# Network-Device-Independent Certification of Causal Nonseparability

Hippolyte Dourdent  ICFO-Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology,
08860 Castelldefels, Barcelona, Spain  Alastair A. Abbott  Univ. Grenoble
Alpes, Inria, 38000 Grenoble, France  Ivan Šupić  LIP6, Sorbonne Université,
CNRS, 4 Place Jussieu, 75005 Paris, France  Cyril Branciard  Univ. Grenoble
Alpes, CNRS, Grenoble INP111Institute of Engineering Univ. Grenoble Alpes,
Institut Néel, 38000 Grenoble, France

(August 24, 2023)

###### Abstract

Causal nonseparability is the property underlying quantum processes
incompatible with a definite causal order. So far it has remained a central
open question as to whether any process with a clear physical realisation can
violate a causal inequality, so that its causal nonseparability can be
certified in a device-independent way, as originally conceived. Here we
present a method solely based on the observed correlations, which certifies
the causal nonseparability of all the processes that can induce a causally
nonseparable distributed measurement in a scenario with trusted quantum input
states, as defined in [Dourdent _et al._ , Phys. Rev. Lett. 129, 090402
(2022)]. This notably includes the celebrated quantum switch. This device-
independent certification is achieved by introducing a network of untrusted
operations, allowing one to self-test the quantum inputs on which the
effective distributed measurement induced by the process is performed.

Introduction.—For both quantum and classical processes, it is usually assumed
that if two parties interact only once with a physical medium, then only one-
way influences are possible. Remarkably, this turns out to be an unnecessary
restriction on the possible observable correlations. Formalised in the process
matrix framework, where quantum theory is taken to hold locally but no global
causal structure is assumed, some processes allow parties to establish
correlations incompatible with a definite causal order [1].

Causal nonseparability, the property underlying such processes [1, 2], bears a
foundational significance and is also the basis for advantages in a number of
different tasks [3, 4, 5, 6, 7, 8]. It can be certified in a device-
independent (DI) way, i.e. requiring knowledge of the observed statistics
only, based on the violation of “causal inequalities” by noncausal
correlations [1, 9]. However, not all causally nonseparable process matrices
are noncausal in this strong sense [10, 11, 12]. Indeed, a large class of
quantum-realisable processes cannot violate causal inequalities [13, 14], and
it is still unclear whether any process with a clear physical realisation is
able to. This includes in particular the canonical “quantum switch” [3], the
resource behind most known advantages arising from causal indefiniteness [3,
4, 5, 6, 7, 8].

In a previous work [15], we showed (with another colleague) that the quantum
switch, as well as a large class of bipartite causally nonseparable processes
(and even all of them under a reasonable additional assumption), can display a
new form of noncausality in a semi-DI with quantum inputs (SDI-QI) scenario.
In the present work, we show how the gap between this SDI-QI certification and
a fully DI certification can be bridged by self-testing the quantum inputs in
a network scenario, where each party that receives a quantum input in the SDI-
QI scenario shares an additional uncharacterised bipartite state with an extra
untrusted party. We call this certification network-device-independent (NDI)
to differentiate it from the standard DI scenario based on causal
inequalities. This approach is motivated by the ability to exploit self-
testing to transform a universal certification of entanglement in a SDI-QI
scenario into a NDI certification [16, 17]. The apparent analogy between
entanglement and causal nonseparability, however, was already seen to have
severe limitations in the SDI-QI scenario [15], making it unclear whether NDI
certification could be applied to this latter resource.

Causal (non)separability in the SDI-QI scenario.—The approach we present is a
general method for transforming SDI-QI certification of causal nonseparability
into NDI certification, applicable to a range of scenarios. For concreteness
and clarity, however, we focus our initial presentation here on the bipartite
scenario, before making some comments about its applicability to other
scenarios and processes later.

Two parties, Alice and Bob, control separate quantum labs with input and
output Hilbert spaces $\mathcal{H}^{A_{I}}$ and $\mathcal{H}^{A_{O}}$ for
Alice, and $\mathcal{H}^{B_{I}}$ and $\mathcal{H}^{B_{O}}$ for Bob. They may
also receive some auxiliary quantum states in Hilbert spaces
$\mathcal{H}^{{A^{\prime}}},\mathcal{H}^{{B^{\prime}}}$.222All Hilbert spaces
we consider are taken to be finite-dimensional. Notations:
$\mathcal{L}(\mathcal{H}^{X})$ denotes the space of linear operators on
$\mathcal{H}^{X}$; superscripts are used to indicate on what spaces operators
act; we shall write concisely
$\mathcal{H}^{XY}=\mathcal{H}^{X}\otimes\mathcal{H}^{Y}$,
$\mathcal{H}^{A}=\mathcal{H}^{A_{I}A_{O}}$, etc.
Note that in [15], the auxiliary spaces in which the quantum inputs live were
denoted with tildes (e.g., $\mathcal{H}^{\tilde{A}}$); here we switch to
primes: the tildes will be used for the untrusted physical systems in the
self-testing procedure below. Also, in [15] Alice and Bob’s quantum input
states were labeled by subscripts $x$ and $y$; here we change these to $z$ and
$w$, as $x$ and $y$ will be introduced later as classical inputs for the self-
testing procedure. Let us consider the SDI-QI scenario introduced in [15]
where Alice and Bob are provided with quantum input states
$\rho_{z}^{A^{\prime}}\in\mathcal{L}(\mathcal{H}^{A^{\prime}})$ and
$\rho_{w}^{B^{\prime}}\in\mathcal{L}(\mathcal{H}^{B^{\prime}})$, respectively,
indexed by the labels $z$ and $w$. They each perform some fixed quantum
operations described as quantum instruments [18], i.e., sets of completely
positive (CP) maps
$\mathcal{M}_{a}:\mathcal{L}(\mathcal{H}^{{A^{\prime}}A_{I}})\to\mathcal{L}(\mathcal{H}^{A_{O}})$
and
$\mathcal{M}_{b}:\mathcal{L}(\mathcal{H}^{{B^{\prime}}B_{I}})\to\mathcal{L}(\mathcal{H}^{B_{O}})$,
whose indices $a,b$ refer to some (classical) outcomes for Alice and Bob, and
whose sums $\sum_{a}\mathcal{M}_{a}$ and $\sum_{b}\mathcal{M}_{b}$ are trace-
preserving (TP). Using the Choi isomorphism [19] (see the Supplemental
Material (SM) of [15], Sec. A), the CP maps $\mathcal{M}_{a}$,
$\mathcal{M}_{b}$ can be represented as positive semidefinite matrices
$M_{a}^{{A^{\prime}}A}$ and $M_{b}^{{B^{\prime}}B}$.

Within the process matrix framework, the correlations established by Alice and
Bob in such a scenario with quantum inputs are given by the probabilities

$\displaystyle P(a,b|\rho_{z}^{A^{\prime}},\rho_{w}^{B^{\prime}})$
$\displaystyle=\big{(}M_{a}^{{A^{\prime}}A}\otimes
M_{b}^{{B^{\prime}}B}\big{)}*\big{(}\rho_{z}^{A^{\prime}}\\!\otimes\rho_{w}^{B^{\prime}}\\!\otimes
W^{AB}\big{)}$
$\displaystyle=E_{a,b}^{{A^{\prime}}{B^{\prime}}}\\!*\\!\big{(}\rho_{z}^{A^{\prime}}\\!\\!\otimes\rho_{w}^{B^{\prime}}\big{)}=\operatorname{Tr}\left[\big{(}E_{a,b}^{{A^{\prime}}{B^{\prime}}}\big{)}^{T}\big{(}\rho_{z}^{A^{\prime}}\\!\\!\otimes\rho_{w}^{B^{\prime}}\big{)}\right]$
(1)

with

$\displaystyle
E_{a,b}^{{A^{\prime}}{B^{\prime}}}=\big{(}M_{a}^{{A^{\prime}}A}\otimes
M_{b}^{{B^{\prime}}B}\big{)}*W^{AB}.$ (2)

Here, $W^{AB}\in\mathcal{L}(\mathcal{H}^{AB})$ is the so-called “process
matrix”, a positive semidefinite matrix satisfying nontrivial linear
constraints in order to always generate valid probabilities [1] (see e.g.
[15], SM, Sec. B); and $*$ is the “link product” [20, 21], a convenient tool
for calculations defined for any matrices
$M^{XY}\in\mathcal{L}(\mathcal{H}^{XY})$,
$N^{YZ}\in\mathcal{L}(\mathcal{H}^{YZ})$ as
$M^{XY}*N^{YZ}=\operatorname{Tr}_{Y}[(M^{XY}\otimes\mathbbm{1}^{Z})^{T_{Y}}(\mathbbm{1}^{X}\otimes
N^{YZ})]\in\mathcal{L}(\mathcal{H}^{XZ})$ (where $T_{Y}$ is the partial
transpose over $\mathcal{H}^{Y}$; see also [15], SM, Sec. A).333A full trace
$\operatorname{Tr}[(M^{Y})^{T}N^{Y}]$ and a tensor product $M^{X}\otimes
N^{Z}$ can both be written as a link product. Moreover, the link product is
commutative and associative. The family
$\mathbb{E}^{{A^{\prime}}{B^{\prime}}}\coloneqq(E_{a,b}^{{A^{\prime}}{B^{\prime}}})_{a,b}$
defines an effective distributed measurement [22, 23] on the quantum inputs,
termed a “distributed positive-operator-valued measure” (D-POVM); see Fig. 1.
Note that by choosing $\\{\rho_{z}^{A^{\prime}}\\}_{z}$ and
$\\{\rho_{w}^{B^{\prime}}\\}_{w}$ to be tomographically complete sets of
states, one can completely reconstruct $\mathbb{E}^{{A^{\prime}}{B^{\prime}}}$
from the statistics obtained through Eq. (1).

Figure 1: The SDI-QI scenario: A process matrix $W^{AB}$ connects two parties
who receive trusted (orange) quantum inputs $\rho_{z}^{A^{\prime}}$ and
$\rho_{w}^{B^{\prime}}$, resp. They each perform an untrusted (black) joint
operation ($(M_{a}^{{A^{\prime}}A})_{a}$ and $(M_{b}^{{B^{\prime}}B})_{b}$,
resp.), producing the classical outcomes $a$ and $b$. The purple semicircle
represents the D-POVM $(E_{a,b}^{{A^{\prime}}{B^{\prime}}})_{a,b}$ induced by
these instruments and the process matrix.

The process matrix formalism makes no _a priori_ assumption of a global causal
structure relating Alice and Bob. In fact, assuming such a structure imposes
further constraints, due to the inability for a party to signal to the causal
past. Process matrices compatible, for example, with Alice acting before Bob
(denoted $A\prec B$) are of the form $W^{A\prec B}=W^{A\prec
B_{I}}\otimes\mathbbm{1}^{B_{O}}$, and similarly $W^{B\prec A}=W^{B\prec
A_{I}}\otimes\mathbbm{1}^{A_{O}}$ for Bob before Alice ($B\prec A$), with
$W^{A\prec B_{I}}\in\mathcal{L}(\mathcal{H}^{AB_{I}})$ and $W^{B\prec
A_{I}}\in\mathcal{L}(\mathcal{H}^{A_{I}B})$ being themselves valid process
matrices [1]. Process matrices that can be written as a convex mixture of
matrices compatible with $A\prec B$ and $B\prec A$, i.e., of the form

$\displaystyle W^{AB}$ $\displaystyle=q\,W^{A\prec
B_{I}}\otimes\mathbbm{1}^{B_{O}}+(1{-}q)\,W^{B\prec
A_{I}}\otimes\mathbbm{1}^{A_{O}},$ (3)

with $q\in[0,1]$, are said to be “causally separable”. They can be interpreted
as being compatible with a definite (although perhaps probabilistic) causal
order. Remarkably, there exist “causally nonseparable” process matrices that
cannot be decomposed as in Eq. (3), and are thus incompatible with any
definite causal order [1].

Causal nonseparability can always be certified in a device-dependent manner
using a “causal witness” [10, 24], while only some processes can be certified
in a DI way through the violation of a causal inequality [1, 10, 11, 12].
Here, we start by considering a relaxation of this DI scenario, by providing
the parties with quantum inputs instead of classical ones. In [15], we showed
that causally separable processes necessarily generate D-POVMs that are
themselves causally separable in the sense that they can be written

$\displaystyle\mathbb{E}^{{A^{\prime}}{B^{\prime}}}=q\,\mathbb{E}^{{A^{\prime}}\prec{B^{\prime}}}+(1{-}q)\,\mathbb{E}^{{B^{\prime}}\prec{A^{\prime}}}$
(4)

with $q\in[0,1]$, where
$\mathbb{E}^{{A^{\prime}}\prec{B^{\prime}}}=(E_{a,b}^{{A^{\prime}}\prec{B^{\prime}}})_{a,b}$
and
$\mathbb{E}^{{B^{\prime}}\prec{A^{\prime}}}=(E_{a,b}^{{B^{\prime}}\prec{A^{\prime}}})_{a,b}$
are D-POVMs satisfying

$\displaystyle\sum_{b}E_{a,b}^{{A^{\prime}}\prec{B^{\prime}}}=E_{a}^{A^{\prime}}\otimes\mathbbm{1}^{B^{\prime}}\hskip
8.53581pt\forall a,\hskip
8.53581pt\sum_{a}E_{a}^{A^{\prime}}=\mathbbm{1}^{A^{\prime}},$ (5)
$\displaystyle\sum_{a}E_{a,b}^{{B^{\prime}}\prec{A^{\prime}}}=E_{b}^{B^{\prime}}\otimes\mathbbm{1}^{A^{\prime}}\hskip
8.53581pt\forall b,\hskip
8.53581pt\sum_{b}E_{b}^{B^{\prime}}=\mathbbm{1}^{B^{\prime}},$ (6)

and are respectively compatible with the causal order where Alice receives her
quantum input and acts before Bob (${A^{\prime}}\prec{B^{\prime}}$) and _vice
versa_ (${B^{\prime}}\prec{A^{\prime}}$). Crucially, some causally
nonseparable processes which cannot violate any causal inequality in the DI
setting can still generate causally nonseparable D-POVMs. This causal
nonseparability—and hence that of the process matrix inducing it through Eq.
(2)—can then be certified by reconstructing the D-POVM (as mentioned above)
and verifying that it cannot be decomposed as in Eq. (4) or, more directly,
using these quantum inputs to violate a witness inequality of the form
$\mathcal{J}\geq 0$, where $\mathcal{J}$ denotes some linear combination of
the probabilities $P(a,b|\rho_{z}^{A^{\prime}},\rho_{w}^{B^{\prime}})$ [15].

NSDI-QI and NDI causal scenarios.—A bipartite SDI-QI scenario can be
transformed into a 4-partite network scenario (NSDI-QI) where two additional
separated parties—Charlie and Daisy—remotely prepare the quantum inputs. They
control separate labs with input Hilbert spaces $\mathcal{H}^{C^{\prime}}$ and
$\mathcal{H}^{D^{\prime}}$, and share maximally entangled pairs of
d-dimensional states,444Here d is the dimension of the quantum input spaces,
which for simplicity we assume to be the same for both parties (although this
can be easily relaxed without consequence).
$\mathinner{|{\phi^{+}}\rangle}^{{C^{\prime}}{A^{\prime}}}$ and
$\mathinner{|{\phi^{+}}\rangle}^{{B^{\prime}}{D^{\prime}}}$ (with
$\mathinner{|{\phi^{+}}\rangle}=\frac{1}{\sqrt{\textit{{d}}}}\sum_{j=0}^{\textit{{d}}-1}\mathinner{|{j,j}\rangle}$)
with Alice and Bob, respectively; see Fig. 2 (top). Charlie and Daisy can then
remotely prepare the quantum inputs $\rho_{z}^{A^{\prime}}$ and
$\rho_{w}^{B^{\prime}}$ by applying measurements with Choi representations
$(M_{c|z}^{C^{\prime}})_{c}=(\rho_{z}^{C^{\prime}},\mathbbm{1}^{C^{\prime}}-\rho_{z}^{C^{\prime}})$
and
$(M_{d|w}^{D^{\prime}})_{d}=(\rho_{w}^{D^{\prime}},\mathbbm{1}^{D^{\prime}}-\rho_{w}^{D^{\prime}})$
on their shares of entangled qudit pairs. When they observe the outcomes $c=0$
and $d=0$ corresponding to the POVM elements $\rho_{z}^{C^{\prime}}$ and
$\rho_{w}^{D^{\prime}}$, respectively, then Alice and Bob receive precisely
the desired quantum inputs since $\rho_{z}^{A^{\prime}}\propto
M_{0|z}^{C^{\prime}}*\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{C^{\prime}}{A^{\prime}}}$
and $\rho_{w}^{B^{\prime}}\propto
M_{0|w}^{D^{\prime}}*\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{B^{\prime}}{D^{\prime}}}$.
Thereby, the observed correlations

$\displaystyle P(c,a,b,d|z,w)=$
$\displaystyle\left(M_{c|z}^{C^{\prime}}\otimes
E_{a,b}^{{A^{\prime}}{B^{\prime}}}\otimes M_{d|w}^{D^{\prime}}\right)$
$\displaystyle*\left(\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{C^{\prime}}{A^{\prime}}}\otimes\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{B^{\prime}}{D^{\prime}}}\right)$
(7)

satisfy
$P(0,a,b,0|z,w)=P(0|z)P(0|w)P(a,b|\rho_{z}^{A^{\prime}},\rho_{w}^{B^{\prime}})$
and the causal nonseparability of $\mathbb{E}^{{A^{\prime}}{B^{\prime}}}$ can
be certified by the violation of a witness inequality as in the SDI-QI
scenario.

This NSDI-QI scenario so far assumes the sharing of maximally entangled states
and trusts Charlie’s and Daisy’s quantum measurements in the “reference
scenario” described above. However, these assumptions can be relaxed by
exploiting self-testing techniques. Self-testing exploits the fact that
quantum correlations which maximally violate a Bell inequality can sometimes
be used to uniquely determine—up to local transformations—which state and
measurements produced them [25, 26]. To self-test the maximally entangled
states and Charlie and Daisy’s measurements, Alice, Bob, Charlie and Daisy are
all given (new) classical inputs $x,y,z,w$, resp., and perform untrusted
operations
$(\mathsf{M}_{a|x}^{{\tilde{A}}A})_{a},(\mathsf{M}_{b|y}^{{\tilde{B}}B})_{b},(\mathsf{M}_{c|z}^{\tilde{C}})_{c},(\mathsf{M}_{d|w}^{\tilde{D}})_{d}$,
producing outcomes $a,b,c,d$, resp. on the process matrix $W^{AB}$ and the
untrusted auxiliary states $\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}$ and
$\uppsi_{2}^{{\tilde{B}}{\tilde{D}}}$ to be self-tested, which are now in the
(untrusted) “physical” Hilbert spaces
$\mathcal{H}^{{\tilde{A}}},\mathcal{H}^{{\tilde{B}}},\mathcal{H}^{{\tilde{C}}},\mathcal{H}^{{\tilde{D}}}$
(denoted with tildes)—that, _a priori_ , may differ from those in the
reference scenario
$\mathcal{H}^{{A^{\prime}}},\mathcal{H}^{{B^{\prime}}},\mathcal{H}^{{C^{\prime}}},\mathcal{H}^{{D^{\prime}}}$
(with primes); see Fig. 2 (bottom). When applied to the process matrix, Alice
and Bob’s instruments generate a D-POVM
$\mathbb{E}_{x,y}^{{\tilde{A}}{\tilde{B}}}=(\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}})_{a,b}$
for each $x$ and $y$ with elements
$\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}=\big{(}\mathsf{M}_{a|x}^{{\tilde{A}}A}\otimes\mathsf{M}_{b|y}^{{\tilde{B}}B}\big{)}*W^{AB}$,
similarly to Eq. (2). We will show that in this scenario, the causal
nonseparability of a D-POVM generated from a causally nonseparable process
matrix $W^{AB}$ can be certified solely based on the correlations
$P(c,a,b,d|z,x,y,w)$, thus achieving a NDI certification of causal
nonseparability.

