acterised the 2-causal polytope for the simplest non-
trivial tripartite scenario and found that almost all of
the 473 nontrivial classes of 2-causal inequalities can
be violated by process matrix correlations. However,
we were unable to find any violation for 2 of those in-
equalities; this stands in contrast to previous studies
of causal inequalities, where violations with process
matrices were always found
11
[4–7, 9]. Although it
remains to be confirmed whether this is simply a fail-
ure of the search method we used, we provided some
intuition why such a violation would in fact be a sur-
prise.
Our definition of genuinely N-partite noncausality
is analogous to the corresponding notion for nonlocal-
ity originally due to Svetlichny [19–21]. It is known,
however, that that notion harbours some issues: for
example, it is not robust under local operations, a nec-
essary requirement for an operational resource theory
of nonlocality [22, 23]. In order to overcome these is-
sues, additional constraints must be imposed on the
correlations shared by subsets of parties when defin-
ing correlations that are not genuinely multipartite
nonlocal. In the case of noncausality, however, there
appears to be no clear reason to impose any addi-
tional such constraints. For nonlocal resources, issues
arise in particular from the possibility that different
parties might access the resource at different times,
with an earlier party then communicating with a later
one. This type of issue is not pertinent for noncausal
resources, where the causal order (be it definite or
indefinite) between parties is determined by the re-
source itself, and additional communication beyond
what the resource specifies seems to fall outside the
relevant framework. More generally, however, an op-
erational framework and understanding of the rele-
vant “free operations” for noncausal resources remains
to be properly developed.
Finally, in this paper we only considered correla-
tions from a fully theory- and device-independent per-
spective; it would be interesting to develop similar no-
tions within specific physical theories like the process
matrix framework, where quantum theory holds lo-
cally for each party. Process matrices that cannot be
realised with a definite causal order are called causally
nonseparable [4], and it would be interesting to study
a notion of genuinely multipartite causal nonsepara-
bility. It should, however, be noted that different pos-
sible notions of multipartite causal (non)separability
have been proposed [8, 14], so a better understand-
ing of their significance would be necessary in order
to extend the notions we have developed here to that
framework.
11
At least for standard causal inequalities that bound prob-
abilities directly; for entropic causal inequalities, which only
provide a relaxed characterisation of the set of causal correla-
tions, no violations were found so far [18]. It would nevertheless
also be interesting to investigate how genuinely multipartite
noncausality can be characterised with the entropic approach.
Acknowledgements
A.A.A., J.W. and C.B. acknowledge support through
the ‘Retour Post-Doctorants’ program (ANR-13-
PDOC-0026) of the French National Research Agency.
F.C. acknowledges support through an Australian Re-
search Council Discovery Early Career Researcher
Award (DE170100712). This publication was made
possible through the support of a grant from the John
Templeton Foundation. The opinions expressed in
this publication are those of the authors and do not
necessarily reflect the views of the John Templeton
Foundation. F.C. acknowledges the traditional own-
ers of the land on which the University of Queensland
is situated, the Turrbal and Jagera people.
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Accepted in Quantum 2017-12-06, click title to verify 13