p_{\pi}\\}$ is a probability distribution with $\sum_{\pi}p_{\pi}\ =1$,
and $W^{\pi(N)}$ denotes a process with final party $X^{(\pi(N))}$. Using the
convexity of the signalling robustness, we obtain

$\displaystyle\mathcal{R}_{s}(W)=\mathcal{R}_{s}(W)\Big{(}\sum_{\pi}p_{\pi}W^{\pi(N)}\Big{)}\leq\sum_{\pi}p_{\pi}\mathcal{R}_{s}(W^{\pi(N)})\leq\underset{\pi}{\max}\mathcal{R}_{s}(W^{\pi(N)})=\underset{\bar{O}}{\max}\\{d_{\bar{O}}^{2}-1\\}.$
(157)

∎

A priori, this bound is not tight. However, under the assumption that there is
a causal ordering of parties such that $d_{\bar{O}}^{2}$ is maximized and the
dimensions of all input spaces are lower bounded by the dimension of their
respective preceding output spaces (i.e., $d_{X^{(i)}_{O}}\leq
d_{X^{(i+1)}_{I}}$ for all $i=1,\dots,N-1$), we have the following
Proposition:

######  Proposition 7.

Let $W$ be a process matrix defined on the $N$ laboratories $\\{X^{(i)}\\}$.
Let $X^{(1)}\prec X^{(2)}\prec\cdots\prec X^{(N)}$ be an order in which
$d_{\bar{O}}^{2}$ is maximized, and let $d_{X^{(i)}_{O}}\leq
d_{X^{(i+1)}_{I}}$ for all $i=1,\dots,N-1$. Then there exists a causally
ordered Markovian process $W^{X^{(1)}\prec\cdots\prec X^{(N)}}$ that maximizes
the signalling robustness on $\textup{{Proc}}_{\pi(N)}^{\textup{{conv}}}$,
i.e., it satisfies

$\displaystyle\mathcal{R}_{s}(W^{X^{(1)}\prec\cdots\prec
X^{(N)}})=\underset{\bar{O}}{\max}\\{d_{\bar{O}}^{2}-1\\}\,.$ (158)

###### Proof.

First, assume that the dimension of neighboring input and output spaces is the
same, i.e., $d_{X^{(i)}_{O}}=d_{X^{(i+1)}_{I}}$ for all $i=1,\dots,N-1$. Then
the above statement is directly implied by from Cor. 1

The case where $d_{X^{(i)}_{O}}<d_{X^{(i+1)}_{I}}$ for some (or possibly all)
adjacent laboratories follows in a similar vein: First, we define a
‘generalized identity map’ $\mathcal{I}_{X^{(i)}_{O}\rightarrow
X^{(i+1)}_{I}}$ by its action on all states
$\rho\in\mathcal{B}(\mathcal{H}_{X^{(i)}_{O}})$:

$\displaystyle\mathcal{I}_{X^{(i)}_{O}\rightarrow
X^{(i+1)}_{I}}[\rho]=\begin{pmatrix}\rho&0\\\
0&0\end{pmatrix}\in\mathcal{B}(\mathcal{H}_{X^{(i+1)}_{I}})\,,$ (159)

i.e., $\mathcal{I}_{X^{(i)}_{O}\rightarrow X^{(i+1)}_{I}}[\rho]$ transmits
$\rho$ and merely pads it out with zeros such that the output lives in the
right space. The corresponding Choi matrix of
$\mathcal{I}_{X^{(i)}_{O}\rightarrow X^{(i+1)}_{I}}$ is given by

$\displaystyle\Phi_{X^{(i)}_{O}\rightarrow
X^{(i+1)}_{I}}^{+}:=\begin{pmatrix}\Phi_{X^{(i)}_{O}X^{(i)^{\prime}}_{O}}^{+}&0\\\
0&0\end{pmatrix}\,,$ (160)

where $\mathcal{H}_{X^{(i)}_{O}}\cong\mathcal{H}_{X^{(i)^{\prime}}_{O}}$.
Using this generalized identity map, the corresponding witness $S$ and the
causally ordered process $W^{X^{(1)}\rightarrow\cdots\rightarrow X^{(N)}}$ can
be defined in the same vein as in Eqs (154) and (156) with the same result for
the signalling robustness. ∎

Since the signalling robustness of causally non-separable processes does not
seem to exceed that of the processes of Eq. (154), allowing for communication
in several directions beyond probabilistic mixing does not appear increase the
amount of causal connection. The fact that the upper bound cannot necessarily
be achieved in the case where the input dimension of an input space is not
larger than or equal to its preceding output space then also possesses a
straightforward explanation. In this case, there is no identity channel from
one laboratory to the other, leading to lower causal connection and an
unattainability of the bound
$\underset{\bar{O}}{\max}\\{d_{\bar{O}}^{2}-1\\}$.