Figure 2: From the NSDI-QI to the NDI scenario: In a NSDI-QI scenario (top),
the quantum inputs of the SDI-QI scenario (Fig. 1) are prepared remotely by
two other separated parties—Charlie and Daisy—performing suitable operations
on their part of a maximally entangled state shared with Alice and Bob, so
that $\rho_{z}^{A^{\prime}}\propto
M_{0|z}^{C^{\prime}}*\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{C^{\prime}}{A^{\prime}}}$
and $\rho_{w}^{B^{\prime}}\propto
M_{0|w}^{D^{\prime}}*\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{B^{\prime}}{D^{\prime}}}$
resp. (orange). These quantum inputs can be self-tested (bottom): Alice, Bob,
Charlie and Daisy are given classical inputs $x,y,z,w$, and perform untrusted
operations that produce outcomes $a,b,c,d$, resp. Verifying that the
correlations $P(c,a|z,x)$ and $P(b,d|y,w)$ both maximally violate a Bell
inequality allows one to self-test the preparation of the quantum inputs; from
this, one can certify the causal nonseparability of the D-POVM
$(\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}})_{a,b}$ (purple semicircle)
for $x=y=\star$, and thus the causal nonseparability of the process matrix
$W^{AB}$, in a NDI scenario.

NDI certification of causal nonseparability.—As suggested above, the idea to
achieve this goal is to combine the SDI-QI certification with self-testing.
For the protocol we present to function, it will be necessary for the quantum
inputs in the initial SDI-QI setting to be chosen such that that the reference
measurements
$(M_{c|z}^{C^{\prime}})_{c}=(\rho_{z}^{C^{\prime}},\mathbbm{1}^{C^{\prime}}-\rho_{z}^{C^{\prime}})$
and
$(M_{d|w}^{D^{\prime}})_{d}=(\rho_{w}^{D^{\prime}},\mathbbm{1}^{D^{\prime}}-\rho_{w}^{D^{\prime}})$
of the NSDI-QI setting are precisely those involved in the self-testing
procedure we use. For simplicity, let us assume that $\textit{{d}}=2^{m}$ for
some $m\geq 1$. This can be done without loss of generality by embedding the
reference experiment, i.e., the spaces $\mathcal{H}^{{C^{\prime}}}$ and
$\mathcal{H}^{{D^{\prime}}}$, into larger $2^{m}$-dimensional spaces if needed
(see Appendix A). We then require not only that $\rho_{z}^{C^{\prime}}$ and
$\rho_{w}^{D^{\prime}}$ form tomographically complete sets, but more precisely
that they are (projectors onto the) eigenstates of orthogonal Pauli
measurements (or, beyond the 2-dimensional case, tensor products thereof). We
will assume henceforth that this is indeed the case.

When Alice and Bob receive generic classical inputs $x,y$ (other than the
specific $x,y=\star$ that we shall consider below), we interpret the
experiment as a self-testing round, in which the correlations are used to test
a Bell inequality: reaching its maximal quantum violation certifies that the
untrusted states
$\uppsi_{1}^{{\tilde{C}}{\tilde{A}}},\uppsi_{2}^{{\tilde{B}}{\tilde{D}}}$ and
measurements
$(\mathsf{M}_{c|z}^{\tilde{C}})_{c},(\mathsf{M}_{d|w}^{\tilde{D}})_{d}$ have
the form of those used in the NSDI-QI setup. Such a statement can only be true
up to local transformations, but these transformations relating the
“physical”, uncharacterised, states and measurements to their idealised
“reference” counterparts can be explicitly incorporated into the analysis.
When they both receive the special input $x=y=\star$, we instead interpret it
as a certification round, in which one follows the same procedure as in a
NSDI-QI certification of causal nonseparability using these self-tested
quantum inputs. Crucially, the aforementioned local transformations preserve
causal (non)separability, allowing the causal nonseparability of the physical
process to be certified, despite lacking any knowledge of the Hilbert spaces
upon which it acts.

The general protocol can thus be described as follows.

  * •

(Correlation generation) Alice, Bob, Charlie and Daisy perform local
operations and measurements on their respective subsystems to obtain the
statistics $P(c,a,b,d|z,x,y,w)$.

  * •

(Self-testing) One certifies that the auxiliary states each contain a
maximally entangled state and that Charlie and Daisy each effectively perform
the desired tomographically complete, orthogonal Pauli measurements on their
subsystems by verifying that the distributions $P_{y}(c,a|z,x):=P(c,a|z,x,y)$
and $P_{x}(b,d|y,w):=P(b,d|x,y,w)$ both maximally violate a Bell inequality
$\mathcal{I}\geq 0$ for some $y$ and $x$, resp.555Note that these
distributions are independent of $w$ and $z$, respectively, due to the network
structure: e.g., Daisy only shares a (nonsignalling) quantum state with Alice,
Bob and Charlie. However, contrary to the scenario of entanglement
certification [16, 17], these may still depend on $y$ and $x$, resp., as the
process matrix may allow for signalling.

  * •

(Causal nonseparability certification) One verifies that the correlations
$P_{\star}(a,b|z,w):=P(a,b|z,x{=}\star,y{=}\star,w,c{=}0,d{=}0)$ violate a
witness inequality $\mathcal{J}\geq 0$, thus certifying that the process
matrix is causally nonseparable.

Let us now explain these steps in more detail. We use the fact that, for
$\textit{{d}}=2^{m}$, one can self-test a d-dimensional maximally entangled
state and orthogonal Pauli measurements (or tensor products thereof, for
$\textit{{d}}>2$) by the maximal quantum violation of a Bell inequality
$\mathcal{I}\geq 0$ [17]. Further discussion of the inequality is provided in
Appendix A, but its exact form (which depends on d) is not important for the
argument here.

The first step of the protocol simply consists in generating the correlations
$P(c,a,b,d|z,x,y,w)$, which can be written as

$\displaystyle P(c,a,b,d|z,x,y,w)$
$\displaystyle\\!=\\!\left(\\!\mathsf{M}_{c|z}^{\tilde{C}}\otimes\mathsf{M}_{a|x}^{{\tilde{A}}A}\otimes\mathsf{M}_{b|y}^{{\tilde{B}}B}\otimes\mathsf{M}_{d|w}^{\tilde{D}}\\!\right)\\!*\\!\left(\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}\\!\otimes\\!W^{AB}\\!\otimes\\!\uppsi_{2}^{{\tilde{B}}{\tilde{D}}}\right)$
$\displaystyle=P(c|z)\,P(d|w)\,\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}*\left(\uprho_{c|z}^{\tilde{A}}\otimes\uprho_{d|w}^{\tilde{B}}\right)$
(8)

with the remotely prepared states satisfying
$P(c|z)\,\uprho_{c|z}^{\tilde{A}}=\mathsf{M}_{c|z}^{\tilde{C}}*\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}$
and
$P(d|w)\,\uprho_{d|w}^{\tilde{B}}=\mathsf{M}_{d|w}^{\tilde{D}}*\uppsi_{2}^{{\tilde{B}}{\tilde{D}}}$,
and with the D-POVM elements
$\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}=\big{(}\mathsf{M}_{a|x}^{{\tilde{A}}A}\otimes\mathsf{M}_{b|y}^{{\tilde{B}}B}\big{)}*W^{AB}$,
as given above.

The second step is the self-testing procedure. The players verify that the
correlations $P_{y}(c,a|z,x)$ and $P_{x}(b,d|y,w)$ reach the maximum quantum
violation of the Bell inequality $\mathcal{I}\geq 0$. As we recall in Appendix
A, it is known that if this is the case, then the physical states
$\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}$ and
$\uppsi_{2}^{{\tilde{B}}{\tilde{D}}}$ and measurements
$(\mathsf{M}_{c|z}^{\tilde{C}})_{c}$ and $(\mathsf{M}_{d|w}^{\tilde{D}})_{d}$
can be related to the ideal reference states and operations of the NSDI-QI
scenario in the d-dimensional spaces
$\mathcal{H}^{C^{\prime}},\mathcal{H}^{A^{\prime}},\mathcal{H}^{B^{\prime}},\mathcal{H}^{D^{\prime}}$
via local isometries [16, 17]. We then show (see Appendix A) that these local
isometries also relate the remotely prepared inputs666In fact, we show that
each $\uprho_{c|z}^{\tilde{A}}$ and $\uprho_{d|w}^{\tilde{B}}$ (for all $c,d$)
can be related to a corresponding state in the reference scenario, but this
stronger result is not needed here. $\uprho_{0|z}^{\tilde{A}}$ and
$\uprho_{0|w}^{\tilde{B}}$ to the reference quantum inputs
$\rho_{z}^{A^{\prime}}\propto
M_{0|z}^{C^{\prime}}*\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{C^{\prime}}{A^{\prime}}}$
and $\rho_{w}^{B^{\prime}}\propto
M_{0|w}^{D^{\prime}}*\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{B^{\prime}}{D^{\prime}}}$.
For example, for Alice’s quantum inputs we prove that

$\uprho_{0|z}^{\tilde{A}}={V^{\mathbf{A}}}^{\dagger}\left(\rho_{z}^{A^{\prime}}\otimes\xi_{0}^{A^{\textup{jf}}}+(\rho_{z}^{A^{\prime}})^{T}\otimes\xi_{1}^{A^{\textup{jf}}}\right)V^{\mathbf{A}},$
(9)

where
$V^{\mathbf{A}}:\mathcal{H}^{\tilde{A}}\to\mathcal{H}^{{A^{\prime}}A^{\textup{jf}}}$
is an isometry and $\xi_{0}^{A^{\textup{jf}}}$, $\xi_{1}^{A^{\textup{jf}}}$
are (subnormalised) “junk and flag” states on a Hilbert space
$\mathcal{H}^{A^{\textup{jf}}}$; a similar relation holds for Bob. This allows
us to write Eq. (8), when $c=d=0$, in terms of the reference scenario as

$\displaystyle P(0,a,b,0|z,x,y,w)$
$\displaystyle\qquad=P(0|z)\,P(0|w)\,E_{a,b|x,y}^{{A^{\prime}}{B^{\prime}}}*\big{(}\rho_{z}^{A^{\prime}}\otimes\rho_{w}^{B^{\prime}}\big{)},$
(10)

where the $E_{a,b|x,y}^{{A^{\prime}}{B^{\prime}}}$ are elements of an
effective D-POVM (which $\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}$ can
likewise be related to through local isometries; see Eq. (23) in Appendix B).
Eq. (Network-Device-Independent Certification of Causal Nonseparability) thus
relates directly the observed correlations to those obtainable in the NSDI-QI
setup.

The last step consists in certifying causal nonseparability. The self-testing
step ensures that, when Charlie and Daisy get the outputs $c=d=0$, Alice and
Bob receive, up to local isometries, the desired quantum inputs
$\rho_{z}^{A^{\prime}}$ and $\rho_{w}^{B^{\prime}}$ from the SDI-QI scenario.
We thus focus on the correlations observed when $x=y=\star$ which, in the
reference experiment, should correspond to the measurements used to obtain a
causally non-separable D-POVM. These correlations, following Eq. (Network-
Device-Independent Certification of Causal Nonseparability), are given by

$\displaystyle P_{\star}(a,b|z,w)$
$\displaystyle=P(a,b|z,x{=}\star,y{=}\star,w,c{=}0,d{=}0)$
$\displaystyle=E_{a,b|\star,\star}^{{A^{\prime}}{B^{\prime}}}*\big{(}\rho_{z}^{A^{\prime}}\otimes\rho_{w}^{B^{\prime}}\big{)},$
(11)

which precisely defines, following Eq. (1), a distribution
$P^{\prime}(a,b|\rho_{z}^{A^{\prime}},\rho_{w}^{B^{\prime}})$ in a SDI-QI
setting with the reference quantum inputs for the certification of the
effective D-POVM
$\mathbb{E}_{\text{eff}}^{{A^{\prime}}{B^{\prime}}}:=(E_{a,b|\star,\star}^{{A^{\prime}}{B^{\prime}}})_{a,b}$.
In Appendix B we show that if $W^{AB}$ is causally separable then
$\mathbb{E}_{\text{eff}}^{{A^{\prime}}{B^{\prime}}}$ is a causally separable
D-POVM. Hence, if
$P^{\prime}(a,b|\rho_{z}^{A^{\prime}},\rho_{w}^{B^{\prime}})$ violates the
witness inequality, yielding $\mathcal{J}<0$, this certifies the causally
nonseparability of $W^{AB}$. By following the ideal reference protocol of the
NSDI-QI scenario, one can thus obtain such a NDI certification for any process
that can be certified in the bipartite SDI-QI scenario, i.e., which can
generate a causally nonseparable D-POVM.

The NDI certification of causal nonseparability described above assumes that
the self-testing is perfect: the maximal quantum violation of the Bell
inequality $\mathcal{I}\geq 0$ is obtained, meaning that noiseless maximally
entangled states and perfect measurements are realised (up to local
isometries). Such requirements are experimentally unobtainable. However, the
self-testing procedure and subsequent certification of causal nonseparability
can be rendered robust to noise in a similar way to that of [17]; see Appendix
C for further explanation.

Applications and scope.—The protocol described in the previous section
exploits a general approach to transforming a SDI-QI scenario into a NDI one.
Indeed, it can be readily generalised to provide NDI certification in settings
beyond the basic bipartite one we described. For instance, in [15] we showed
that all bipartite causally nonseparable process matrices can be certified in
a SDI-QI way if we assume an additional natural assumption on the form of
Alice’s and Bob’s instruments: that they factorise as a joint measurement on
one subsystem of their quantum input and the system they receive from the
process matrix, and a channel taking the rest of their quantum input into the
process matrix. By assuming the same “measurement device and channel
independent” (MDCI) structure for the physical instruments
$({\mathsf{M}}_{a|\star}^{{\tilde{A}}A})_{a}$ and
$({\mathsf{M}}_{b|\star}^{{\tilde{B}}B})_{b}$, and using independent auxiliary
sources to self-test Alice and Bob’s quantum input subsystems, our method here
can readily be used to show that all bipartite causally nonseparable processes
can also be certified in a NDI way (modulo these assumptions); see Appendix D
for details on the soundness of the NDI certification in this MDCI scenario.

A process of particular interest is the quantum switch [3], which can be
described in either a “($2$+$F$)-partite” or “($P$+$2$+$F$)-partite scenario”
in which, in addition to Alice and Bob, a third party Fiona with no output
Hilbert space performs a measurement in their causal future, and (in the
($P$+$2$+$F$)-partite scenario), a fourth party Phil with no input Hilbert
space acts in their causal past. In the quantum switch, Phil prepares a target
system and a control qubit, the latter of which coherently controls the order
that Alice and Bob then act on the target system, before both the target and
control are received by Fiona. (In the ($2$+$F$)-partite scenario, Phil’s
operations are fixed and absorbed into the process matrix.) The quantum switch
is the canonical example of a physically realisable causally nonseparable
process [13], and the resource behind most known advantages arising from
causal indefiniteness [3, 4, 5, 6, 7, 8]. Certifying its causal
nonseparability is thus a key problem, and several experiments have done this
in a device-dependent way using causal witnesses [27, 28]. In [15], the SDI-QI
scenario was extended to the ($2$+$F$)-scenario to certify the causal
nonseparability of the quantum switch, and the generality of our NDI approach
can again be illustrated in these scenarios. In Appendix E we further extend
the SDI-QI framework to the ($P$+$2$+$F$)-partite scenario—in which the SDI-QI
approach can then be transformed into a NDI one by having Phil and Fiona also
share maximally entangled states with additional parties, and following the
natural generalisation of our bipartite protocol. In fact, for the quantum
switch, in the SDI-QI setting it is not necessary to provide all parties with
a quantum input: some parties may do with classical inputs, or no input at
all. For example, in [15] the case where only Alice and Bob have quantum
inputs was considered; in Appendix E we show that one can also certify the
causal nonseparability of the quantum switch in a SDI-QI manner while
providing only Phil with quantum inputs—a scenario that can then be turned
into a NDI one by introducing just one additional party. We thus see that,
despite its inability to violate any causal inequality [10, 11], the quantum
switch can be certified in a particular type of DI scenario; see also the
discussion below.

Discussion.—As we emphasised, our certification of causal nonseparability is
device-independent, in the sense that it does not assume any knowledge about
the underlying process or operations involved. Every element involved in the
protocol is treated as a (quantum) black box with classical inputs and
outputs, and all conclusions are drawn from the observed correlations.
However, we called it network-device-independent to differentiate it from the
standard approach to DI certification, which requires the violation of a
causal inequality [1]. Indeed, our approach differs from the standard one as
it assumes a network structure involving two separated untrusted parties, each
independently performing operations on a system that is independently shared
with a party involved in the process being certified. Moreover, unlike the
standard approach which certifies causal indefiniteness in a theory-
independent manner, our method certifies the theory-dependent notion of causal
nonseparability. Indeed, it is based on the violation of a witness inequality
by a causally nonseparable process matrix and on the self-testing of quantum
input states through the maximal quantum violation of Bell inequalities. It
thus assumes the validity of quantum theory and of its extension through the
process matrix formalism. This, and the simple network structural assumption,
is what allows one to certify—without trusting any device—the causal
nonseparability of process matrices which cannot violate a causal inequality,
such as the quantum switch [10, 11].

