Device-independent certification of indefinite causal order in the quantum switch

Quantum theory is compatible with scenarios in which the order of operations is indefinite. Experimental investigations of such scenarios, all of which have been based on a process known as the quantum switch, have provided demonstrations of indefinite causal order conditioned on assumptions on the devices used in the laboratory. But is a device-independent certification possible, similar to the certification of Bell nonlocality through the violation of Bell inequalities? Previous results have shown that the answer is negative if the switch is considered in isolation. Here, however, we present an inequality that can be used to device-independently certify indefinite causal order in the quantum switch in the presence of an additional spacelike-separated observer under an assumption asserting the impossibility of superluminal and retrocausal influences.

In recent years, a number of strategies to certify indefinite causal order in the quantum switch have been developed [5,[26][27][28] and adopted in experimental investigations [29][30][31][32][33][34][35].A common characteristic of these strategies is that they are device-dependent, in the sense that they rely on assumptions on the devices used in the laboratory and the physical theory that governs them.To provide stronger evidence of indefinite causal order, it is desirable to have a device-independent certification, which only relies on the statistics of measurement outcomes, in the same way as violation of a Bell inequality certifies Bell nonlocality.
For some causally nonseparable processes, such deviceindependent certification is possible through the violation of causal inequalities [3,6,[36][37][38][39]; however, the physicality of these processes is still unclear [39][40][41][42].The quantum switch, on the other hand-the only causally nonseparable process to have been studied experimentally-has been shown not to violate any such inequality [5,6], a result that was recently extended to the broader class of quantum circuits with quantum control of causal order [43,44].As a consequence, a device-independent certification of indefinite causal order for the quantum switch has so far been missing, leaving open the question whether it is compatible with a hidden variable description in which the order is well-defined.
In this paper we extend the standard causal inequality scenario by adding a spacelike-separated party.We derive a set of device-independent inequalities satisfied by all correlations observed in experiments satisfying the three assumptions of 'Definite Causal Order', 'Relativistic Causality', and 'Free Interventions', the second of which rules out causal influences outside the future lightcone.We then show that these inequalities are violated by a quantum process involving the quantum switch and an additional system entangled to the switch's control qubit.This establishes a device-independent certification of indefinite causal order for the quantum switch, under the assumptions of Relativistic Causality and Free Interventions.Crucially, our notion of Relativistic Causality is strictly weaker than Bell Locality, which is already known to be violated by quantum physics [45,46]; in particular, it (together with Free Interventions) only entails parameter independence, while Bell Locality also requires outcome independence [47].In addition to deriving the inequalities and their violation, we begin to unravel the structure of the corresponding correlation polytope, which shares features with causal polytopes, no-signalling polytopes, and Bell-local polytopes.