##  Appendix J All monotones of the resource theory of causal connection

As we have seen, neither the signalling nor the causal robustness are
sufficient to decide whether a process can be converted into another by means
of free adapters. Intuitively though, if one knew all monotones of the
resource theory of causal connection, one could unambiguously decide whether
or not two processes can be connected via free adapters. Here, using results
from Ref. [40], we make this intuition concrete and provide all monotones of
the resource theory of causal connection. These monotones, in turn, provide a
necessary and sufficient condition for the convertibility of processes.

In Ref. [40], the complete set of monotones for a wide class of quantum
resource theories of processes has been derived. There, the authors consider
sets of free maps that map CPTP maps onto CPTP maps, and, under weak
assumptions on the set of free CPTP maps, derive all monotones of such a
resource theory. Here, we follow these ideas to provide a characterization of
all monotones in a resource theory where the free processes are given by
$\texttt{Free}=\\{\rho_{A_{I}B_{I}}\otimes\mathbbm{1}_{A_{O}B_{O}}\in\mathcal{B}(\mathcal{H}_{AB})\\}$.
Unlike in the previous sections, we will not work with the Choi matrices of
adapters, but consider the free adapters as maps $\widetilde{\Upsilon}$ that
map Choi matrices $W$ on $\mathcal{B}(\mathcal{H}_{AB})$ to Choi matrices
$W^{\prime}$ on $\mathcal{B}(\mathcal{H}_{A^{\prime}B^{\prime}})$. Concretely,
with this, we get the set of free transformations

$\displaystyle\bar{\Theta}_{\texttt{NS}}(AB\rightarrow
A^{\prime}B^{\prime})=\\{\widetilde{\Upsilon}|\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}\in\Theta_{\texttt{NS}}\\}\,,$
(161)

where
$\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}$ is
the Choi matrix of $\widetilde{\Upsilon}$.

Considering the map $\widetilde{\Upsilon}$ instead of its Choi matrix
$\Upsilon_{\mathbbm{A}_{I}\mathbbm{A}_{O}\mathbbm{B}_{I}\mathbbm{B}_{O}}$ will
lead to some notational simplifications in what follows. Naturally, everything
can be phrased entirely in terms of Choi matrices (or entirely in terms of
maps, as is done in Ref. [40]). In what follows, whenever there is no risk of
confusion, we will drop the explicit input and output spaces and simply write
$\bar{\Theta}_{\texttt{NS}}$ instead of
$\bar{\Theta}_{\texttt{NS}}(AB\rightarrow A^{\prime}B^{\prime})$.

Following Ref. [40], we can define the following monotones for our resource
theory:

$\displaystyle
f_{Q}(W)=\underset{\widetilde{\Upsilon}\in\bar{\Theta}_{\texttt{NS}}}{\max}\operatorname{tr}(Q\widetilde{\Upsilon}(W))\,,$
(162)

where $W$ is a proper process matrix on $\mathcal{B}(\mathcal{H}_{AB})$ and
$Q$ is a proper process matrix on
$\mathcal{B}(\mathcal{H}_{A^{\prime}B^{\prime}})$. Note that we can use the
maximum instead of the supremum in the above equation, as the set of free
adapters we consider is compact, i.e., the maximum exists. Since the
concatenation of free adapters still yields a free adapter, it is easy to see
that Eq. (162) indeed defines a monotone of the resource theory. With this, we
obtain an analogous characterization of a complete set of monotones as the one
found in [40]:

######  Proposition 8.

Let $W_{AB}$ be a proper process matrix on $\mathcal{B}(\mathcal{H}_{AB})$ and
$W^{\prime}_{A^{\prime}B^{\prime}}$ be a proper process matrix on
$\mathcal{B}(\mathcal{H}_{A^{\prime}B^{\prime}})$. Then there exists a free
adapter $\widetilde{\Upsilon}\in\bar{\Theta}_{\textup{{NS}}}$ such that
$W^{\prime}_{A^{\prime}B^{\prime}}=\widetilde{\Upsilon}(W_{AB})$ iff

$\displaystyle f_{Q}(W_{AB})\geq
f_{Q}(W^{\prime}_{A^{\prime}B^{\prime}})\quad\forall
Q\in\textup{{Proc}}_{A^{\prime}B^{\prime}}\,,$ (163)

where $\textup{{Proc}}_{A^{\prime}B^{\prime}}$ is the set of valid process
matrices on $A^{\prime}B^{\prime}$.