Finally, we note that some recent works have also proposed another approach to
certify the indefinite causal order in the quantum switch in a device-
independent way [29, 30]. In the scenario presented in [30] (also considered
in [29]), an additional space-like separated party is introduced along with
Alice, Bob and Fiona, and an inequality is derived based on the assumptions of
“Definite causal order, Relativistic causality, and Free interventions” (DRF).
It is then shown that the correlations generated from a quantum switch in
which the control system is entangled with the newly introduced party can
violate this “DRF inequality”. Notably, our approach allows us to consider a
scenario very close to that of [30] by transforming a SDI-QI test of a
($P$+$2$+$F$)-partite quantum switch, with only Phil (in the causal past)
being given a quantum input, into a NDI test (as briefly considered above, and
further discussed in Appendix E). It is then interesting to compare the
assumptions and conclusions one may draw from these two approaches in which
the causal indefiniteness of the quantum switch can be certified in a DI
manner (see Appendix F for further discussion). A first important difference
is that the approach of [30], and the inequality they derive, is not only
device-independent, but also theory-independent—like standard causal
inequalities, but contrary to our approach. Furthermore, while [30] explicitly
introduced some form of no-signalling assumption at the ontological level
(“Relativistic causality”) between the new party and those involved in the
switch, we did not need such an assumption. Instead, we make the simple
network structure assumption which implies just the weaker operational notion
of no-signalling. With a violation of a “DRF inequality”, the conclusion that
there is indefinite causal order can hence be escaped by rejecting this
relativistic causality assumption; in our case, one would need to reject
entirely the network structure assumption to salvage definite causal order
(or, in both cases, one could reject the “Free interventions assumption”,
which we indeed also made implicitly). It will be insightful to further
investigate the connections between the two approaches, possibly in different
scenarios. Recall indeed that our approach allows one to turn any SDI-QI
certification of causal nonseparability into a NDI one; it would be
interesting to see if the approach of [30] can similarly be made more
systematic.

Acknowledgements.—We thank Jean-Daniel Bancal, Nicolas Brunner and Tein van
der Lugt for enlightening discussions, and acknowledge financial support from
the EU NextGen Funds, the Government of Spain (FIS2020-TRANQI and Severo Ochoa
CEX2019-000910-S), Fundació Cellex, Fundació Mir-Puig, Generalitat de
Catalunya (CERCA program), the PEPR integrated project EPiQ ANR-22-PETQ-0007
part of Plan France 2030, the ERC Starting grant QUSCO, and the Agence
Nationale de la Recherche (project ANR-22-CE47-0012).

For the purpose of open access, the authors have applied a CC-BY public
copyright licence to any Author Accepted Manuscript (AAM) version arising from
this submission.

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oneΔ

## Appendix

##  Appendix A Self-testing the quantum inputs

In this appendix we begin by recalling briefly the framework of self-testing
[25, 26], before considering in more detail the self-testing of maximally
entangled states and tomographically complete sets of orthogonal Pauli
measurements, and how this can be used to self-test the remotely prepared
quantum inputs in the NDI scenario.

We will focus here on Charlie and Alice, who share a quantum state
$\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}$ and perform local measurements on their
share of the state described by POVMs $({\mathsf{M}}_{c|z}^{{\tilde{C}}})_{c}$
and $({\mathsf{M}}_{a|x}^{{\tilde{A}}})_{a}$, respectively. The self-testing
results for Bob and Daisy can be derived in an entirely analogous way.

The general goal of self-testing is to use the observed correlations
$P(c,a|z,x)$ to show that the state $\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}$
and/or the measurements $({\mathsf{M}}_{c|z}^{{\tilde{C}}})_{c}$,
$({\mathsf{M}}_{a|x}^{{\tilde{A}}})_{a}$ must be of a specific, desired form.
More precisely, the observed correlations are given by the Born rule as777Note
that as we work here with the Choi representation, the measurement operators
are transposed with respect to those that one would usually write (e.g., the
Choi representation for a projector onto a state $\mathinner{|{\phi}\rangle}$
is $(\mathinner{|{\phi}\rangle\\!\langle{\phi}|})^{T}$). This also explains
the presence of transposes on $({\mathsf{M}}_{c|z}^{\tilde{C}})^{T}$ and
$(M_{c|z}^{C^{\prime}})^{T}$ in the self-testing statement of Eq. (16) and in
the subsequent equations below, compared to the notations used in the
literature [16, 17].

$P(c,a|z,x)=({\mathsf{M}}_{c|z}^{{\tilde{C}}}\otimes{\mathsf{M}}_{a|x}^{{\tilde{A}}})*\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}=\operatorname{Tr}\big{[}({\mathsf{M}}_{c|z}^{{\tilde{C}}}\otimes{\mathsf{M}}_{a|x}^{{\tilde{A}}})^{T}\,\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}\big{]},\qquad\forall
z,x,c,a,$ (12)

in a “physical” experiment described by
$(\uppsi_{1}^{{\tilde{C}}{\tilde{A}}},({\mathsf{M}}_{c|z}^{{\tilde{C}}})_{c},({\mathsf{M}}_{a|x}^{{\tilde{A}}})_{a})$,
which we would typically like to relate (in part or wholly) to a “reference”
experiment
$(\mathinner{|{\psi_{1}}\rangle\\!\langle{\psi_{1}}|}^{{C^{\prime}}{A^{\prime}}},(M_{c|z}^{{C^{\prime}}})_{c},(M_{a|x}^{{A^{\prime}}})_{a})$.
In general, one cannot identify a unique reference experiment compatible with
the observed correlations, but only an equivalence class of such experiments.
Indeed, there are two classes of transformations of quantum states and
measurements which do not change the observed correlations. The first are
local isometries, encompassing local rotations and embeddings in Hilbert
spaces of higher dimension. The other is a simultaneous transposition of the
underlying quantum state and measurements. We thus seek to show a relation
between the physical and reference experiments up to such local isometries and
transpositions.

### Self-testing the maximally entangled state and Pauli measurements

For simplicity, let us focus here on the case of $\textit{{d}}=2$ where the
quantum inputs are two-dimensional. We will discuss at the end of the section
how the self-testing procedure can be generalised to arbitrary
$\textit{{d}}>2$.

In order to self-test the remotely prepared quantum inputs, we will first show
how Alice and Charlie can self-test the fact that they effectively share a
maximally entangled state
$\mathinner{|{\phi^{+}}\rangle}^{{C^{\prime}}{A^{\prime}}}=(\mathinner{|{00}\rangle}+\mathinner{|{11}\rangle})/\sqrt{2}$
and that Charlie’s reference measurements $(M_{c|z}^{{C^{\prime}}})_{c}$
effectively correspond to the tomographically complete Pauli measurements
$\sigma_{x},\sigma_{y},\sigma_{z}$.

Although the physical state $\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}$ that Charlie
and Alice share may not be pure, it will be convenient to work with a
purification $\mathinner{|{\uppsi_{1}}\rangle}^{{\tilde{C}}{\tilde{A}}R}$ of
$\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}$, where $\mathcal{H}^{R}$ is a
purification system that the measurements
$({\mathsf{M}}_{c|z}^{{\tilde{C}}})_{c}$ and
$({\mathsf{M}}_{a|x}^{{\tilde{A}}})_{a}$ act, by definition, trivially upon.
Since no operations act nontrivially on this space, we will generally, in a
slight abuse of notation, omit writing explicitly this space, writing simply
$\mathinner{|{\uppsi_{1}}\rangle}^{{\tilde{C}}{\tilde{A}}}$ and leave
identities acting on $\mathcal{H}^{R}$ implicit. For simplicity, we will also
assume in what follows that the physical measurements
$({\mathsf{M}}_{c|z}^{{\tilde{C}}})_{c}$ and
$({\mathsf{M}}_{a|x}^{{\tilde{A}}})_{a}$ are projective. Although common in
self-testing proofs (and justifiable on reasonable grounds [26]), this
assumption is not a fundamental one in our NDI protocol, and could be removed
using the techniques of [31] to _derive_ this projectivity from the maximal
violation of a Bell inequality.

The self-testing of a maximally entangled state and the three mutually
orthogonal Pauli measurements can be obtained using an extended form of the
CHSH inequality, as shown in Lemma 1 of [16, 17]. Here we simply restate the
elements of the results necessary for what follows, directing the interested
reader to the proofs in [16, 17].

Let us consider an experiment involving three measurements for Charlie and six
for Alice, labelled $z=1,\dots,3$ and $x=1,\dots,6$ respectively, all of which
are binary valued, and denote $\mathcal{E}_{z,x}=P(c=a|z,x)-P(c\neq a|z,x)$
the expectation value of the correlation between the outputs $c$ and $a$ for
inputs $z$ and $x$. Starting from the extended CHSH inequality

$\mathcal{E}:=\mathcal{E}_{1,1}+\mathcal{E}_{1,2}+\mathcal{E}_{2,1}-\mathcal{E}_{2,2}+\mathcal{E}_{1,3}+\mathcal{E}_{1,4}-\mathcal{E}_{3,3}+\mathcal{E}_{3,4}+\mathcal{E}_{2,5}+\mathcal{E}_{2,6}-\mathcal{E}_{3,5}+\mathcal{E}_{3,6}\leq
6$ (13)

obtained by the sum of three CHSH inequalities, each with a local bound of 2,
let us consider the Bell inequality $\mathcal{I}=6-\mathcal{E}\geq 0$. In the
experiment we consider (described by quantum theory), the correlators
$\mathcal{E}_{z,x}$ are obtained as

$\displaystyle\mathcal{E}_{z,x}$
$\displaystyle=(\mathsf{M}_{0|z}^{{\tilde{C}}}\otimes\mathsf{M}_{0|x}^{{\tilde{A}}})*\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}+(\mathsf{M}_{1|z}^{{\tilde{C}}}\otimes\mathsf{M}_{1|x}^{{\tilde{A}}})*\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}-(\mathsf{M}_{0|z}^{{\tilde{C}}}\otimes\mathsf{M}_{1|x}^{{\tilde{A}}})*\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}-(\mathsf{M}_{1|z}^{{\tilde{C}}}\otimes\mathsf{M}_{0|x}^{{\tilde{A}}})*\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}.$
(14)

Since $\mathcal{E}$ is the sum of three CHSH inequalities, its maximum quantum
value is $\mathcal{E}_{Q}=6\sqrt{2}$, and the maximum quantum violation of
$\mathcal{I}\geq 0$ is $\mathcal{I}_{Q}=6-6\sqrt{2}<0$. If this value is
reached by Alice and Charlie’s correlations $P_{y}(c,a|z,x)$ (for some input
$y$ of Bob) in their physical experiment, then by Lemma 1 of Refs. [16, 17]
there exist local isometries
$V^{\mathbf{C}}:\mathcal{H}^{\tilde{C}}\to\mathcal{H}^{{C^{\prime}}C^{\textup{j}}C^{\textup{f}}}$
and
$V^{\mathbf{A}}:\mathcal{H}^{\tilde{A}}\to\mathcal{H}^{{A^{\prime}}A^{\textup{j}}A^{\textup{f}}}$,
where $C^{\textup{j}}$ and $A^{\textup{j}}$ are “junk” systems and
$C^{\textup{f}}$ and $A^{\textup{f}}$ are (2-dimensional) “flag” systems, such
that

$\displaystyle V^{\mathbf{C}}\otimes
V^{\mathbf{A}}\mathinner{|{\uppsi_{1}}\rangle}^{{\tilde{C}}{\tilde{A}}}$
$\displaystyle=\mathinner{|{\phi^{+}}\rangle}^{{C^{\prime}}{A^{\prime}}}\otimes\mathinner{|{\xi_{0}}\rangle}^{C^{\textup{j}}A^{\textup{j}}}\otimes\mathinner{|{00}\rangle}^{C^{\textup{f}}A^{\textup{f}}}+\mathinner{|{\phi^{+}}\rangle}^{{C^{\prime}}{A^{\prime}}}\otimes\mathinner{|{\xi_{1}}\rangle}^{C^{\textup{j}}A^{\textup{j}}}\otimes\mathinner{|{11}\rangle}^{C^{\textup{f}}A^{\textup{f}}}$
(15)

and

$\displaystyle V^{\mathbf{C}}\otimes
V^{\mathbf{A}}\left(({\mathsf{M}}_{c|z}^{\tilde{C}})^{T}\mathinner{|{\uppsi_{1}}\rangle}^{{\tilde{C}}{\tilde{A}}}\right)$
$\displaystyle=\left((M_{c|z}^{C^{\prime}})^{T}\mathinner{|{\phi^{+}}\rangle}^{{C^{\prime}}{A^{\prime}}}\right)\otimes\mathinner{|{\xi_{0}}\rangle}^{C^{\textup{j}}A^{\textup{j}}}\otimes\mathinner{|{00}\rangle}^{C^{\textup{f}}A^{\textup{f}}}+\left(M_{c|z}^{C^{\prime}}\mathinner{|{\phi^{+}}\rangle}^{{C^{\prime}}{A^{\prime}}}\right)\otimes\mathinner{|{\xi_{1}}\rangle}^{C^{\textup{j}}A^{\textup{j}}}\otimes\mathinner{|{11}\rangle}^{C^{\textup{f}}A^{\textup{f}}},$
(16)

where we omit identity operators acting on Alice’s side, the junk states
$\mathinner{|{\xi_{0}}\rangle}^{C^{\textup{j}}A^{\textup{j}}}$ and
$\mathinner{|{\xi_{1}}\rangle}^{C^{\textup{j}}A^{\textup{j}}}$ satisfy
$\|\mathinner{|{\xi_{0}}\rangle}\|^{2}+\|\mathinner{|{\xi_{1}}\rangle}\|^{2}=1$,
and Charlie’s reference measurements $M_{c|z}^{C^{\prime}}$ are precisely the
tomographically complete set of projectors

$(M_{c|1}^{C^{\prime}})^{T}=\tfrac{1}{2}(\mathbbm{1}+(-1)^{c}\sigma_{Z}),\quad(M_{c|2}^{C^{\prime}})^{T}=\tfrac{1}{2}(\mathbbm{1}+(-1)^{c}\sigma_{X}),\quad(M_{c|3}^{C^{\prime}})^{T}=\tfrac{1}{2}(\mathbbm{1}+(-1)^{c}\sigma_{Y}).$
(17)

From this self-testing result, we can determine how the remotely prepared
inputs
$\uprho_{c|z}^{{\tilde{A}}}\propto{\mathsf{M}}_{c|z}^{{\tilde{A}}}*\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}$
relate to the corresponding reference quantum inputs
$\rho_{c|z}^{{A^{\prime}}}\propto
M_{c|z}^{{C^{\prime}}}*\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{C^{\prime}}{A^{\prime}}}$,
which here are precisely the tomographically complete states
$\rho_{c|z}^{{A^{\prime}}}=(M_{c|z}^{C^{\prime}})^{T}$ specified in Eq. (17).
Multiplying Eqs. (15) and (16) by
${V^{\mathbf{C}}}^{\dagger}\otimes{V^{\mathbf{A}}}^{\dagger}$, and recalling
that
$\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}=\operatorname{Tr}_{R}\mathinner{|{\uppsi_{1}}\rangle\\!\langle{\uppsi_{1}}|}^{{\tilde{C}}{\tilde{A}}R}$
(with the implicit purification space made explicit here, but left implicit
below on both the state and in the partial trace), it follows that (again with
some implicit identity operators)

$\displaystyle{\mathsf{M}}_{c|z}^{\tilde{C}}*\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}$
$\displaystyle=\operatorname{Tr}_{\tilde{C}}\left(({\mathsf{M}}_{c|z}^{\tilde{C}})^{T}\mathinner{|{\uppsi_{1}}\rangle\\!\langle{\uppsi_{1}}|}^{{\tilde{C}}{\tilde{A}}}\right)$
$\displaystyle={V^{\mathbf{A}}}^{\dagger}\\!\left(\operatorname{Tr}_{\tilde{C}}\left[{V^{\mathbf{C}}}^{\dagger}\left(\sum_{k,l=0,1}\left((M_{c|z}^{C^{\prime}})^{T^{\bar{k}}}\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{C^{\prime}}{A^{\prime}}}\right)\otimes\mathinner{|{\xi_{k}}\rangle\\!\langle{\xi_{l}}|}^{C^{\textup{j}}A^{\textup{j}}}\otimes\mathinner{|{kk}\rangle\\!\langle{ll}|}^{C^{\textup{f}}A^{\textup{f}}}\right)V^{\mathbf{C}}\right]\right)\\!V^{\mathbf{A}}$
$\displaystyle={V^{\mathbf{A}}}^{\dagger}\\!\left(\\!\operatorname{Tr}_{{C^{\prime}}C^{\textup{j}}C^{\textup{f}}}\\!\left[V^{\mathbf{C}}{V^{\mathbf{C}}}^{\dagger}\\!\left(\sum_{k,l=0,1}\\!\left((M_{c|z}^{C^{\prime}})^{T^{\bar{k}}}\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{C^{\prime}}\\!{A^{\prime}}}\right)\otimes\mathinner{|{\xi_{k}}\rangle\\!\langle{\xi_{l}}|}^{C^{\textup{j}}\\!A^{\textup{j}}}\otimes\mathinner{|{kk}\rangle\\!\langle{ll}|}^{C^{\textup{f}}\\!A^{\textup{f}}}\right)\right]\right)\\!V^{\mathbf{A}},$
(18)

where $(\cdot)^{T^{\bar{k}}}$ denotes the “transpose to the power
$\bar{k}:=1-k$”. Noting now that $V^{\mathbf{C}}{V^{\mathbf{C}}}^{\dagger}$
effectively acts as the identity on the state of Eq. (16), it can be removed
from the previous equation. The partial trace then kills the cross terms with
$k\neq l$ and we obtain

$\displaystyle
P(c|z)\,\uprho_{c|z}^{\tilde{A}}={\mathsf{M}}_{c|z}^{\tilde{C}}*\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}=P(c|z)\,{V^{\mathbf{A}}}^{\dagger}\left(\sum_{k=0,1}(\rho_{c|z}^{A^{\prime}})^{T^{k}}\otimes\xi_{k}^{A^{\textup{jf}}}\right)V^{\mathbf{A}},$
(19)

with $A^{\textup{jf}}=A^{\textup{j}}A^{\textup{f}}$,
$\xi_{k}^{A^{\textup{jf}}}=\operatorname{Tr}_{C^{\textup{j}}}\mathinner{|{\xi_{k}}\rangle\\!\langle{\xi_{k}}|}^{C^{\textup{j}}A^{\textup{j}}}\otimes\mathinner{|{k}\rangle\\!\langle{k}|}^{A^{\textup{f}}}$
such that $\sum_{k}\operatorname{Tr}(\xi_{k}^{A^{\textup{jf}}})=1$, and
$P(c|z)\,\rho_{c|z}^{A^{\prime}}=\operatorname{Tr}_{{C^{\prime}}}\left((M_{c|z}^{C^{\prime}})^{T}\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{C^{\prime}}{A^{\prime}}}\right)=M_{c|z}^{C^{\prime}}*\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{C^{\prime}}{A^{\prime}}}$.888One
may note that up to a proportionality constant, $\rho_{c|z}^{A^{\prime}}$ is
formally the same matrix as $M_{c|z}^{C^{\prime}}$, in the space
${A^{\prime}}$ rather than ${C^{\prime}}$. (This can indeed be interpreted as
some form of quantum teleportation from system ${C^{\prime}}$ to system
${A^{\prime}}$.)

