
are made about some parties but not others.
Semi-device-independent schemes have been de-
veloped and extensively studied for the certific-
ation of entanglement [14] and measurement in-
compatibility [15, 16], known as EPR-steering,
and applied to quantum key distribution proto-
cols where some but not all parties can be trusted
[17].
A close analogy can be developed with regard
to the certification of indefinite causal order, as
encoded in a process matrix [18]. A process mat-
rix is a higher-order operation [19, 20, 21] – i.e.
a transformation of quantum operations – that
acts on independent sets of operations. Funda-
mental laws of quantum theory predict the exist-
ence of process matrices that act on these opera-
tions in a such a way that a well-defined causal
order cannot be established among them. Pro-
cess matrices with indefinite causal order were
proven to be a powerful resource, outperforming
causally ordered ones in tasks such as quantum
channel discrimination [22], communication com-
plexity [23, 24], quantum computation [25], and
inverting unknown unitary operations [26].
To certify that a process matrix in fact does
not act in a causally ordered way, there are two
standard methods available in the literature. The
first is to evaluate a causal witness [27, 28]. Ana-
logous to the evaluation of an entanglement wit-
ness, this method relies on detailed knowledge of
the quantum operations being implemented, and
as such it allows for a device-dependent certifica-
tion. All experimental certifications of indefinite
causal order to date either measure a causal wit-
ness [29, 30] or rely on similar device-dependent
assumptions [31, 32, 33]. The second method is
the violation of a causal inequality, phenomenon
which is also predicted by quantum mechanics
[18, 34]. Analogous to the violation of a Bell in-
equality, this method does not rely on detailed
knowledge of the quantum operations implemen-
ted by the parties, but rather only that they com-
pose under a tensor product. As such, it allows
for a device-independent certification. When a
causal inequality is violated, it is verified that
least one of the principles used to derived it is
not respected. Moreover, this claim holds true in-
dependently of the physical theory that supports
the experiment that led to the violation. Hence,
although we focus on quantum theory, indefinite
causal order could in principle be certified even
without relying on the laws of quantum mechan-
ics. Although it would be highly desirable to per-
form such device-independent certification of in-
definite causal order, no physical implementation
of process matrices that would violate a causal
inequality is currently known.
In this work, we introduce a semi-device-
independent framework for certifying noncausal
properties of process matrices that allows for an
experimental certification of indefinite causal or-
der that relies on fewer assumptions than previ-
ous ones. In our semi-device-independent scen-
ario, the operations of some parties are fully char-
acterized while no assumptions are made about
the others.
We begin by considering the bipartite case,
for which we construct a general framework
for certifying indefinite causal order in a semi-
device-independent scenario, while contextualiz-
ing previously developed device-dependent and -
independent ones. We then extend our framework
to a tripartite case in which the third party is al-
ways in the future of the other two, and provide
an extensive machinery that may be generalized
to other multipartite scenarios. We apply our
methods to the notorious quantum switch [22, 35],
a process matrix that, despite having an indefin-
ite causal order, only leads to causal correlations
in a device-independent scenario. We show that
the noncausal properties of the quantum switch
can be certified in a semi-device-independent way,
proving that it generates stronger noncausal cor-
relations than it was previously known.
1 Preliminaries
In our certification scheme, we will deal with
statistical data in the form of behaviours.
A general bipartite behaviour {p
(
ab|xy
)
} is a set
of joint probability distributions, that is, a set in
which each element p
(
ab|xy
)
is a real non-negative
number such that
P
a,b
p
(
ab|xy
) = 1
for all x, y,
where a ∈ {
1
, . . . , O
A
} and b ∈ {
1
. . . , O
B
} are
labels for outcomes and x ∈ {
1
, . . . , I
A
} and y ∈
{
1
, . . . , I
B
} are labels for inputs, for parties Alice
and Bob, respectively.
The most general operation allowed by
quantum theory is modelled by a quantum instru-
ment, which is a set of completely-positive (CP)
maps that sum to a completely-positive trace-
preserving (CPTP) map [36].
Accepted in Quantum 2019-07-07, click title to verify. Published under CC-BY 4.0. 2