The proof proceeds analogous to the original proof of Thm. 4 in Ref. [40].
First, using convexity and closedness of the set of all free adapters, we will
show that Eq. (163) provides all monotones if $Q$ is Hermitian. In the second
step, we show that it is sufficient to restrict $Q$ to be a proper process
matrix on $\mathcal{B}(\mathcal{H}_{A^{\prime}B^{\prime}})$.

###### Proof.

Step 1: We denote the orbit of a process matrix $W_{AB}$ on
$\mathcal{B}(\mathcal{H}_{AB})$ under free transformations by
$\mathcal{O}_{W}=\\{\widetilde{\Upsilon}(W_{AB})|\widetilde{\Upsilon}\in\bar{\Theta}_{\texttt{NS}}\\}$.
As $\bar{\Theta}_{\texttt{NS}}$ is convex and closed, so is $\mathcal{O}_{W}$.
Consequently, any process matrix $W^{\prime}_{A^{\prime}B^{\prime}}$ is not an
element of $\mathcal{O}_{W}$ iff there exists a Hermitian matrix
$Q_{A^{\prime}B^{\prime}}$ on
$\mathcal{B}(\mathcal{H}_{A^{\prime}B^{\prime}})$ such that

$\displaystyle\operatorname{tr}(Q_{A^{\prime}B^{\prime}}W^{\prime}_{A^{\prime}B^{\prime}})>\underset{\Upsilon\in\bar{\Theta}_{\texttt{NS}}}{\max}\operatorname{tr}(Q_{A^{\prime}B^{\prime}}\widetilde{\Upsilon}[W_{AB}])\,,$
(164)

where the above follows from the hyperplane separation theorem for convex
sets. On the other hand, $W^{\prime}_{A^{\prime}B^{\prime}}$ is inside the set
$\mathcal{O}_{W}$ iff for all Hermitian matrices $Q_{A^{\prime}B^{\prime}}$ on
$\mathcal{B}(\mathcal{H}_{A^{\prime}B^{\prime}})$, we have

$\displaystyle\operatorname{tr}(Q_{A^{\prime}B^{\prime}}W^{\prime}_{A^{\prime}B^{\prime}})\leq\underset{\Upsilon\in\bar{\Theta}_{\texttt{NS}}}{\max}\operatorname{tr}(Q_{A^{\prime}B^{\prime}}\Upsilon(W_{AB}))\,.$
(165)

Naturally, the ‘only if’ direction of the above equation holds trivially for
any set, while the ‘if’ direction requires the convexity of the set. Eq. (165)
allows one to check whether or not there exists an
$\Upsilon\in\bar{\Theta}_{\texttt{NS}}$ such that
$\Upsilon(W_{AB})=W^{\prime}_{A^{\prime}B^{\prime}}$; this is exactly the case
when $W^{\prime}_{A^{\prime}B^{\prime}}$ satisfies Eq. (165). Now, we can show
that Eq. (165) is equivalent to Eq. (163) when we allow
$Q_{A^{\prime}B^{\prime}}$ to be Hermitian (instead of a proper process
matrix).

First, if Eq. (163) holds (for all Hermitian $Q_{A^{\prime}B^{\prime}}$), then
we have

$\displaystyle\underset{\widetilde{\Upsilon}^{\prime}\in\bar{\Theta}_{\texttt{NS}}(A^{\prime}B^{\prime}\rightarrow
A^{\prime}B^{\prime})}{\max}\operatorname{tr}[Q_{A^{\prime}B^{\prime}}\widetilde{\Upsilon}^{\prime}(W^{\prime}_{A^{\prime}B^{\prime}})]\leq\underset{\Upsilon\in\bar{\Theta}_{\texttt{NS}}(AB\rightarrow
A^{\prime}B^{\prime})}{\max}\operatorname{tr}[Q_{A^{\prime}B^{\prime}}\widetilde{\Upsilon}(W_{AB})]\quad\forall
Q_{A^{\prime}B^{\prime}}\,.$ (166)

Choosing $\widetilde{\Upsilon}^{\prime}$ to be the identity map (which is a
free adapter), we see that Eq. (165) is satisfied, and consequently
$W^{\prime}_{A^{\prime}B^{\prime}}\in\mathcal{O}_{W}$.