By the symmetry of the considered network scenario (cf. Eq. (8) and Fig. 2),
we obtain a similar self-testing statement for Bob and Daisy: provided that
their correlations $P_{x}(b,d|y,w)$ also violate the Bell inequality Eq. (13)
up to its quantum maximum (for some input $x$ of Alice), there exists a local
isometry of the form
$V^{\mathbf{B}}:\mathcal{H}^{\tilde{B}}\to\mathcal{H}^{{B^{\prime}}B^{\textup{jf}}}$
such that we can write

$\displaystyle
P(d|w)\,\uprho_{d|w}^{\tilde{B}}={\mathsf{M}}_{d|w}^{\tilde{D}}*\uppsi_{2}^{{\tilde{B}}{\tilde{D}}}=P(d|w)\,{V^{\mathbf{B}}}^{\dagger}\left(\sum_{l=0,1}(\rho_{d|w}^{B^{\prime}})^{T^{l}}\otimes\zeta_{l}^{B^{\textup{jf}}}\right)V^{\mathbf{B}},$
(20)

with $\sum_{l}\operatorname{Tr}(\zeta_{l}^{B^{\textup{jf}}})=1$ and
$P(d|w)\,\rho_{d|w}^{B^{\prime}}=\operatorname{Tr}_{{D^{\prime}}}\left((M_{d|w}^{D^{\prime}})^{T}\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{B^{\prime}}{D^{\prime}}}\right)=M_{d|w}^{D^{\prime}}*\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{B^{\prime}}{D^{\prime}}}$.

Thus, we have shown that both remotely prepared inputs
$\uprho_{c|z}^{\tilde{A}}\propto{\mathsf{M}}_{c|z}^{\tilde{C}}*\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}$
and
$\uprho_{d|w}^{\tilde{B}}\propto{\mathsf{M}}_{d|w}^{\tilde{D}}*\uppsi_{2}^{{\tilde{B}}{\tilde{D}}}$
can be related to the reference quantum inputs $\rho_{c|z}^{A^{\prime}}\propto
M_{c|z}^{C^{\prime}}*\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{C^{\prime}}{A^{\prime}}}$
and $\rho_{d|w}^{B^{\prime}}\propto
M_{d|w}^{D^{\prime}}*\mathinner{|{\phi^{+}}\rangle\\!\langle{\phi^{+}}|}^{{B^{\prime}}{D^{\prime}}}$
via local isometries. That is, we are able to turn the self-test of the
maximally entangled state that Charlie and Alice share, along with Charlie’s
tomographically complete set of orthogonal Pauli measurements, into a self-
test of the quantum inputs used in the reference NSDI-QI certification
protocol (and likewise for Bob and Daisy). Note that in the NDI protocol
presented in the main text, we assumed that Charlie’s (tomographically
complete) quantum inputs in the reference scenario are obtained as the
remotely prepared states when $c=0$ for the different inputs $z$, whereas here
the $c=0$ outcomes only give one projector for each measurement, which is not
quite a tomographically complete set. This detail can be overcome by
considering some additional measurements in the NDI protocol obtained by
relabelling some of the above measurements’ outcomes, so as to complete the
set of reference quantum inputs (and likewise for Daisy’s measurements).

Finally, while the above approach, presented explicitly for the case of two-
dimensional quantum inputs (i.e., $\textit{{d}}=2$), it can be readily
generalised to $\textit{{d}}=2^{m}$, for $m>1$. Indeed, the maximally
entangled state
$\mathinner{|{\phi^{+}_{\textit{{d}}}}\rangle}^{{C^{\prime}}{A^{\prime}}}=\frac{1}{\sqrt{\textit{{d}}}}\sum_{j=0}^{\textit{{d}}-1}\mathinner{|{j,j}\rangle}$,
for $\textit{{d}}=2^{m}$, can be self-tested through the parallel repetition
of the extended CHSH inequality (13) [17]. In this case, Charlie has $3^{m}$
measurement settings corresponding to the nontrivial tensor products of
orthogonal single-qubit Pauli measurements, each of which has $2^{m}$ outcomes
so that the $M_{c|z}^{C^{\prime}}$ one self-tests are precisely the projectors
onto the $6^{m}$ eigenstates of $m$-fold Pauli measurements. As in the
$\textit{{d}}=2$ case, the quantum inputs prepared on the $c=0$ outcomes can
be extended to a tomographically complete set by adding some additional
measurements with relabelled outcomes.

For $\textit{{d}}\,\neq 2^{m}$, the above procedure can again be utilised by
first embedding the reference scenario into some
$\textit{{d}}\,^{\prime}$-dimensional spaces for some
$\textit{{d}}\,^{\prime}=2^{m}>\textit{{d}}$. By writing the D-POVM
$\mathbb{E}^{{A^{\prime}}{B^{\prime}}}$ in these
$\textit{{d}}\,^{\prime}$-dimensional spaces (and taking
$\mathcal{H}^{{C^{\prime}}}$ and $\mathcal{H}^{{D^{\prime}}}$ to hence also be
$\textit{{d}}\,^{\prime}$-dimensional) one can then proceed using the general
approach described above for $\textit{{d}}\,^{\prime}=2^{m}$ before
restricting ourselves to quantum inputs in the d-dimensional subspace of
interest.

##  Appendix B Certifying causal nonseparability in a network-device-
independent manner

The self-testing statements above allow us to directly relate the observed
correlations to those of the ideal reference systems in the NSDI-QI setup. To
see this, let us define for convenience
$\mathinner{|{{V^{\mathbf{A}}}^{\dagger}}\rangle\\!\rangle}^{{A^{\prime}}A^{\textup{jf}}{\tilde{A}}}\coloneqq\sum_{i}\mathinner{|{i}\rangle}^{{A^{\prime}}A^{\textup{jf}}}\otimes{V^{\mathbf{A}}}^{\dagger}\mathinner{|{i}\rangle}^{{A^{\prime}}A^{\textup{jf}}}$,
where $\\{\mathinner{|{i}\rangle}^{{A^{\prime}}A^{\textup{jf}}}\\}_{i}$ is the
computational basis of $\mathcal{H}^{{A^{\prime}}A^{\textup{jf}}}$. With these
notations one can write, using the link product,

$\displaystyle{V^{\mathbf{A}}}^{\dagger}\big{[}(\rho_{c|z}^{A^{\prime}})^{T^{k}}\otimes\xi_{k}^{A^{\textup{jf}}}\big{]}V^{\mathbf{A}}$
$\displaystyle=\mathinner{|{{V^{\mathbf{A}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{A}}}^{\dagger}}|}^{{A^{\prime}}A^{\textup{jf}}{\tilde{A}}}*\big{[}(\rho_{c|z}^{A^{\prime}})^{T^{k}}\otimes\xi_{k}^{A^{\textup{jf}}}\big{]}=\Big{(}\mathinner{|{{V^{\mathbf{A}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{A}}}^{\dagger}}|}^{{A^{\prime}}A^{\textup{jf}}{\tilde{A}}}*\xi_{k}^{A^{\textup{jf}}}\Big{)}^{T_{{A^{\prime}}}^{k}}*\rho_{c|z}^{A^{\prime}},$
(21)

and similarly for
${V^{\mathbf{B}}}^{\dagger}\big{[}(\rho_{d|w}^{B^{\prime}})^{T^{l}}\otimes\zeta_{l}^{B^{\textup{jf}}}\big{]}V^{\mathbf{B}}$,
introducing
$\mathinner{|{{V^{\mathbf{B}}}^{\dagger}}\rangle\\!\rangle}^{{B^{\prime}}B^{\textup{jf}}{\tilde{B}}}$
in a similar way to
$\mathinner{|{{V^{\mathbf{A}}}^{\dagger}}\rangle\\!\rangle}^{{A^{\prime}}A^{\textup{jf}}{\tilde{A}}}$
above. From Eqs. (19) and (20), the correlations $P(c,a,b,d|z,x,y,w)$ can then
be written as

$\displaystyle
P(c,a,b,d|z,x,y,w)=\left({\mathsf{M}}_{c|z}^{\tilde{C}}\otimes{\mathsf{M}}_{a|x}^{{\tilde{A}}A}\otimes{\mathsf{M}}_{b|y}^{{\tilde{B}}B}\otimes{\mathsf{M}}_{d|w}^{\tilde{D}}\right)*\left(\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}\otimes
W^{AB}\otimes\uppsi_{2}^{{\tilde{B}}{\tilde{D}}}\right)$
$\displaystyle\quad=P(c|z)\,P(d|w)\left({\mathsf{M}}_{a|x}^{{\tilde{A}}A}\otimes{\mathsf{M}}_{b|y}^{{\tilde{B}}B}\right)*W^{AB}*\Bigg{[}\Bigg{(}\sum_{k=0,1}{V^{\mathbf{A}}}^{\dagger}\big{[}(\rho_{c|z}^{A^{\prime}})^{T^{k}}\otimes\xi_{k}^{A^{\textup{jf}}}\big{]}V^{\mathbf{A}}\Bigg{)}\otimes\Bigg{(}\sum_{l=0,1}{V^{\mathbf{B}}}^{\dagger}\big{[}(\rho_{d|w}^{B^{\prime}})^{T^{l}}\otimes\zeta_{l}^{B^{\textup{jf}}}\big{]}V^{\mathbf{B}}\Bigg{)}\Bigg{]}$
$\displaystyle\quad=P(c|z)\,P(d|w)\,\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}*\\!\Bigg{[}\\!\Bigg{(}\\!\sum_{k=0,1}\\!\\!\Big{(}\\!\mathinner{|{{V^{\mathbf{A}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{A}}}^{\dagger}}|}^{{A^{\prime}}A^{\textup{jf}}{\tilde{A}}}\\!*\xi_{k}^{A^{\textup{jf}}}\Big{)}^{\\!T_{{A^{\prime}}}^{k}}\\!\\!*\rho_{c|z}^{A^{\prime}}\\!\Bigg{)}\\!\otimes\\!\Bigg{(}\\!\sum_{l=0,1}\\!\\!\Big{(}\\!\mathinner{|{{V^{\mathbf{B}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}}}^{\dagger}}|}^{{B^{\prime}}B^{\textup{jf}}{\tilde{B}}}\\!*\zeta_{l}^{B^{\textup{jf}}}\Big{)}^{\\!T_{{B^{\prime}}}^{l}}\\!\\!*\rho_{d|w}^{B^{\prime}}\\!\Bigg{)}\\!\Bigg{]}\\!$
$\displaystyle\quad=P(c|z)\,P(d|w)\
E_{a,b|x,y}^{{A^{\prime}}{B^{\prime}}}*\left(\rho_{c|z}^{A^{\prime}}\otimes\rho_{d|w}^{B^{\prime}}\right)$
(22)

with the “physical” D-POVM elements
$\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}=\big{(}\mathsf{M}_{a|x}^{{\tilde{A}}A}\otimes\mathsf{M}_{b|y}^{{\tilde{B}}B}\big{)}*W^{AB}$
and the new D-POVM elements effectively acting on the “reference” quantum
input states

$\displaystyle E_{a,b|x,y}^{{A^{\prime}}{B^{\prime}}}$
$\displaystyle\coloneqq\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}*\Bigg{[}\sum_{k=0,1}\left(\mathinner{|{{V^{\mathbf{A}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{A}}}^{\dagger}}|}^{{A^{\prime}}A^{\textup{jf}}{\tilde{A}}}*\xi_{k}^{A^{\textup{jf}}}\right)^{T_{{A^{\prime}}}^{k}}\otimes\sum_{l=0,1}\left(\mathinner{|{{V^{\mathbf{B}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}}}^{\dagger}}|}^{{B^{\prime}}B^{\textup{jf}}{\tilde{B}}}*\zeta_{l}^{B^{\textup{jf}}}\right)^{T_{{B^{\prime}}}^{l}}\Bigg{]}.$
(23)

The correlations (cf. Eq. (11))

$P_{\star}(a,b|z,w)=P(a,b|z,x{=}\star,y{=}\star,w,c{=}0,d{=}0)=E_{a,b|\star,\star}^{{A^{\prime}}{B^{\prime}}}*\left(\rho_{z}^{A^{\prime}}\otimes\rho_{w}^{B^{\prime}}\right)=P^{\prime}(a,b|\rho_{z}^{A^{\prime}},\rho_{w}^{B^{\prime}})$
(24)

thus describe precisely the correlations in a SDI-QI setting with the
reference quantum inputs $\rho_{z}^{A^{\prime}}$ and $\rho_{w}^{B^{\prime}}$
and the effective D-POVM
$\mathbb{E}_{\text{eff}}^{{A^{\prime}}{B^{\prime}}}=(E_{a,b|*,*}^{{A^{\prime}}{B^{\prime}}})_{a,b}$
involving the D-POVM elements defined in Eq. (23).

We now show that the local transformations used in this self-testing procedure
preserve causal separability, thus allowing us to certify the causal
nonseparability of the physical D-POVM
$\mathbb{E}^{{\tilde{A}}{\tilde{B}}}=(\mathsf{E}_{a,b|*,*}^{{\tilde{A}}{\tilde{B}}})_{a,b}$—and
hence the underlying physical process $W^{AB}$—despite lacking any knowledge
of the Hilbert spaces upon which it acts.

Let us for that assume that the process matrix is compatible with the causal
order $A\prec B$. Such a process matrix takes the form $W=W^{A\prec
B_{I}}\otimes\mathbbm{1}^{B_{O}}$, with $\operatorname{Tr}_{B_{I}}W^{A\prec
B_{I}}=W^{A_{I}}\otimes\mathbbm{1}^{A_{O}}$ and
$\operatorname{Tr}W^{A_{I}}=1$. Using the fact that
$\sum_{b}{\mathsf{M}}_{b|y}^{{\tilde{B}}B}*\mathbbm{1}^{B_{O}}=\sum_{b}\operatorname{Tr}_{B_{O}}{\mathsf{M}}_{b|y}^{{\tilde{B}}B}=\mathbbm{1}^{{\tilde{B}}B_{I}}$
according to the overall trace-preserving condition for a quantum instrument,
we have

$\displaystyle\sum_{b}\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}=\sum_{b}\big{(}\mathsf{M}_{a|x}^{{\tilde{A}}A}\otimes\mathsf{M}_{b|y}^{{\tilde{B}}B}\big{)}*W^{A\prec
B_{I}}\otimes\mathbbm{1}^{B_{O}}=\big{(}\mathsf{M}_{a|x}^{{\tilde{A}}A}\otimes\mathbbm{1}^{{\tilde{B}}B_{I}}\big{)}*W^{A\prec
B_{I}}=\mathsf{E}_{a|x}^{{\tilde{A}}}\otimes\mathbbm{1}^{{\tilde{B}}}$ (25)

with
$\mathsf{E}_{a|x}^{{\tilde{A}}}\coloneqq\mathsf{M}_{a|x}^{{\tilde{A}}A}*\operatorname{Tr}_{B_{I}}W^{A\prec
B_{I}}=\operatorname{Tr}_{A_{O}}\mathsf{M}_{a|x}^{{\tilde{A}}A}*W^{A_{I}}$
(such that $(\mathsf{E}_{a|x}^{{\tilde{A}}})_{a}$ forms Alice’s marginal POVM
applied to her physical quantum input). From Eq. (23), we then obtain999Here
we use the fact that
$\operatorname{Tr}_{{\tilde{B}}}\mathinner{|{{V^{\mathbf{B}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}}}^{\dagger}}|}^{{B^{\prime}}B^{\textup{jf}}{\tilde{B}}}=(V^{\mathbf{B}}{V^{\mathbf{B}}}^{\dagger})^{T}$,
so that
$\operatorname{Tr}_{{\tilde{B}}}\mathinner{|{{V^{\mathbf{B}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}}}^{\dagger}}|}^{{B^{\prime}}B^{\textup{jf}}{\tilde{B}}}*\zeta_{l}^{B^{\textup{jf}}}=\operatorname{Tr}_{B^{\textup{jf}}}\big{(}V^{\mathbf{B}}{V^{\mathbf{B}}}^{\dagger}(\mathbbm{1}^{{B^{\prime}}}\otimes\zeta_{l}^{B^{\textup{jf}}})\big{)}^{T}$.
To evaluate this further, note that $V^{\mathbf{A}}{V^{\mathbf{A}}}^{\dagger}$
acts as the identity on the state of Eq. (16), and therefore on the state
obtained after tracing out over ${C^{\prime}}C^{\textup{jf}}$—that is, on
$\sum_{k}(\rho_{c|z}^{A^{\prime}})^{T^{k}}\otimes\xi_{k}^{A^{\textup{jf}}}$,
for some tomographically complete set of states $\rho_{c|z}^{A^{\prime}}$. The
same holds on Bob’s side, so that
$V^{\mathbf{B}}{V^{\mathbf{B}}}^{\dagger}[\sum_{l}(\rho^{{B^{\prime}}})^{T^{\bar{l}}}\otimes\zeta_{l}^{B^{\textup{jf}}}]=\sum_{l}(\rho^{{B^{\prime}}})^{T^{\bar{l}}}\otimes\zeta_{l}^{B^{\textup{jf}}}$
for all $\rho^{{B^{\prime}}}$. It then follows that
$\operatorname{Tr}\big{[}\big{(}\sum_{l}\big{(}\operatorname{Tr}_{{\tilde{B}}}\mathinner{|{{V^{\mathbf{B}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}}}^{\dagger}}|}^{{B^{\prime}}B^{\textup{jf}}{\tilde{B}}}*\zeta_{l}^{B^{\textup{jf}}}\big{)}^{T^{l}}\big{)}\rho^{{B^{\prime}}}\big{]}=\operatorname{Tr}\big{[}\sum_{l}\operatorname{Tr}_{B^{\textup{jf}}}\big{(}V^{\mathbf{B}}{V^{\mathbf{B}}}^{\dagger}(\mathbbm{1}^{{B^{\prime}}}\otimes\zeta_{l}^{B^{\textup{jf}}})\big{)}^{T^{\bar{l}}}\rho^{{B^{\prime}}}\big{]}=\operatorname{Tr}\big{[}V^{\mathbf{B}}{V^{\mathbf{B}}}^{\dagger}[\sum_{l}(\rho^{{B^{\prime}}})^{T^{\bar{l}}}\otimes\zeta_{l}^{B^{\textup{jf}}}]\big{]}=\operatorname{Tr}\big{[}\sum_{l}(\rho^{{B^{\prime}}})^{T^{\bar{l}}}\otimes\zeta_{l}^{B^{\textup{jf}}}\big{]}=(\operatorname{Tr}\rho^{{B^{\prime}}})(\sum_{l}\operatorname{Tr}\zeta_{l}^{B^{\textup{jf}}})=\operatorname{Tr}\rho^{{B^{\prime}}}$
for all $\rho^{{B^{\prime}}}$, which implies that
$\sum_{l}\big{(}\operatorname{Tr}_{{\tilde{B}}}\mathinner{|{{V^{\mathbf{B}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}}}^{\dagger}}|}^{{B^{\prime}}B^{\textup{jf}}{\tilde{B}}}*\zeta_{l}^{B^{\textup{jf}}}\big{)}^{T^{l}}=\mathbbm{1}^{{B^{\prime}}}$.

$\displaystyle\sum_{b}E_{a,b|x,y}^{{A^{\prime}}{B^{\prime}}}$
$\displaystyle=\mathsf{E}_{a|x}^{{\tilde{A}}}\otimes\mathbbm{1}^{{\tilde{B}}}*\Bigg{[}\sum_{k=0,1}\left(\mathinner{|{{V^{\mathbf{A}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{A}}}^{\dagger}}|}^{{A^{\prime}}A^{\textup{jf}}{\tilde{A}}}*\xi_{k}^{A^{\textup{jf}}}\right)^{T_{{A^{\prime}}}^{k}}\otimes\sum_{l=0,1}\left(\mathinner{|{{V^{\mathbf{B}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}}}^{\dagger}}|}^{{B^{\prime}}B^{\textup{jf}}{\tilde{B}}}*\zeta_{l}^{B^{\textup{jf}}}\right)^{T_{{B^{\prime}}}^{l}}\Bigg{]}$
$\displaystyle=\Bigg{[}\mathsf{E}_{a|x}^{{\tilde{A}}}*\sum_{k=0,1}\left(\mathinner{|{{V^{\mathbf{A}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{A}}}^{\dagger}}|}^{{A^{\prime}}A^{\textup{jf}}{\tilde{A}}}*\xi_{k}^{A^{\textup{jf}}}\right)^{T_{{A^{\prime}}}^{k}}\Bigg{]}\otimes\sum_{l=0,1}\left(\operatorname{Tr}_{{\tilde{B}}}\mathinner{|{{V^{\mathbf{B}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}}}^{\dagger}}|}^{{B^{\prime}}B^{\textup{jf}}{\tilde{B}}}*\zeta_{l}^{B^{\textup{jf}}}\right)^{T^{l}}$
$\displaystyle=E_{a|x}^{{A^{\prime}}}\otimes\mathbbm{1}^{{B^{\prime}}}.$ (26)

with

$\displaystyle E_{a|x}^{{A^{\prime}}}$
$\displaystyle\coloneqq\mathsf{E}_{a|x}^{{\tilde{A}}}*\sum_{k=0,1}\left(\mathinner{|{{V^{\mathbf{A}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{A}}}^{\dagger}}|}^{{A^{\prime}}A^{\textup{jf}}{\tilde{A}}}*\xi_{k}^{A^{\textup{jf}}}\right)^{T_{{A^{\prime}}}^{k}},$
(27)

which now forms a single-partite POVM acting on Alice’s reference quantum
input (one can in particular check that
$\sum_{a}\mathsf{E}_{a|x}^{{\tilde{A}}}=\mathbbm{1}^{{\tilde{A}}}$ and
$\sum_{a}E_{a|x}^{{A^{\prime}}}=\mathbbm{1}^{{A^{\prime}}}$).

Thus, from Eq. (26) we can conclude that, starting from a causally ordered
$W=W^{A\prec B_{I}}\otimes\mathbbm{1}^{B_{O}}$ compatible with $A\prec B$, we
obtain a D-POVM $(E_{a,b|x,y}^{{A^{\prime}}{B^{\prime}}})_{a,b}$ compatible
with the order ${A^{\prime}}\prec{B^{\prime}}$. Similarly, from a process
matrix compatible with the causal order $B\prec A$ we obtain a D-POVM
$(E_{a,b|x,y}^{{A^{\prime}}{B^{\prime}}})_{a,b}$ compatible with the order
${B^{\prime}}\prec{A^{\prime}}$. We can then conclude that any D-POVM
$(E_{a,b|x,y}^{{A^{\prime}}{B^{\prime}}})_{a,b}$ induced by a causally
separable process matrix is causally separable, as defined in [15].

This is the case, in particular, when $x=y=\star$, so that the effective
D-POVM
$\mathbb{E}_{\text{eff}}^{{A^{\prime}}{B^{\prime}}}=(E_{a,b|*,*}^{{A^{\prime}}{B^{\prime}}})_{a,b}$
is causally separable whenever $W^{AB}$ is causally separable. As a result,
the correlations $P^{\prime}(a,b|\rho_{z}^{A^{\prime}},\rho_{w}^{B^{\prime}})$
defined in Eq. (24) obtained from a causally separable process $W^{AB}$ must
satisfy any witness inequality $\mathcal{J}\geq 0$. If these correlations
violate such an inequality, giving $\mathcal{J}<0$, then this therefore proves
the causal nonseparability of the process in a NDI manner.

Such a violation—and hence NDI certification—can moreover be obtained for any
process $W^{AB}$ that can (for appropriately chosen reference instruments
$(M_{a}^{{A^{\prime}}A})_{a}$ and $(M_{b}^{{B^{\prime}}B})_{b}$) induce a
causally nonseparable D-POVM
$\mathbb{E}^{{A^{\prime}}{B^{\prime}}}=(E_{a,b}^{{A^{\prime}}{B^{\prime}}})_{a,b}$
in the reference scenario, i.e., which cannot be decomposed according to Eqs.
(4)–(6). Indeed, as outlined in [15], the causal nonseparability of
$\mathbb{E}^{{A^{\prime}}{B^{\prime}}}$ can always be witnessed by a family
$\mathbb{S}^{{A^{\prime}}{B^{\prime}}}=(S_{a,b}^{{A^{\prime}}{B^{\prime}}})_{a,b}$
of operators (a “witness”) such that
$\sum_{a,b}S_{a,b}^{{A^{\prime}}{B^{\prime}}}*E_{a,b}^{{A^{\prime}}{B^{\prime}}}<0$
only if $\mathbb{E}^{{A^{\prime}}{B^{\prime}}}$ is causally nonseparable.
Concretely, writing
$S_{a,b}^{{A^{\prime}}{B^{\prime}}}=\sum_{z,w}s_{a,b}^{(z,w)}\rho_{z}^{A^{\prime}}\otimes\rho_{w}^{B^{\prime}}$
using the tomographically complete sets of states from the NDI-protocol, one
obtains the witness inequality

$\mathcal{J}:=\sum_{a,b,z,w}s_{a,b}^{(z,w)}P^{\prime}(a,b|\rho_{z}^{A^{\prime}},\rho_{w}^{B^{\prime}})\geq
0.$ (28)

By implementing precisely the reference NSDI-QI protocol, one simply has
$E_{a,b}^{{A^{\prime}}{B^{\prime}}}=E_{a,b|*,*}^{{A^{\prime}}{B^{\prime}}}={\mathsf{E}}_{a,b|*,*}^{{\tilde{A}}{\tilde{B}}}$
for all $a,b$ so that the reference, effective, and physical D-POVMs all
coincide, and the witness inequality can indeed, as expected, be violated.

##  Appendix C Robustness of the procedure

The exact self-testing conditions (i.e., strictly reaching the maximal quantum
violation of the Bell inequality under consideration) are never met in any
realistic scenario. Thus, to make self-testing applicable in practice one
needs to ensure that it is “robust”. In particular, the reference and physical
experiments should be sufficiently close if the self-testing correlations are
approximately reproduced by the physical experiment. In this section we
discuss how our NDI certification procedure is affected if the states shared
by Alice and Charlie and by Bob and Daisy approximately reproduce the self-
testing correlations.

The robust self-testing of the maximally entangled pair of qubits (resp.,
qudits) is discussed in Appendix B (resp., E) of [17]. Let us restrict
ourselves again here, for simplicity, to the case of qubits (for qudits one
can adapt the approach of [17] in a similar manner). The robust self-testing
statements—i.e., the robust version of Eq. (16)—then take the form101010Recall
that we assume, for simplicity, that the physical measurements
$({\mathsf{M}}_{c|z}^{{\tilde{C}}})_{c}$ and
$({\mathsf{M}}_{a|x}^{{\tilde{A}}})_{a}$ are projective. In the noisy case
this projectivity property cannot be derived directly by using the techniques
of [31], which require the maximal violation of a Bell inequality, although we
expect that the noise-robust statements of [31] should still allow one to
evade the strict requirement for the projectivity assumption (working out the
details is left as an open question).

$\bigg{\|}V^{\mathbf{C}}\otimes
V^{\mathbf{A}}\left(({{\mathsf{M}}}_{c|z}^{\tilde{C}})^{T}\mathinner{|{\uppsi_{1}}\rangle}^{{\tilde{C}}{\tilde{A}}}\right)\\\
-\bigg{(}\left((M_{c|z}^{{\color[rgb]{1,.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,.5,0}T}})^{C^{\prime}}\mathinner{|{\phi^{+}}\rangle}^{{C^{\prime}}{A^{\prime}}}\right)\otimes\mathinner{|{\xi_{0}}\rangle}^{C^{\textup{j}}A^{\textup{j}}}\otimes\mathinner{|{00}\rangle}^{C^{\textup{f}}A^{\textup{f}}}+\left(M_{c|z}^{C^{\prime}}\mathinner{|{\phi^{+}}\rangle}^{{C^{\prime}}{A^{\prime}}}\right)\otimes\mathinner{|{\xi_{1}}\rangle}^{C^{\textup{j}}A^{\textup{j}}}\otimes\mathinner{|{11}\rangle}^{C^{\textup{f}}A^{\textup{f}}}\bigg{)}\bigg{\|}\leq\theta,$
(29)

where $\theta$ depends on the distance between observed and reference
correlations (e.g., in the $\textit{{d}}=2$ case, if the violation of the Bell
inequality (13) reaches some value $6\sqrt{2}-\epsilon$, one can take
$\theta=\kappa\sqrt{\epsilon}$, for some multiplicative factor $\kappa$ [17]).
Denoting by $\mathinner{|{\Psi_{c|z}}\rangle}$ the state in the right-hand
side of Eq. (16) (in the noiseless case), this equation can be rewritten as

$\displaystyle V^{\mathbf{C}}\otimes
V^{\mathbf{A}}\left(({\mathsf{M}}_{c|z}^{\tilde{C}})^{T}\mathinner{|{\uppsi_{1}}\rangle}^{{\tilde{C}}{\tilde{A}}}\right)=\mathinner{|{\Psi_{c|z}}\rangle}^{{C^{\prime}}C^{\textup{jf}}{A^{\prime}}A^{\textup{jf}}}+\mathinner{|{\omega_{c|z}}\rangle}^{{C^{\prime}}C^{\textup{jf}}{A^{\prime}}A^{\textup{jf}}},$
(30)

where $\mathinner{|{\omega_{c|z}}\rangle}$ is an unknown state with vector
norm smaller than or equal to $\theta$. Eq. (19) then becomes, in the noisy
case,

$\displaystyle
P(c|z)\,\uprho_{c|z}^{\tilde{A}}={\mathsf{M}}_{c|z}^{\tilde{C}}*\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}$
$\displaystyle={V^{\mathbf{A}}}^{\dagger}\operatorname{Tr}_{{C^{\prime}}C^{\textup{jf}}}\left((\mathinner{|{\Psi_{c|z}}\rangle}+\mathinner{|{\omega_{c|z}}\rangle})(\mathinner{\langle{\Psi_{c|z}}|}+\mathinner{\langle{\omega_{c|z}}|})\right)V^{\mathbf{A}}$
$\displaystyle=P(c|z)\,{V^{\mathbf{A}}}^{\dagger}\left(\sum_{k=0,1}(\rho_{c|z}^{A^{\prime}})^{T^{k}}\otimes\xi_{k}^{A^{\textup{jf}}}\right)V^{\mathbf{A}}+\delta_{c|z}^{{\tilde{A}}},$
(31)

with

$\displaystyle\delta_{c|z}^{{\tilde{A}}}={V^{\mathbf{A}}}^{\dagger}\operatorname{Tr}_{{C^{\prime}}C^{\textup{jf}}}\left(\mathinner{|{\Psi_{c|z}}\rangle\\!\langle{\omega_{c|z}}|}+\mathinner{|{\omega_{c|z}}\rangle\\!\langle{\Psi_{c|z}}|}+\mathinner{|{\omega_{c|z}}\rangle\\!\langle{\omega_{c|z}}|}\right)V^{\mathbf{A}}.$
(32)

The norm of this correction term can be bounded in the same way as it is done
in Appendix G of [17]. Namely, using the triangle inequality and noting that
$\|\mathinner{|{\Psi_{c|z}}\rangle}\|=\frac{1}{\sqrt{2}}$, we get

$\left\|\delta_{c|z}^{{\tilde{A}}}\right\|_{1}\leq\sqrt{2}\,\theta+\theta^{2},$
(33)

where $\|\cdot\|_{1}$ is the trace norm.

Similar statements can be made on Bob and Daisy’s side, allowing one to write
${\mathsf{M}}_{d|w}^{\tilde{D}}*\uppsi_{2}^{{\tilde{B}}{\tilde{D}}}$ in a
similar way to Eq. (31), in terms of a correction term
$\delta_{d|w}^{{\tilde{B}}}$ whose trace norm is also upper-bounded as in Eq.
(33) (for simplicity we take the same value of $\theta$ on both sides). The
observed correlations $P(c,a,b,d|z,x,y,w)$, following Eq. (22), then become

$\displaystyle P(c,a,b,d|z,x,y,w)$
$\displaystyle=\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}*\left(\big{(}{\mathsf{M}}_{c|z}^{\tilde{C}}*\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}\big{)}\otimes\big{(}{\mathsf{M}}_{d|w}^{\tilde{D}}*\uppsi_{2}^{{\tilde{B}}{\tilde{D}}}\big{)}\right)$
$\displaystyle=\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}*\bigg{(}\\!P(c|z)\sum_{k=0,1}\Big{(}\mathinner{|{{V^{\mathbf{A}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{A}}}^{\dagger}}|}^{{A^{\prime}}A^{\textup{jf}}{\tilde{A}}}\\!*\xi_{k}^{A^{\textup{jf}}}\Big{)}^{T_{{A^{\prime}}}^{k}}*\rho_{c|z}^{A^{\prime}}+\delta_{c|z}^{{\tilde{A}}}\bigg{)}$
$\displaystyle\hskip
85.35826pt\otimes\bigg{(}\\!P(d|w)\sum_{l=0,1}\Big{(}\mathinner{|{{V^{\mathbf{B}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}}}^{\dagger}}|}^{{B^{\prime}}B^{\textup{jf}}{\tilde{B}}}\\!*\zeta_{l}^{B^{\textup{jf}}}\Big{)}^{T_{{B^{\prime}}}^{l}}*\rho_{d|w}^{B^{\prime}}+\delta_{d|w}^{{\tilde{B}}}\bigg{)}$
$\displaystyle=P(c|z)\,P(d|w)\
E_{a,b|x,y}^{{A^{\prime}}{B^{\prime}}}*\left(\rho_{c|z}^{A^{\prime}}\otimes\rho_{d|w}^{B^{\prime}}\right)+P(c|z)\,\mathsf{E}_{a,b|x,y}^{{A^{\prime}}{\tilde{B}}}*\left(\rho_{c|z}^{A^{\prime}}\otimes\delta_{d|w}^{{\tilde{B}}}\right)$
$\displaystyle\hskip
42.67912pt+P(d|w)\,\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{B^{\prime}}}*\left(\delta_{c|z}^{{\tilde{A}}}\otimes\rho_{d|w}^{B^{\prime}}\right)+\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}*\left(\delta_{c|z}^{{\tilde{A}}}\otimes\delta_{d|w}^{{\tilde{B}}}\right)$
(34)

with the “physical” D-POVM elements defined as
$\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}=\big{(}\mathsf{M}_{a|x}^{{\tilde{A}}A}\otimes\mathsf{M}_{b|y}^{{\tilde{B}}B}\big{)}*W^{AB}$,
the D-POVM elements $E_{a,b|x,y}^{{A^{\prime}}{B^{\prime}}}$ acting on the
“reference” input states as given in Eq. (23), and with the new D-POVM
elements effectively acting on the “reference” and the “physical” quantum
input states

$\displaystyle\mathsf{E}_{a,b|x,y}^{{A^{\prime}}{\tilde{B}}}$
$\displaystyle\coloneqq\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}*\sum_{k=0,1}\Big{(}\mathinner{|{{V^{\mathbf{A}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{A}}}^{\dagger}}|}^{{A^{\prime}}A^{\textup{jf}}{\tilde{A}}}\\!*\xi_{k}^{A^{\textup{jf}}}\Big{)}^{T_{{A^{\prime}}}^{k}},$
$\displaystyle\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{B^{\prime}}}$
$\displaystyle\coloneqq\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}*\sum_{l=0,1}\Big{(}\mathinner{|{{V^{\mathbf{B}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}}}^{\dagger}}|}^{{B^{\prime}}B^{\textup{jf}}{\tilde{B}}}\\!*\zeta_{l}^{B^{\textup{jf}}}\Big{)}^{T_{{B^{\prime}}}^{l}}.$
(35)

The correlations $P_{\star}(a,b|z,w)$ used in our protocol to certify causal
nonseparability are then

$\displaystyle P_{\star}(a,b|z,w)$
$\displaystyle=P(a,b|z,x{=}\star,y{=}\star,w,c{=}0,d{=}0)$
$\displaystyle=P^{\prime}(a,b|\rho_{z}^{A^{\prime}},\rho_{w}^{B^{\prime}})+\mathsf{E}_{a,b|\star,\star}^{{A^{\prime}}{\tilde{B}}}*\left(\rho_{z}^{A^{\prime}}\otimes\frac{\delta_{0|w}^{{\tilde{B}}}}{P(0|w)}\right)$
$\displaystyle\hskip
42.67912pt+\mathsf{E}_{a,b|\star,\star}^{{\tilde{A}}{B^{\prime}}}*\left(\frac{\delta_{0|z}^{{\tilde{A}}}}{P(0|z)}\otimes\rho_{w}^{B^{\prime}}\right)+\mathsf{E}_{a,b|\star,\star}^{{\tilde{A}}{\tilde{B}}}*\left(\frac{\delta_{0|z}^{{\tilde{A}}}}{P(0|z)}\otimes\frac{\delta_{0|w}^{{\tilde{B}}}}{P(0|w)}\right),$
(36)

with
$P^{\prime}(a,b|\rho_{z}^{A^{\prime}},\rho_{w}^{B^{\prime}})=E_{a,b|\star,\star}^{{A^{\prime}}{B^{\prime}}}*\left(\rho_{z}^{A^{\prime}}\otimes\rho_{w}^{B^{\prime}}\right)$.
We can bound the second term in the right-hand side of Eq. (36) as

$\displaystyle\left|\operatorname{Tr}\left[\left(\mathsf{E}_{a,b|\star,\star}^{{A^{\prime}}{\tilde{B}}}\right)^{T}\left(\rho_{z}^{A^{\prime}}\otimes\frac{\delta_{0|w}^{{\tilde{B}}}}{P(0|w)}\right)\right]\right|$
$\displaystyle\leq\left\|\left(\mathsf{E}_{a,b|\star,\star}^{{A^{\prime}}{\tilde{B}}}\right)^{T}\left(\rho_{z}^{A^{\prime}}\otimes\frac{\delta_{0|w}^{{\tilde{B}}}}{P(0|w)}\right)\right\|_{1}\leq\left\|\mathsf{E}_{a,b|\star,\star}^{{A^{\prime}}{\tilde{B}}}\right\|_{\infty}\left\|\rho_{z}^{A^{\prime}}\otimes\frac{\delta_{0|w}^{{\tilde{B}}}}{P(0|w)}\right\|_{1}$
$\displaystyle\leq\frac{1}{P(0|w)}\left\|\rho_{z}^{A^{\prime}}\right\|_{1}\left\|\delta_{0|w}^{{\tilde{B}}}\right\|_{1}\leq\frac{1}{P(0|w)}\left(\sqrt{2}\,\theta+\theta^{2}\right),$
(37)

where in the first inequality we used the fact that
$\left|\operatorname{Tr}(A)\right|\leq\operatorname{Tr}\left|A\right|=\|A\|_{1}$,
the second one follows from Hölder’s inequality, in the third inequality we
used the fact that the maximal eigenvalue of
$\mathsf{E}_{a,b|\star,\star}^{{A^{\prime}}{\tilde{B}}}$ cannot be larger than
1, and in the last inequality we used the normalisation of
$\rho_{z}^{A^{\prime}}$ together with the analogue of the bound (33) above,
for $\left\|\delta_{0|w}^{{\tilde{B}}}\right\|_{1}$. In a similar way, for the
last two terms in Eq. (36), we can write