Conversely, if Eq. (165) holds, then for any
$\widetilde{\Upsilon}^{\prime}\in\bar{\Theta}_{\texttt{NS}}(A^{\prime}B^{\prime}\rightarrow
A^{\prime}B^{\prime})$ we have

$\displaystyle\begin{split}\operatorname{tr}(Q_{A^{\prime}B^{\prime}}\widetilde{\Upsilon}^{\prime}(W^{\prime}_{A^{\prime}B^{\prime}}))&=\operatorname{tr}(\widetilde{\Upsilon}^{\prime\dagger}(Q_{A^{\prime}B^{\prime}})W^{\prime}_{A^{\prime}B^{\prime}})\leq\underset{\Upsilon\in\bar{\Theta}_{\texttt{NS}}(AB\rightarrow
A^{\prime}B^{\prime})}{\max}\operatorname{tr}[\widetilde{\Upsilon}^{\prime\dagger}(Q_{A^{\prime}B^{\prime}})\widetilde{\Upsilon}(W_{AB})]\\\
&=\underset{\Upsilon\in\bar{\Theta}_{\texttt{NS}}(AB\rightarrow
A^{\prime}B^{\prime})}{\max}\operatorname{tr}[Q_{A^{\prime}B^{\prime}}\widetilde{\Upsilon}^{\prime}\circ\widetilde{\Upsilon}(W_{AB})]\\\
&\leq\underset{\Upsilon\in\bar{\Theta}_{\texttt{NS}}(AB\rightarrow
A^{\prime}B^{\prime})}{\max}\operatorname{tr}[Q_{A^{\prime}B^{\prime}}\widetilde{\Upsilon}(W_{AB})]=f_{Q}(W)\,,\end{split}$
(167)

where we used Eq. (165) in the second line,
$\widetilde{\Upsilon}^{\prime\dagger}$ is the dual map of
$\widetilde{\Upsilon}$, and we have employed the fact that
$\widetilde{\Upsilon}^{\prime}\circ\widetilde{\Upsilon}$ is still a free
operation. As the above equation holds for all $Q_{A^{\prime}B^{\prime}}$ and
free adapters $\widetilde{\Upsilon}^{\prime}$, it implies $f_{Q}(W_{AB})\geq
f_{Q}(W^{\prime}_{A^{\prime}B^{\prime}})$. Now, it remains to show that we can
restrict the set of matrices $Q_{A^{\prime}B^{\prime}}$ for which we have to
check the monotones to the set of proper process matrices.

Step 2: To show that proper process matrices are sufficient, we consider the
matrix

$\displaystyle\bar{Q}_{A^{\prime}B^{\prime}}=(1-\varepsilon)\frac{\mathbbm{1}_{A^{\prime}B^{\prime}}}{d_{A_{I}^{\prime}B_{I}^{\prime}}}+\varepsilon
L_{V}(Q_{A^{\prime}B^{\prime}}).$ (168)

This matrix satisfies
$L_{V}(\bar{Q}_{A^{\prime}B^{\prime}})=\bar{Q}_{A^{\prime}B^{\prime}}$, and
for $\varepsilon$ small enough, we have $\bar{Q}_{A^{\prime}B^{\prime}}\geq
0$, i.e., $\bar{Q}_{A^{\prime}B^{\prime}}$ is – up to normalization – a proper
process matrix. For this matrix, we obtain

$\displaystyle\begin{split}\operatorname{tr}(\bar{Q}_{A^{\prime}B^{\prime}}W^{\prime}_{A^{\prime}B^{\prime}})&=d_{A_{O}^{\prime}B_{O}^{\prime}}^{2}(1-\varepsilon)+\varepsilon\operatorname{tr}(L_{V}(Q_{A^{\prime}B^{\prime}})W^{\prime}_{A^{\prime}B^{\prime}})\\\
&=d_{A_{O}^{\prime}B_{O}^{\prime}}^{2}(1-\varepsilon)+\varepsilon\operatorname{tr}(Q_{A^{\prime}B^{\prime}}W^{\prime}_{A^{\prime}B^{\prime}})\,,\end{split}$
(169)

where we have used the self-duality of $L_{V}$ and the fact that
$W^{\prime}_{A^{\prime}B^{\prime}}$ is a proper process matrix, i.e., it
satisfies
$L_{V}(W^{\prime}_{A^{\prime}B^{\prime}})=W^{\prime}_{A^{\prime}B^{\prime}}$.
From the above equation we see that up to a constant offset that only depends
on the dimension $d_{A_{O}^{\prime}B_{O}^{\prime}}$, and a positive constant
pre-factor, $f_{\bar{Q}}$ yields the same value as $f_{Q}$. Consequently,
checking Eq. (163) for proper process matrices $Q_{A^{\prime}B^{\prime}}$ is
sufficient. This concludes the proof. ∎

Since we did not rely in the proof on the fact that there are only two