$\displaystyle\left|\operatorname{Tr}\left[\left(\mathsf{E}_{a,b|\star,\star}^{{\tilde{A}}{B^{\prime}}}\right)^{T}\left(\frac{\delta_{0|z}^{{\tilde{A}}}}{P(0|z)}\otimes\rho_{w}^{B^{\prime}}\right)\right]\right|\leq\frac{1}{P(0|z)}\left(\sqrt{2}\,\theta+\theta^{2}\right),$
(38)

and

$\displaystyle\left|\operatorname{Tr}\left[\left(\mathsf{E}_{a,b|\star,\star}^{{\tilde{A}}{\tilde{B}}}\right)^{T}\left(\frac{\delta_{0|z}^{{\tilde{A}}}}{P(0|z)}\otimes\frac{\delta_{0|w}^{{\tilde{B}}}}{P(0|w)}\right)\right]\right|\leq\frac{1}{P(0|w)P(0|z)}\left(\sqrt{2}\,\theta+\theta^{2}\right)^{2}.$
(39)

Consider now a witness inequality of the form of Eq. (28),
$\mathcal{J}_{\text{ref.}}:=\sum_{a,b,z,w}s_{a,b}^{(z,w)}P^{\prime}(a,b|\rho_{z}^{A^{\prime}},\rho_{w}^{B^{\prime}})\geq
0$. Evaluating the same witness expression with the experimentally obtained
probabilities $P_{\star}(a,b|z,w)$ instead of
$P^{\prime}(a,b|\rho_{z}^{A^{\prime}},\rho_{w}^{B^{\prime}})$, so as to define
$\mathcal{J}_{\text{exp.}}:=\sum_{a,b,z,w}s_{a,b}^{(z,w)}P_{\star}(a,b|z,w)$,
we can write

$\displaystyle\mathcal{J}_{\text{exp.}}=\mathcal{J}_{\text{ref.}}+\mathcal{J}_{\text{correction}}$
(40)

with

$\displaystyle\mathcal{J}_{\text{correction}}=\sum_{a,b,z,w}s_{a,b}^{(z,w)}\left\\{\mathsf{E}_{a,b|\star,\star}^{{A^{\prime}}{\tilde{B}}}*\left(\rho_{z}^{A^{\prime}}\otimes\frac{\delta_{0|w}^{{\tilde{B}}}}{P(0|w)}\right)+\mathsf{E}_{a,b|\star,\star}^{{\tilde{A}}{B^{\prime}}}*\left(\frac{\delta_{0|z}^{{\tilde{A}}}}{P(0|z)}\otimes\rho_{w}^{B^{\prime}}\right)+\mathsf{E}_{a,b|\star,\star}^{{\tilde{A}}{\tilde{B}}}*\left(\frac{\delta_{0|z}^{{\tilde{A}}}}{P(0|z)}\otimes\frac{\delta_{0|w}^{{\tilde{B}}}}{P(0|w)}\right)\right\\}.$
(41)

The previous analysis allows us to bound $\mathcal{J}_{\text{correction}}$ as
$\left|\mathcal{J}_{\text{correction}}\right|\leq\mathcal{J}_{\text{correction}}^{\text{max}}$,
with an upper bound $\mathcal{J}_{\text{correction}}^{\text{max}}$ on the
maximum correction that can be evaluated explicitly given
$(s_{a,b}^{(z,w)})_{a,b,z,w}$ using Eqs. (37)–(39), and that is (for small
$\theta$) of order $\mathcal{O}(\theta)$—with
$\mathcal{J}_{\text{correction}}=0$ when $\theta=0$. Given the witness
inequality $\mathcal{J}_{\text{ref.}}\geq 0$, which implies

$\displaystyle\mathcal{J}_{\text{exp.}}\geq-\mathcal{J}_{\text{correction}}^{\text{max}},$
(42)

we thus find that a violation of Eq. (42) certifies causal nonseparability in
a manner that is robust to noise.

##  Appendix D “Network-MDCI” certification of all causally nonseparable
process matrices

It was shown in [15] that all bipartite causally nonseparable process matrices
can be certified with trusted quantum inputs, if we assume that Alice and
Bob’s (trusted) ancillary Hilbert spaces have a bipartite structure of the
form
$\mathcal{H}^{{A^{\prime}}}=\mathcal{H}^{{A_{I}^{\prime}}{A_{O}^{\prime}}}$
and
$\mathcal{H}^{{B^{\prime}}}=\mathcal{H}^{{B_{I}^{\prime}}{B_{O}^{\prime}}}$,
and that their instruments have a specific structure as well—namely, that they
factorise as a joint measurement of system ${A_{I}^{\prime}}$ together with
the system $A_{I}$ received from the process matrix, and a channel taking
system ${A_{O}^{\prime}}$ into the system $A_{O}$ that is fed into the process
matrix (and similarly on Bob’s side). For this certification, it suffices to
use separate (product) quantum inputs in systems ${A_{I}^{\prime}}$ and
${A_{O}^{\prime}}$ (and in systems ${B_{I}^{\prime}}$ and ${B_{O}^{\prime}}$).
This certification, called the measurement-device-and-channel-independent
(MDCI) certification, can be transformed into a NDI certification by making
similar assumptions on the (now untrusted) physical spaces and instruments.

Let us first recall that the above assumptions imply that the Choi matrices of
Alice’s instruments decompose as
$\mathsf{M}_{a|x}^{{\tilde{A}}A}=\mathsf{M}_{a|x}^{{\tilde{A}_{I}}A_{I}}\otimes\mathsf{M}_{|x}^{{\tilde{A}_{O}}A_{O}}$,
for some POVM $(\mathsf{M}_{a|x}^{{\tilde{A}_{I}}A_{I}})_{a}$ and a channel
$\mathsf{M}_{|x}^{{\tilde{A}_{O}}A_{O}}$, and similarly for Bob. Given that
the channels $\mathsf{M}_{|x}^{{\tilde{A}_{O}}A_{O}}$ and
$\mathsf{M}_{|y}^{{\tilde{B}_{O}}B_{O}}$ must be trace-preserving, these
decompositions imply that

$\displaystyle\operatorname{Tr}_{A_{O}}\mathsf{M}_{a|x}^{{\tilde{A}}A}=\mathsf{M}_{a|x}^{{\tilde{A}_{I}}A_{I}}\otimes\mathbbm{1}^{{\tilde{A}_{O}}},\quad\operatorname{Tr}_{B_{O}}\mathsf{M}_{b|y}^{{\tilde{B}}B}=\mathsf{M}_{b|y}^{{\tilde{B}_{I}}B_{I}}\otimes\mathbbm{1}^{{\tilde{B}_{O}}},$
(43)

for all $a,b$, with
$\sum_{a}\mathsf{M}_{a|x}^{{\tilde{A}_{I}}A_{I}}=\mathbbm{1}^{{\tilde{A}_{I}}A_{I}}$
and
$\sum_{b}\mathsf{M}_{b|y}^{{\tilde{B}_{I}}B_{I}}=\mathbbm{1}^{{\tilde{B}_{I}}B_{I}}$.
To obtain our results below, it is in fact sufficient to assume that Eq. (43)
holds, for some particular values of $a$ and $b$.

Consider a causally ordered process matrix $W=W^{A\prec
B_{I}}\otimes\mathbbm{1}^{B_{O}}$. The structure of Eq. (43) implies that the
physical D-POVM elements
$\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}=\big{(}\mathsf{M}_{a|x}^{{\tilde{A}}A}\otimes\mathsf{M}_{b|y}^{{\tilde{B}}B}\big{)}*W^{AB}$
satisfy

$\displaystyle\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}=\big{(}\mathsf{M}_{a|x}^{{\tilde{A}}A}\otimes\operatorname{Tr}_{B_{O}}\mathsf{M}_{b|y}^{{\tilde{B}}B}\big{)}*W^{A\prec
B_{I}}=\big{(}\mathsf{M}_{a|x}^{{\tilde{A}}A}\otimes\mathsf{M}_{b|y}^{{\tilde{B}_{I}}B_{I}}\big{)}*W^{A\prec
B_{I}}\otimes\mathbbm{1}^{{\tilde{B}_{O}}}=\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}_{I}}}\otimes\mathbbm{1}^{{\tilde{B}_{O}}}$
(44)

with
$\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}_{I}}}=\big{(}\mathsf{M}_{a|x}^{{\tilde{A}}A}\otimes\mathsf{M}_{b|y}^{{\tilde{B}_{I}}B_{I}}\big{)}*W^{A\prec
B_{I}}$.

Moving on to the NDI scenario, let us focus, for now, on the MDCI assumption
on Bob’s side. Instead of introducing just one source shared between Bob and
Daisy, we now introduce two independent sources shared between Bob and two new
parties, $\text{Daisy}_{I}$, with Hilbert space
$\mathcal{H}^{{\tilde{D}_{I}}}$, and $\text{Daisy}_{O}$, with Hilbert space
$\mathcal{H}^{{\tilde{D}_{O}}}$ (we use the same notations as before, just
adding the subscripts I and O appropriately). Self-testing the quantum inputs
in spaces $\mathcal{H}^{{\tilde{B}_{I}}}$ and $\mathcal{H}^{{\tilde{B}_{O}}}$
separately, we can write, similarly to Eq. (20),

$\displaystyle P(d_{I}|w_{I})\,\uprho_{d_{I}|w_{I}}^{\tilde{B}_{I}}$
$\displaystyle={\mathsf{M}}_{d_{I}|w_{I}}^{\tilde{D}_{I}}*\uppsi_{2,I}^{{\tilde{B}_{I}}{\tilde{D}_{I}}}=P(d_{I}|w_{I})\,{V^{\mathbf{B}_{I}}}^{\dagger}\left(\sum_{l=0,1}(\rho_{d_{I}|w_{I}}^{B_{I}^{\prime}})^{T^{l}}\otimes\zeta_{l}^{B^{\textup{jf}}_{I}}\right)V^{\mathbf{B}_{I}},$
$\displaystyle P(d_{O}|w_{O})\,\uprho_{d_{O}|w_{O}}^{\tilde{B}_{O}}$
$\displaystyle={\mathsf{M}}_{d_{O}|w_{O}}^{\tilde{D}_{O}}*\uppsi_{2,O}^{{\tilde{B}_{O}}{\tilde{D}_{O}}}=P(d_{O}|w_{O})\,{V^{\mathbf{B}_{O}}}^{\dagger}\left(\sum_{l^{\prime}=0,1}(\rho_{d_{O}|w_{O}}^{B_{O}^{\prime}})^{T^{l^{\prime}}}\otimes\zeta_{l^{\prime}}^{B^{\textup{jf}}_{O}}\right)V^{\mathbf{B}_{O}},$
(45)

for some reference quantum input states
$\rho_{d_{I}|w_{I}}^{B_{I}^{\prime}}\in\mathcal{L}(\mathcal{H}^{B_{I}^{\prime}})$
and
$\rho_{d_{O}|w_{O}}^{B_{O}^{\prime}}\in\mathcal{L}(\mathcal{H}^{B_{O}^{\prime}})$.

Then as in Eq. (22), and still for a causally ordered $W=W^{A\prec
B_{I}}\otimes\mathbbm{1}^{B_{O}}$, the correlations
$P(c,a,b,d_{I},d_{O}|z,x,y,w_{I},w_{O})$ can then be written (using the above
equations together with Eq. (19)) as

$\displaystyle
P(c,a,b,d_{I},d_{O}|z,x,y,w_{I},w_{O})=\left(\\!{\mathsf{M}}_{c|z}^{\tilde{C}}\\!\otimes\\!{\mathsf{M}}_{a|x}^{{\tilde{A}}A}\\!\otimes\\!{\mathsf{M}}_{b|y}^{{\tilde{B}}B}\\!\otimes\\!{\mathsf{M}}_{d_{I}|w_{I}}^{\tilde{D}_{I}}\\!\otimes\\!{\mathsf{M}}_{d_{O}|w_{O}}^{\tilde{D}_{O}}\\!\right)\\!*\\!\left(\\!\uppsi_{1}^{{\tilde{C}}{\tilde{A}}}\\!\otimes\\!W^{A\prec
B_{I}}\\!\otimes\\!\mathbbm{1}^{B_{O}}\\!\otimes\\!\uppsi_{2,I}^{{\tilde{B}_{I}}{\tilde{D}_{I}}}\\!\otimes\\!\uppsi_{2,O}^{{\tilde{B}_{O}}{\tilde{D}_{O}}}\\!\right)$
$\displaystyle\quad=P(c|z)\,P(d_{I}|w_{I})\,P(d_{O}|w_{O})\,\left(\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}_{I}}}\otimes\mathbbm{1}^{{\tilde{B}_{O}}}\right)*\Bigg{(}\sum_{k=0,1}\Big{(}\mathinner{|{{V^{\mathbf{A}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{A}}}^{\dagger}}|}^{{A^{\prime}}A^{\textup{jf}}{\tilde{A}}}*\xi_{k}^{A^{\textup{jf}}}\Big{)}^{T_{{A^{\prime}}}^{k}}*\rho_{c|z}^{A^{\prime}}$
$\displaystyle\hskip
241.84842pt\otimes\sum_{l=0,1}\Big{(}\mathinner{|{{V^{\mathbf{B}_{I}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}_{I}}}^{\dagger}}|}^{{B_{I}^{\prime}}B^{\textup{jf}}_{I}{\tilde{B}_{I}}}*\zeta_{l}^{B^{\textup{jf}}_{I}}\Big{)}^{T_{{B_{I}^{\prime}}}^{l}}*\rho_{d_{I}|w_{I}}^{B_{I}^{\prime}}$
$\displaystyle\hskip
256.0748pt\otimes\sum_{l^{\prime}=0,1}\Big{(}\mathinner{|{{V^{\mathbf{B}_{O}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}_{O}}}^{\dagger}}|}^{{B_{O}^{\prime}}B^{\textup{jf}}_{O}{\tilde{B}_{O}}}*\zeta_{l^{\prime}}^{B^{\textup{jf}}_{O}}\Big{)}^{T_{{B_{O}^{\prime}}}^{l^{\prime}}}*\rho_{d_{O}|w_{O}}^{B_{O}^{\prime}}\Bigg{)}$
$\displaystyle\quad=P(c|z)\,P(d_{I}|w_{I})\,P(d_{O}|w_{O})\
\left(E_{a,b|x,y}^{{A^{\prime}}{B_{I}^{\prime}}}\otimes\mathbbm{1}^{B_{O}^{\prime}}\right)*\left(\rho_{c|z}^{A^{\prime}}\otimes\rho_{d_{I}|w_{I}}^{B_{I}^{\prime}}\otimes\rho_{d_{O}|w_{O}}^{B_{O}^{\prime}}\right)$
(46)

with

$\displaystyle E_{a,b|x,y}^{{A^{\prime}}{B_{I}^{\prime}}}$
$\displaystyle=\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}_{I}}}*\Bigg{[}\sum_{k=0,1}\left(\mathinner{|{{V^{\mathbf{A}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{A}}}^{\dagger}}|}^{{A^{\prime}}A^{\textup{jf}}{\tilde{A}}}*\xi_{k}^{A^{\textup{jf}}}\right)^{T_{{A^{\prime}}}^{k}}\otimes\sum_{l=0,1}\left(\mathinner{|{{V^{\mathbf{B}_{I}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}_{I}}}^{\dagger}}|}^{{B_{I}^{\prime}}B^{\textup{jf}}_{I}{\tilde{B}_{I}}}*\zeta_{l}^{B^{\textup{jf}}_{I}}\right)^{T_{{B_{I}^{\prime}}}^{l}}\Bigg{]},$
(47)

where we used the fact that
$\sum_{l^{\prime}}\Big{(}\operatorname{Tr}_{{\tilde{B}_{O}}}\mathinner{|{{V^{\mathbf{B}_{O}}}^{\dagger}}\rangle\\!\rangle\\!\langle\\!\langle{{V^{\mathbf{B}_{O}}}^{\dagger}}|}^{{B_{O}^{\prime}}B^{\textup{jf}}_{O}{\tilde{B}_{O}}}*\zeta_{l^{\prime}}^{B^{\textup{jf}}_{O}}\Big{)}^{T_{{B_{O}^{\prime}}}^{l^{\prime}}}=\mathbbm{1}^{{B_{O}^{\prime}}}$,
cf. Footnote 9.

Hence, we see that for a causally ordered $W=W^{A\prec
B_{I}}\otimes\mathbbm{1}^{B_{O}}$, the structure
$\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}=\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}_{I}}}\otimes\mathbbm{1}^{{\tilde{B}_{O}}}$
that the MDCI assumption (on Bob’s side) implies on the physical D-POVM
elements $\mathsf{E}_{a,b|x,y}^{{\tilde{A}}{\tilde{B}}}$ gets transferred to
the effective D-POVM elements
$E_{a,b|x,y}^{{A^{\prime}}{B^{\prime}}}=E_{a,b|x,y}^{{A^{\prime}}{B_{I}^{\prime}}}\otimes\mathbbm{1}^{B_{O}^{\prime}}$
that act on the reference input states. By symmetry, the same structure
preservation applies for a causally ordered $W=W^{B\prec
A_{I}}\otimes\mathbbm{1}^{A_{O}}$ (using the MDCI assumption on Alice’s side,
introducing now two independent sources shared by two new parties
$\text{Charlie}_{I}$ and $\text{Charlie}_{O}$). It follows that for a causally
separable process matrix $W^{AB}=q\,W^{A\prec
B_{I}}\otimes\mathbbm{1}^{B_{O}}+(1{-}q)\,W^{B\prec
A_{I}}\otimes\mathbbm{1}^{A_{O}}$, the D-POVM elements acting on the reference
input states (in some reference Hilbert spaces
$\mathcal{H}^{{A_{I}^{\prime}}}$, $\mathcal{H}^{{A_{O}^{\prime}}}$,
$\mathcal{H}^{{B_{I}^{\prime}}}$ and $\mathcal{H}^{{B_{O}^{\prime}}}$)
decompose as

$\displaystyle
E_{a,b|x,y}^{{A_{I}^{\prime}}{A_{O}^{\prime}}{B_{I}^{\prime}}{B_{O}^{\prime}}}$
$\displaystyle=q\,E_{a,b|x,y}^{{A_{I}^{\prime}}{A_{O}^{\prime}}{B_{I}^{\prime}}}\otimes\mathbbm{1}^{B_{O}^{\prime}}+(1-q)\,E_{a,b|x,y}^{{A_{I}^{\prime}}{B_{I}^{\prime}}{B_{O}^{\prime}}}\otimes\mathbbm{1}^{A_{O}^{\prime}}.$
(48)

In this scenario (involving all four additional parties $\text{Charlie}_{I}$,
$\text{Charlie}_{O}$, $\text{Daisy}_{I}$ and $\text{Daisy}_{O}$ together with
Alice and Bob), the correlations

$\displaystyle P_{\star}(a,b|z,w)$
$\displaystyle=P(a,b|z_{I},z_{O},x{=}\star,y{=}\star,w_{I},w_{O},c_{I}{=}0,c_{O}{=}0,d_{I}{=}0,d_{O}{=}0)$
$\displaystyle=E_{a,b|\star,\star}^{{A_{I}^{\prime}}{A_{O}^{\prime}}{B_{I}^{\prime}}{B_{O}^{\prime}}}*\left(\rho_{z_{I}}^{A_{I}^{\prime}}\otimes\rho_{z_{O}}^{A_{O}^{\prime}}\otimes\rho_{w_{I}}^{B_{I}^{\prime}}\otimes\rho_{w_{O}}^{B_{O}^{\prime}}\right)$
(49)

then define a distribution
$P^{\prime}(a,b|\rho_{z_{I}}^{A_{I}^{\prime}},\rho_{z_{O}}^{A_{O}^{\prime}},\rho_{w_{I}}^{B_{I}^{\prime}},\rho_{w_{O}}^{B_{O}^{\prime}})$
in a SDI-QI setting with the reference quantum inputs, that must respect any
witness inequality for correlations induced by D-POVM elements of the form of
Eq. (48). Now, we have shown in [15] that any bipartite causally nonseparable
process matrix can violate such a witness inequality. We thus conclude here
that any bipartite causally nonseparable process matrix can be certified in
the scenario considered here, which combines the NDI approach with the MDCI
structure assumption—i.e., in a “N-MDCI” (“Network-MDCI”) manner.

Recall that in [15] we also studied MDCI certification in the
“($2$+$F$)-partite” scenario (which involves an additional party, Fiona, in
the causal future of Alice and Bob and who has no output Hilbert space),
showing that a similar result holds for all “TTU-” and “TUU-noncausal”
processes as defined in [32]—that is, causally nonseparable processes that can
be certified with trusted operations for Alice and Bob (but not for Fiona), or
for Alice only. In particular, we showed that all “TTU-” and “TUU-noncausal”
processes could be certified in a “MDCI-MDCI-DI” and “MDCI-DI-DI” manner,
respectively, when imposing the extra assumption of Eq. (43) to the parties
with formally trusted operations (and still with trusted quantum inputs).
Following the same arguments as above, we can conclude that all such
processes—including the quantum switch–can also be certified in a N-MDCI
manner for those parties, thus relaxing the need for trusted quantum inputs.

Although we leave it as an open problem, we expect it to be rather
straightforward to further generalise all these results to the
($P$+$2$+$F$)-partite scenario, which also includes a party in the global past
and that we will consider in Appendix E.

##  Appendix E SDI-QI and NDI certification of causal nonseparability in the
“($P$+$2$+$F$)-partite” scenario

We consider here the “($P$+$2$+$F$)-partite” scenario introduced in the main
text, which extends the ($2$+$F$)-partite scenario (considered in particular
in [15]) by adding a new party, Phil, in the causal past of all other parties.
Phil is only connected to the process matrix through his output Hilbert space
$\mathcal{H}^{P}$ (he has no input Hilbert space). We shall first recall the
definition of causal nonseparability in this scenario, before considering its
SDI-QI certification when all four parties receive quantum inputs, as well as
when only Phil does, and applying this to the quantum switch. As in the
bipartite scenario, this SDI-QI certification can be readily transformed into
NDI certification, and we will briefly discuss this transformation, which is
performed analogously to the bipartite case, at the end of the section.

###  E.1 Causal (non)separability in the “($P$+$2$+$F$)-partite” scenario

Given that Phil is in the global past of the process, and that Fiona is in the
global future, the only relevant orders in this scenario are $P\prec A\prec
B\prec F$ and $P\prec B\prec A\prec F$. The causally separable process
matrices are those that can be decomposed as [2]

$\displaystyle W^{PABF}$ $\displaystyle=W^{P\prec A\prec B\prec F}+W^{P\prec
B\prec A\prec F}$ (50)

such that

$\displaystyle\operatorname{Tr}_{F}W^{P\prec A\prec B\prec F}=W^{P\prec A\prec
B_{I}}\otimes\mathbbm{1}^{B_{O}},$
$\displaystyle\qquad\operatorname{Tr}_{F}W^{P\prec B\prec A\prec F}=W^{P\prec
B\prec A_{I}}\otimes\mathbbm{1}^{A_{O}},$
$\displaystyle\operatorname{Tr}_{B_{I}}W^{P\prec A\prec B_{I}}=W^{P\prec
A_{I}}\otimes\mathbbm{1}^{A_{O}},$
$\displaystyle\qquad\operatorname{Tr}_{A_{I}}W^{P\prec B\prec A_{I}}=W^{P\prec
B_{I}}\otimes\mathbbm{1}^{B_{O}},$
$\displaystyle\operatorname{Tr}_{A_{I}}W^{P\prec A_{I}}+$
$\displaystyle\operatorname{Tr}_{B_{I}}W^{P\prec B_{I}}=\mathbbm{1}^{P},$ (51)

with positive semidefinite matrices $W^{P\prec A\prec
B_{I}}\in\mathcal{L}(\mathcal{H}^{PAB_{I}})$, $W^{P\prec B\prec
A_{I}}\in\mathcal{L}(\mathcal{H}^{PBA_{I}})$, $W^{P\prec
A_{I}}\in\mathcal{L}(\mathcal{H}^{PA_{I}})$, $W^{P\prec
B_{I}}\in\mathcal{L}(\mathcal{H}^{PB_{I}})$.

Note that in general $W^{P\prec A\prec B\prec F}$ and $W^{P\prec B\prec A\prec
F}$ are not necessarily valid (subnormalised) process matrices themselves
(unless $\operatorname{Tr}_{A_{I}}W^{P\prec A_{I}}\propto\mathbbm{1}^{P}$ and
$\operatorname{Tr}_{B_{I}}W^{P\prec B_{I}}\propto\mathbbm{1}^{P}$). Indeed
causally separable processes allow for the order between Alice and Bob to be
influenced by Phil’s action (in a classically controlled manner [13]), rather
than involving only convex combinations of processes with the fixed orders
$P\prec A\prec B\prec F$ and $P\prec B\prec A\prec F$. This explains why Eq.
(50), contrary to Eqs. (3) and (48), does not include convex weights, which
would otherwise suggest such a combination.

###  E.2 SDI-QI certification with quantum inputs for all 4 parties

Figure 3: ($P$+$2$+$F$)-partite SDI-QI scenario with quantum inputs for Phil,
Alice, Bob and Fiona. The trusted quantum inputs (orange)
$\rho_{t}^{{P^{\prime}}}\in\mathcal{L}(\mathcal{H}^{{P^{\prime}}}),\rho_{z}^{{A^{\prime}}}\in\mathcal{L}(\mathcal{H}^{{A^{\prime}}}),\rho_{w}^{{B^{\prime}}}\in\mathcal{L}(\mathcal{H}^{{B^{\prime}}}),\rho_{v}^{{F^{\prime}}}\in\mathcal{L}(\mathcal{H}^{{F^{\prime}}})$
are prepared and effectively measured by the D-POVM
$(E^{{P^{\prime}}{A^{\prime}}{B^{\prime}}{F^{\prime}}}_{p,a,b,f})_{p,a,b,f}$
generated by Phil, Alice, Bob and Fiona’s untrusted operations on the process
matrix $W^{PABF}$.

In a general ($P$+$2$+$F$)-partite SDI-QI scenario, all four parties Phil,
Alice, Bob and Fiona receive some quantum inputs $\rho_{t}^{{P^{\prime}}}$,
$\rho_{z}^{{A^{\prime}}}$, $\rho_{w}^{{B^{\prime}}}$ and
$\rho_{v}^{{F^{\prime}}}$ in some Hilbert spaces
$\mathcal{L}(\mathcal{H}^{P^{\prime}})$,
$\mathcal{L}(\mathcal{H}^{A^{\prime}})$,
$\mathcal{L}(\mathcal{H}^{B^{\prime}})$ and
$\mathcal{L}(\mathcal{H}^{F^{\prime}})$ respectively; see Fig. 3. Generalising
the bipartite case introduced in the main text, the correlations established
by the four parties are

$\displaystyle
P(p,a,b,f|\rho_{t}^{P^{\prime}},\rho_{z}^{A^{\prime}},\rho_{w}^{B^{\prime}},\rho_{v}^{F^{\prime}})$
$\displaystyle=\big{(}M_{p}^{{P^{\prime}}P}\otimes
M_{a}^{{A^{\prime}}A}\otimes M_{b}^{{B^{\prime}}B}\otimes
M_{f}^{{F^{\prime}}F}\big{)}*\big{(}\rho_{t}^{P^{\prime}}\otimes\rho_{z}^{A^{\prime}}\otimes\rho_{w}^{B^{\prime}}\otimes\rho_{v}^{F^{\prime}}\otimes
W^{PABF}\big{)}$
$\displaystyle=E_{p,a,b,f}^{{P^{\prime}}{A^{\prime}}{B^{\prime}}{F^{\prime}}}*\big{(}\rho_{t}^{P^{\prime}}\otimes\rho_{z}^{A^{\prime}}\otimes\rho_{w}^{B^{\prime}}\otimes\rho_{v}^{F^{\prime}}\big{)},$
(52)

where we introduced the D-POVM elements

$\displaystyle
E_{p,a,b,f}^{{P^{\prime}}{A^{\prime}}{B^{\prime}}{F^{\prime}}}=\big{(}M_{p}^{{P^{\prime}}P}\otimes
M_{a}^{{A^{\prime}}A}\otimes M_{b}^{{B^{\prime}}B}\otimes
M_{f}^{{F^{\prime}}F}\big{)}*W^{PABF}.$ (53)

Assuming that $W^{PABF}$ is a causally separable process matrix satisfying
Eqs. (50)–(51), the induced D-POVM
$\mathbb{E}^{{P^{\prime}}{A^{\prime}}{B^{\prime}}{F^{\prime}}}=(E_{p,a,b,f}^{{P^{\prime}}{A^{\prime}}{B^{\prime}}{F^{\prime}}})_{p,a,b,f}$
also decomposes as

$\displaystyle\mathbb{E}^{{P^{\prime}}{A^{\prime}}{B^{\prime}}{F^{\prime}}}=\mathbb{E}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}\prec{F^{\prime}}}+\mathbb{E}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}\prec{F^{\prime}}}$
(54)

with
$\mathbb{E}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}\prec{F^{\prime}}}=(E_{p,a,b,f}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}\prec{F^{\prime}}})_{p,a,b,f}$
and
$\mathbb{E}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}\prec{F^{\prime}}}=(E_{p,a,b,f}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}\prec{F^{\prime}}})_{p,a,b,f}$
such that

$\displaystyle
E_{p,a,b,f}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}\prec{F^{\prime}}}$
$\displaystyle=(M_{p}^{{P^{\prime}}P}\otimes M_{a}^{{A^{\prime}}A}\otimes
M_{b}^{{B^{\prime}}B}\otimes M_{f}^{{F^{\prime}}F})*W^{P\prec A\prec B\prec
F},$
$\displaystyle\sum_{f}E_{p,a,b,f}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}\prec{F^{\prime}}}$
$\displaystyle=(M_{p}^{{P^{\prime}}P}\otimes
M_{a}^{{A^{\prime}}A}\otimes\operatorname{Tr}_{B_{O}}M_{b}^{{B^{\prime}}B})*W^{P\prec
A\prec
B_{I}}\otimes\mathbbm{1}^{{F^{\prime}}}=E_{p,a,b}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}}\otimes\mathbbm{1}^{{F^{\prime}}},$
$\displaystyle\sum_{b}E_{p,a,b}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}}$
$\displaystyle=(M_{p}^{{P^{\prime}}P}\otimes\operatorname{Tr}_{A_{O}}M_{a}^{{A^{\prime}}A})*W^{P\prec
A_{I}}\otimes\mathbbm{1}^{{B^{\prime}}}=E_{p,a}^{{P^{\prime}}\prec{A^{\prime}}}\otimes\mathbbm{1}^{{B^{\prime}}},$
$\displaystyle\sum_{a}E_{p,a}^{{P^{\prime}}\prec{A^{\prime}}}$
$\displaystyle=M_{p}^{{P^{\prime}}P}*\operatorname{Tr}_{A_{I}}W^{P\prec
A_{I}}\otimes\mathbbm{1}^{{A^{\prime}}}=E_{p}^{{P^{\prime}}\,[A\prec
B]}\otimes\mathbbm{1}^{{A^{\prime}}},$ (55)

analogously for
$E_{p,a,b,f}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}\prec{F^{\prime}}}$
and the order $P\prec B\prec A\prec F$, and with

$\displaystyle E_{p}^{{P^{\prime}}\,[A\prec B]}+E_{p}^{{P^{\prime}}\,[B\prec
A]}$
$\displaystyle=M_{p}^{{P^{\prime}}P}*\mathbbm{1}^{P}=\operatorname{Tr}_{P}M_{p}^{{P^{\prime}}P},$
(56)

such that, since $\sum_{p}M_{p}^{{P^{\prime}}P}$ is a CPTP map,

$\displaystyle\sum_{p}(E_{p}^{{P^{\prime}}\,[A\prec
B]}+E_{p}^{{P^{\prime}}\,[B\prec A]})$
$\displaystyle=\mathbbm{1}^{{P^{\prime}}}.$ (57)

Note that the individual sets of positive semidefinite matrices
$\mathbb{E}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}\prec{F^{\prime}}}=(E_{p,a,b,f}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}\prec{F^{\prime}}})_{p,a,b,f}$,
$\mathbb{E}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}\prec{F^{\prime}}}=(E_{p,a,b,f}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}\prec{F^{\prime}}})_{p,a,b,f}$,
$\mathbb{E}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}}=(E_{p,a,b}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}})_{p,a,b}$,
$\mathbb{E}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}}=(E_{p,a,b}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}})_{p,a,b}$,
$\mathbb{E}^{{P^{\prime}}\prec{A^{\prime}}}=(E_{p,a}^{{P^{\prime}}\prec{A^{\prime}}})_{p,a}$,
$\mathbb{E}^{{P^{\prime}}\prec{B^{\prime}}}=(E_{p,b}^{{P^{\prime}}\prec{B^{\prime}}})_{p,b}$,
$\mathbb{E}^{{P^{\prime}}\,[A\prec B]}=(E_{p}^{{P^{\prime}}\,[A\prec
B]})_{p}$, $\mathbb{E}^{{P^{\prime}}\,[B\prec
A]}=(E_{p}^{{P^{\prime}}\,[B\prec A]})_{p}$ are not D-POVMs as they do not sum
to the identity in general. On the other hand, the set
$\mathbb{E}^{{P^{\prime}}\,[A\prec B]}\cup\mathbb{E}^{{P^{\prime}}\,[B\prec
A]}$ formed by all elements of $\mathbb{E}^{{P^{\prime}}\,[A\prec B]}$ and
$\mathbb{E}^{{P^{\prime}}\,[B\prec A]}$ taken together forms a valid POVM.

Analogously to the bipartite and ($2$+$F$)-partite cases, this leads us to the
following definition.

######  Definition 1.

A ($P$+$2$+$F$)-partite D-POVM
$\mathbb{E}^{{P^{\prime}}{A^{\prime}}{B^{\prime}}{F^{\prime}}}=(E_{p,a,b,f}^{{P^{\prime}}{A^{\prime}}{B^{\prime}}{F^{\prime}}})_{p,a,b,f}$
that can be decomposed in the form

$\displaystyle\mathbb{E}^{{P^{\prime}}{A^{\prime}}{B^{\prime}}{F^{\prime}}}=\mathbb{E}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}\prec{F^{\prime}}}+\mathbb{E}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}\prec{F^{\prime}}},$
(58)

where
$\mathbb{E}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}\prec{F^{\prime}}}=(E_{p,a,b,f}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}\prec{F^{\prime}}})_{p,a,b,f}$
and
$\mathbb{E}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}\prec{F^{\prime}}}=(E_{p,a,b,f}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}\prec{F^{\prime}}})_{p,a,b,f}$
satisfy

$\displaystyle\sum_{f}E_{p,a,b,f}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}\prec{F^{\prime}}}=E_{p,a,b}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}}\otimes\mathbbm{1}^{{F^{\prime}}},\quad$
$\displaystyle\sum_{b}E_{p,a,b}^{{P^{\prime}}\prec{A^{\prime}}\prec{B^{\prime}}}=E_{p,a}^{{P^{\prime}}\prec{A^{\prime}}}\otimes\mathbbm{1}^{{B^{\prime}}},\quad\sum_{a}E_{p,a}^{{P^{\prime}}\prec{A^{\prime}}}=E_{p}^{{P^{\prime}}\,[A\prec
B]}\otimes\mathbbm{1}^{{A^{\prime}}},$
$\displaystyle\sum_{f}E_{p,a,b,f}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}\prec{F^{\prime}}}=E_{p,a,b}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}}\otimes\mathbbm{1}^{{F^{\prime}}},\quad$
$\displaystyle\sum_{a}E_{p,a,b}^{{P^{\prime}}\prec{B^{\prime}}\prec{A^{\prime}}}=E_{p,b}^{{P^{\prime}}\prec{B^{\prime}}}\otimes\mathbbm{1}^{{A^{\prime}}},\quad\sum_{b}E_{p,b}^{{P^{\prime}}\prec{B^{\prime}}}=E_{p}^{{P^{\prime}}\,[B\prec
A]}\otimes\mathbbm{1}^{{B^{\prime}}},$
$\displaystyle\sum_{p}(E_{p}^{{P^{\prime}}\,[A\prec
B]}+E_{p}^{{P^{\prime}}\,[B\prec A]})=\mathbbm{1}^{{P^{\prime}}}$ (59)

(with all operators $E_{\cdots}^{\cdots}\geq 0$) is said to be _causally
separable_.

As noted above, a causally separable ($P$+$2$+$F$)-partite process matrix can
only generate a causally separable ($P$+$2$+$F$)-partite D-POVM. _A contrario_
, if one finds that, for some choice of operations for each party, the induced
D-POVM is causally nonseparable, this certifies in a SDI-QI manner that the
process matrix that generated it is itself causally nonseparable. Similarly to
the bipartite and the ($2$+$F$)-partite cases, one can construct a causal
witness—and associated witness inequality—that can be measured by providing
tomographically complete quantum inputs to the parties, thereby allowing one
to certify the causal non-separability of ($P$+$2$+$F$)-partite D-POVMs [10,
24, 15].

###  E.3 SDI-QI certification with quantum inputs only for Phil

SDI-QI scenarios can also be considered, where not all parties receive quantum
inputs: some may receive classical inputs (or no input at all). For example,
in [15] we considered the ($2$+$F$)-partite case where only Alice and Bob (but
not Fiona) receive quantum inputs—a situation that could readily be extended
to the ($P$+$2$+$F$)-partite case.

Figure 4: ($P$+$2$+$F$)-partite SDI-QI scenario with quantum inputs solely for
Phil. The trusted quantum input (orange)
$\rho_{t}\in\mathcal{L}(\mathcal{H}^{{P^{\prime}}})$ is prepared and
effectively measured by the D-POVMs
$(E^{{P^{\prime}}}_{p,a,b,f|z,w,v})_{p,a,b,f}$ generated by Phil, Alice, Bob
and Fiona’s untrusted operations on the process matrix $W^{PABF}$.

Let us here consider instead the case where only Phil receives quantum input
states $\rho_{t}^{{P^{\prime}}}\in\mathcal{L}(\mathcal{H}^{{P^{\prime}}})$,
while Alice, Bob and Fiona are given classical inputs $z$, $w$ and $v$
respectively; see Fig. 4. Conditioning their instruments explicitly on the
classical inputs,111111Classical inputs can also be fitted into the framework
with quantum inputs by using orthogonal quantum input states. Indeed, one can
encode (e.g., for Alice) the classical inputs into some orthonormal basis
$\\{\mathinner{|{z}\rangle}^{A^{\prime}}\\}_{z}$, and introduce the
instruments
$(M_{a}^{{A^{\prime}}A})_{a}=\big{(}\sum_{z}\mathinner{|{z}\rangle\\!\langle{z}|}^{A^{\prime}}\otimes
M_{a|z}^{A}\big{)}_{a}$ [15]. One can use this, for instance, to obtain the
analogous constraints to Eqs. (54)–(57) for classical inputs $z,w,v$, and to
thereby justify the decomposition in Definition 2—and in particular, the “no-
signalling constraints” (e.g., that
$\sum_{f}E_{p,a,b,f|z,w,v}^{{P^{\prime}}\,[A\prec
B]}=E_{p,a,b|z,w}^{{P^{\prime}}\,[A\prec B]}$ does not depend on $v$, that
$\sum_{b}E_{p,a,b|z,w}^{{P^{\prime}}\,[A\prec
B]}=E_{p,a|z}^{{P^{\prime}}\,[A\prec B]}$ does not depend on $w$, etc.). the
correlations established by the four parties are given here by

$\displaystyle P(p,a,b,f|\rho_{t}^{P^{\prime}},z,w,v)$
$\displaystyle=\big{(}M_{p}^{{P^{\prime}}P}\otimes M_{a|z}^{A}\otimes
M_{b|w}^{B}\otimes M_{f|v}^{F}\big{)}*\big{(}\rho_{t}^{P^{\prime}}\otimes
W^{PABF}\big{)}$
$\displaystyle=E_{p,a,b,f|z,w,v}^{{P^{\prime}}}*\rho_{t}^{P^{\prime}},$ (60)

with the D-POVM elements121212While the D-POVM acts on a single Hilbert space
$\mathcal{H}^{P^{\prime}}$, accessed by Phil only, it is still distributed in
the sense that the classical inputs and outputs are associated with
distributed parties, even if no quantum systems are associated with them.

$\displaystyle
E_{p,a,b,f|z,w,v}^{{P^{\prime}}}=\big{(}M_{p}^{{P^{\prime}}P}\otimes
M_{a|z}^{A}\otimes M_{b|w}^{B}\otimes M_{f|v}^{F}\big{)}*W^{PABF}.$ (61)

Assuming that $W^{PABF}$ is causally separable, one obtains a similar
decomposition to Eqs. (54)–(57), for the whole family of D-POVMs
$\\{\mathbb{E}_{|z,w,v}^{{P^{\prime}}}=(E_{p,a,b,f|z,w,v}^{{P^{\prime}}})_{p,a,b,f}\\}_{z,w,v}$
(i.e., involving the D-POVMs for all classical inputs $z,w,v$). Similarly to
the previous case, we introduce here the following definition.

######  Definition 2.

A family of ($P$+$2$+$F$)-partite D-POVMs
$\\{\mathbb{E}_{|z,w,v}^{{P^{\prime}}}=(E_{p,a,b,f|z,w,v}^{{P^{\prime}}})_{p,a,b,f}\\}_{z,w,v}$
that can be decomposed in the form

$\displaystyle\mathbb{E}_{|z,w,v}^{{P^{\prime}}}=\mathbb{E}_{|z,w,v}^{{P^{\prime}}\,[A\prec
B]}+\mathbb{E}_{|z,w,v}^{{P^{\prime}}\,[B\prec A]},$ (62)

for all $z,w,v$, where $\mathbb{E}_{|z,w,v}^{{P^{\prime}}\,[A\prec
B]}=(E_{p,a,b,f|z,w,v}^{{P^{\prime}}\,[A\prec B]})_{p,a,b,f}$ and
$\mathbb{E}_{|z,w,v}^{{P^{\prime}}\,[B\prec
A]}=(E_{p,a,b,f|z,w,v}^{{P^{\prime}}\,[B\prec A]})_{p,a,b,f}$ satisfy

$\displaystyle\sum_{f}E_{p,a,b,f|z,w,v}^{{P^{\prime}}\,[A\prec
B]}=E_{p,a,b|z,w}^{{P^{\prime}}\,[A\prec B]},\quad$
$\displaystyle\sum_{b}E_{p,a,b|z,w}^{{P^{\prime}}\,[A\prec
B]}=E_{p,a|z}^{{P^{\prime}}\,[A\prec
B]},\quad\sum_{a}E_{p,a|z}^{{P^{\prime}}\,[A\prec
B]}=E_{p}^{{P^{\prime}}\,[A\prec B]},$
$\displaystyle\sum_{f}E_{p,a,b,f|z,w,v}^{{P^{\prime}}\,[B\prec
A]}=E_{p,a,b|z,w}^{{P^{\prime}}\,[B\prec A]},\quad$
$\displaystyle\sum_{a}E_{p,a,b|z,w}^{{P^{\prime}}\,[B\prec
A]}=E_{p,b|w}^{{P^{\prime}}\,[B\prec
A]},\quad\sum_{b}E_{p,b|w}^{{P^{\prime}}\,[B\prec
A]}=E_{p}^{{P^{\prime}}\,[B\prec A]},$
$\displaystyle\sum_{p}(E_{p}^{{P^{\prime}}\,[A\prec
B]}+E_{p}^{{P^{\prime}}\,[B\prec A]})=\mathbbm{1}^{{P^{\prime}}}$ (63)

(with all operators $E_{\cdots}^{\cdots}\geq 0$) is said to be _causally
separable_.

As always, a causally separable ($P$+$2$+$F$)-partite process matrix
$W^{PABF}$ can only generate a causally separable family of
($P$+$2$+$F$)-partite D-POVMs, so that obtaining a causally nonseparable
family of ($P$+$2$+$F$)-partite D-POVMs (which can be verified using some form
of causal witnesses) certifies in a SDI-QI manner the causal nonseparability
of $W^{PABF}$.

###  E.4 SDI-QI certification of the quantum switch in the
“($P$+$2$+$F$)-partite” scenario

The quantum switch [3] is the canonical example of a quantum process with
indefinite causal order, which can be understood as a quantum circuit with
quantum control of causal order [13]. Despite being causally nonseparable,
however, the quantum switch can only generate so-called causal correlations
[10, 11], so that its causal nonseparability cannot be certified in the
standard DI way.

As recalled previously, in [15] we showed that the causal nonseparability of
the quantum switch could nevertheless be certified in a ($2$+$F$)-partite SDI-
QI scenario where Alice and Bob receive quantum inputs. This scenario can be
seen as a particular case of a ($P$+$2$+$F$)-partite one, with a trivial role
for the past party Phil. Introducing a nontrivial $P$ allows one to also
provide the latter with quantum inputs. We will show here that the causal
nonseparability of the quantum switch can still be certified if only Phil
receives quantum inputs.

In the most general form of the quantum switch, as described in the main text,
Phil sends and Fiona receives both “target” and “control” systems, in spaces
$\mathcal{H}^{P_{\text{t}}}$, $\mathcal{H}^{P_{\text{c}}}$,
$\mathcal{H}^{F_{\text{t}}}$ and $\mathcal{H}^{F_{\text{c}}}$, respectively.
Here, let us take the target system to be initialised in the state
$\mathinner{|{0}\rangle\\!\langle{0}|}^{P_{\text{t}}}\in\mathcal{L}(\mathcal{H}^{P_{\text{t}}})$
and traced out in the future space $\mathcal{H}^{F_{\text{t}}}$, and the qubit
control system to be prepared by Phil in the global past and given to Fiona in
the global future. The quantum switch can then be written as a
($P$+$2$+$F$)-partite process matrix as

$W_{\text{QS}}=\mathinner{|{0}\rangle\\!\langle{0}|}^{P_{t}}*\operatorname{Tr}_{F_{\text{t}}}(\mathinner{|{w_{\text{QS}}}\rangle\\!\langle{w_{\text{QS}}}|}^{PP_{\text{t}}ABF_{\text{t}}F}),$
(64)

with

$\mathinner{|{w_{\text{QS}}}\rangle}^{PP_{\text{t}}ABF_{\text{t}}F}=\mathinner{|{0}\rangle}^{P}\mathinner{|{\mathbbm{1}}\rangle\\!\rangle}^{P_{\text{t}}A_{I}}\mathinner{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}B_{I}}\mathinner{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}F_{\text{t}}}\mathinner{|{0}\rangle}^{F}+\mathinner{|{1}\rangle}^{P}\mathinner{|{\mathbbm{1}}\rangle\\!\rangle}^{P_{\text{t}}B_{I}}\mathinner{|{\mathbbm{1}}\rangle\\!\rangle}^{B_{O}A_{I}}\mathinner{|{\mathbbm{1}}\rangle\\!\rangle}^{A_{O}F_{\text{t}}}\mathinner{|{1}\rangle}^{F}$
(65)

where
$\mathinner{|{\mathbbm{1}}\rangle\\!\rangle}^{XX^{\prime}}=\sum_{i}\mathinner{|{i}\rangle}^{X}\otimes\mathinner{|{i}\rangle}^{X^{\prime}}$
(and with implicit tensor products) [13].131313When considering isomorphic
Hilbert spaces $\mathcal{H}^{X}$ and $\mathcal{H}^{X^{\prime}}$, we take their
computational bases $\\{\mathinner{|{i}\rangle}^{X^{(\prime)}}\\}_{i}$ to be
in one-to-one correspondence.

Let us then consider the operations

$\displaystyle
M^{{P^{\prime}}P}=\mathinner{|{\mathbbm{1}}\rangle\\!\rangle\\!\langle\\!\langle{\mathbbm{1}}|}^{{P^{\prime}}P},\quad
M_{a|z}^{A}=\mathinner{|{a}\rangle\\!\langle{a}|}^{A_{I}}\otimes\mathinner{|{z}\rangle\\!\langle{z}|}^{A_{O}},\quad
M_{b|w}^{B}=\mathinner{|{b}\rangle\\!\langle{b}|}^{B_{I}}\otimes\mathinner{|{w}\rangle\\!\langle{w}|}^{B_{O}},\quad
M_{f=\pm}^{F}=\mathinner{|{\pm}\rangle\\!\langle{\pm}|}^{F},$ (66)

which can be interpreted as follows (see Fig. 5): Phil simply implements an
identity channel that directly forwards the quantum input into the quantum
switch, with no (or just a trivial) output $p$. Alice and Bob measure the
target system (taken here to be a qubit) in the computational basis with
classical outcomes $a,b=0,1$, and re-prepare it in the computational basis
state that corresponds to their input $z,w=0,1$. Fiona has no (or just a
trivial) input $v$ and she simply measures the control qubit in the
$\\{\mathinner{|{\pm}\rangle}=\frac{1}{\sqrt{2}}(\mathinner{|{0}\rangle}\pm\mathinner{|{1}\rangle})\\}_{\pm}$
basis.

Figure 5: D-POVM generated by the quantum instruments of Eq. (66), which can
be used to certify the causal nonseparability of the quantum switch in a SDI-
QI manner.

As it turns out, when implemented on a quantum switch, these instruments
generate a causally nonseparable family of D-POVMs
$\\{\mathbb{E}_{|z,w}^{{P^{\prime}}}=(E_{a,b,f|z,w}^{{P^{\prime}}})_{a,b,f}\\}_{z,w}$,
according to Definition 2 (with the trivial $p$ and $v$ omitted in this
special case). Indeed, using a semidefinite programming approach, we can show
that these D-POVMs cannot be decomposed as in Eqs. (62)–(63), and a causal
witness can be constructed to test this [10, 24, 15]. More specifically, we
find that a “depolarised” version of the quantum switch
$W_{\text{QS}}(r)={\textstyle\frac{1}{1+r}}(W_{\text{QS}}+r\,\mathbbm{1}^{\circ})$,
obtained by mixing it with some fully white noise described by the process
matrix $\mathbbm{1}^{\circ}=\mathbbm{1}^{PABF}/8$ (by some amount $r\geq 0$),
induces with the above instruments a causally nonseparable family of D-POVMs
for all $r<2-\sqrt{2}$ (up to numerical precision).

Hence, as we claimed, the quantum switch can be certified in a SDI-QI manner
by simply letting Phil receive quantum inputs. Of course, providing more
parties with quantum inputs (or here, also providing Fiona with some input, or
letting Phil also have outputs) could only make the certification more
efficient.

Finally, as for any SDI-QI scenario considered in this work, the SDI-QI
certification of a causally nonseparable process in the ($2$+$F$)- and
($P$+$2$+$F$)-partite scenarios can readily be turned into a NDI certification
by introducing new parties that share maximally entangled states with
whichever parties receive quantum inputs, and by self-testing these states and
the new parties’ measurements. For the quantum switch, in the case where only
Phil receives a quantum input that he directly feeds into the quantum switch,
as considered above (see Eq. (65)), this leads to a very similar scenario to
that considered in Ref. [30], which we shall discuss in the next appendix.

##  Appendix F Comparing the NDI and DRF certifications of the quantum switch

In this appendix we provide some further comparison between our NDI
certification of the causal nonseparability of the quantum switch and the DRF
certification of Ref. [30]. This complements the discussion in the main text
of the underlying assumptions behind these two approaches.

As mentioned at the end of Appendix E.4, the SDI-QI certification of the
quantum switch described therein can be transformed into a NDI certification
by following the same procedure as for the bipartite scenario discussed in
detail in the main text. In particular, the SDI-QI certification is
transformed first into a NSDI-QI certification by adding an extra separated
party, Emily, who shares a maximally entangled state with Phil and remotely
prepares his quantum inputs $\rho_{t}^{{P^{\prime}}}$, and then into a NDI
certification by self-testing both the state Phil and Emily share, and Emily’s
measurements. The certification is then obtained by violating a witness
inequality using the observed correlations
$P_{*}(p,a,b,f|t,z,w):=P(p,a,b,f|u{=}*,t,z,w,e{=}0)=P^{\prime}(p,a,b,f|\rho_{t}^{{P^{\prime}}},z,w)$,
analogously to the bipartite setting. This NDI scenario is shown in Fig. 6.

Figure 6: Scenario for the NDI certification of the quantum switch with a
global past. The remote preparation of a quantum input
$\rho_{t}^{{P^{\prime}}}$ is self-tested by first self-testing the (untrusted)
shared state $\uppsi^{{\tilde{P}}\tilde{E}}$ (to contain a maximally entangled
state) and Emily’s measurements ${\mathsf{M}}_{e|t}^{\tilde{E}}$ (to
effectively be tomographically complete projectors) from the correlations
$P(p,e|u,t)$ shared by Phil and Emily. One can then certify the causal
nonseparability of the family of D-POVMs
$\\{(E_{p,a,b,f|u,z,w}^{\tilde{P}})_{p,a,b,f}\\}_{u,z,w}$ and thus the causal
nonseparability of the process matrix $W^{PABF}$.

As we already noted, Ref. [30] showed how to device-independently certify the
causal indefiniteness of the quantum switch by the violation of an inequality
derived from the assumptions of Definite causal order, Relativistic causality,
and Free interventions. The structure of the “DRF scenario” they considered is
very similar to that of our NDI certification, but there are some important
differences. In the DRF scenario, a (space-like) separated Emily is also added
to a standard quantum switch scenario. However, the scenario contains no Phil
and, in contrast to the NDI certification, Fiona must be provided with a
classical input (which we could also have done, but did not need to). By
having Alice share an entangled state with the control system in the global
past of a quantum switch (directly incorporated in the process, instead of
being provided by Phil), the parties can generate correlations violating the
DRF inequality.

The DRF inequality itself, rephrased here in notation consistent with Fig. 6
and with $v$ denoting Fiona’s classical input, has the form [30]

$\displaystyle\mathcal{I}_{\text{DRF}}=P(e=0,b=z|t=0)+P(e=1,a=w|t=0)+P(f\oplus
e=vt|z=w=0)\leq\frac{7}{4}.$ (67)

One can interpret this inequality as certifying (i) a space-like separated
“control of causal order” (the first two terms of $\mathcal{I}_{\text{DRF}}$),
and (ii) the non-classical, indeterministic behavior of the control (the last
term of $\mathcal{I}_{\text{DRF}}$). The latter of these can be interpreted as
a CHSH inequality tested between Fiona (acting after the quantum switch, and
who is thus provided with inputs) and Emily. It can be violated if the control
system passes transparently through the process, which can be ensured to be
the case if Alice and Bob perform appropriate operations (on inputs $z=w=0$).
On the other hand, while the NDI certification also relies on the violation of
a Bell inequality, it is tested between Emily and Phil—who is in the causal
past of the quantum switch and has classical inputs and outputs—to self-test
the quantum state they share. Fiona, in this case, needs no classical input
and performs a fixed measurement. Curiously, however, we were able to obtain
NDI certification using exactly the same measure-and-prepare instruments for
Alice and Bob (see Eq. (66)) as were used to violate the DRF inequality of Eq.
(67) in [30].

Finally, although we saw that our approach can, in the SDI-QI setting, certify
the causal nonseparability of the quantum switch even in the presence of
substantial noise on the quantum switch, the NDI certification is less
tolerant to noise on the entangled state shared between Phil and Emily.
Indeed, although we saw in Appendix C that the NDI certification is “robust”,
the actual reduction in Bell-inequality violation that can be tolerated in the
self-testing procedure is typical small in absolute terms [17].141414We leave
it as an open question to calculate the exact robustness, e.g. for the optimal
witness inequality for the certification of the D-POVM obtained from the
quantum switch from the operations of Eq. (66). Note, however, that with a
noiseless maximally entangled state shared between Phil and Emily, the NDI
certification can tolerate the same amount of noise on the quantum switch as
the SDI-QI certification. The DRF inequality, on the other hand, appears to be
much more robust to noise on the entangled state shared between Emily and
Fiona after passing through the quantum switch, as any violation of the
classical bound of $3/4$ of the last term in $\mathcal{I}_{\text{DRF}}$
(corresponding to a CHSH inequality between them) can potentially lead to a
violation of the DRF inequality. A more precise quantitive comparison of the
robustness of the NDI and DRF certifications under different noise models
would be an interesting direction of future study.