Results
Device-independent inequality We will consider an experiment carried out by four agents, Alice 1 (A 1 ), Alice 2 (A 2 ), Bob (B), and Charlie (C), who each perform one intervention in the course of each run.The experiment is set up in such a way that Charlie's intervention always occurs in the future lightcone of those of Alice 1 and 2, and Bob's intervention is spacelike-separated from those of the other agents (see Figure 1a).Consider the following causal assumptions.
• Definite Causal Order: There is a variable λ, taking a value on each run of the experiment, and associated partial orders ≺ λ on {A 1 , A 2 , B, C}, such that on each run, the four agents are causally ordered according to ≺ λ (cf.Ref. [3]).Without loss of generality, we will assume that λ takes values in {1, 2}, where 1b).(Strictly speaking, Relativistic Causality leaves open the possibility for other causal orders; their contribution to the argument is however already covered by ≺ 1 and ≺ 2 .See Methods and Supplementary Note 2 for a proof and for more formal statements of the assumptions.)We now consider device-independent data in the form of correlations between classical settings x 1 , x 2 , y, z and outcomes a 1 , a 2 , b, c of the agents' interventions.The following third assumption imposes constraints on these correlations on the basis of the purely causal assumptions above.
• Free Interventions: The settings x 1 , x 2 , y, z have no relevant causes.In particular, they are (i) statistically independent of the hidden variable λ, and (ii) conditioned on any value of λ, statistically independent of any outcome variables of agents outside their ≺ λ -future.This means that agents cannot signal outside their ≺ λ -future, even when the value of λ is known.
Part (i) of this assumption implies that the observed correlations, represented by a conditional probability distribution p(a 1 a 2 bc|x 1 x 2 yz) =: p(⃗ abc|⃗ xyz), can be written as The no-signalling conditions of part (ii) can then be expressed as p( • | • λ) ∈ DRF λ , where NS := {q ∈ P ⃗ abc|⃗ xyz : ⃗ ac ⊥ ⊥ q y and b ⊥ ⊥ q ⃗ xz}; (2) (3) Here P ⃗ abc|⃗ xyz is the set of conditional probability distributions, while ⊥ ⊥ q denotes statistical independence: for example, ⃗ ac ⊥ ⊥ q y means ∀⃗ a, c, ⃗ x, y, y ′ , z : b q(⃗ abc|⃗ xyz) = b q(⃗ abc|⃗ xy ′ z).NS is the set of correlations with no signalling between Bob and the other agents.
We will denote by DRF the set of all correlations p(⃗ abc|⃗ xyz) arising in experiments satisfying Definite Causal Order, Relativistic Causality, and Free Interventions-i.e. those of the form (1) with p( • | • λ) ∈ DRF λ .It is a polytope (see Methods), and is given by the convex hull (see Figure 1c).
A few comments about our three assumptions are in order.First of all, note that if a delay between the generation of the setting x 1 and outcome a 1 of Alice 1 is present, and two-way communication with Alice 2 during this period is allowed (or vice versa), then arbitrarily strong two-way signalling correlations between Alice 1 and 2 can arise.This includes correlations not in DRF.Indeed, the Definite Causal Order assumption becomes interesting only when the agents' laboratories are assumed 'closed', in the sense that communication during such a delay (if present) is not allowed [3] Finally, in general one should allow for dynamical causal order, wherein the causal order between agents depends on interventions performed by agents in their causal past [6,37,48].This would contradict part (i) of Free Interventions; however, since by Relativistic Causality no agents are in the causal past of Alice 1 and Alice 2, in our case this does not lead to any more general correlations than those already in the polytope DRF defined above.This is proved in Methods.
From now on, let us consider all variables a 1 , a 2 , b, c, x 1 , x 2 , y, z to take values in {0, 1}.⊕ denotes addition modulo 2.Moreover, to condense notation, we assume that the settings x 1 , x 2 , y, z are independent and uniformly distributed (see Equation ( 14) in Methods for an example).The following inequality, together with its violation by the quantum switch demonstrated in the next section, forms our main result.6) is defined by a 1 = 0, a 2 = x 1 , c = 0 and b = 0; a nondeterministic example is given by setting a 1 = 0, a 2 = x 1 and letting Bob and Charlie use a PR box [50].(PR correlations-which are maximally Bell-nonlocal yet nonsignalling-are allowed in DRF, as we do not assume full Bell Locality.) Violation by the quantum switch The quantum switch is one of the few causally nonseparable processes that has a known physical interpretation, and the only such process to date that has been studied experimentally [29][30][31][32][33][34].Yet, the device-independent correlations that it generates do not violate any causal inequalities as previously considered in literature [5,6].(This is explained in more detail in Supplementary Note 1.)Here we will show that it does violate the inequality in Theorem 1.
The quantum switch can be described as a bipartite supermap [51], i.e. a map S taking two quantum operations E, F on a system T , here taken to be a qubit, to an operation S(E, F) on the joint system CT , which applies E and F to the target system T in an order that is coherently controlled by the state of the control qubit C (see Figure 2).Hence, if these systems are described by Hilbert spaces The quantum switch.Drawn here in blue, it is a bipartite supermap taking two quantum operations on the system T , denoted E and F, to an operation on CT , where C is the control qubit (see Equation ( 7)).The dotted (red) and dashed (blue) lines illustrate the wirings to which the quantum switch reduces upon preparation of C in state |0⟩⟨0| and |1⟩⟨1|, respectively.To see how the four agents discussed in the previous section can violate Inequality (6) when they have access to a quantum switch, we prepare the target system in the initial state |0⟩ T while entangling the input control qubit C to an additional qubit B in the state |Φ + ⟩ := (|00⟩ + |11⟩)/ √ 2 (see Figure 3).Alice 1 and Alice 2, placed inside the two slots of the switch, use measure-and-prepare instruments: for i = 1, 2, Alice i measures the incoming target system T in the computational basis-independently of her setting x i -and records the outcome in a i .She then prepares T in the computational basis state |x i ⟩, before sending it away.Bob has access to the spacelike-separated qubit B, which he measures in the computational (Z) direction if y = 0, and in the X direction if y = 1; he records his outcome in b.Finally, Charlie measures the output control qubit C in the Z +X (for z = 0) or Z −X (for z = 1) direction, recording his outcome in c.The output target system is discarded.
With these choices of instruments and state preparations, the first two terms in Inequality ( 6) are both 1/2: for instance, if y = 0, Bob obtains b = 0 with probability 1/2; and postselecting on that outcome yields the same correlations in the switch as if the control qubit had been prepared in state |0⟩ C .The latter would reduce the switch to a wiring in which Alice 1 is before Alice 2, meaning that a 2 = x 1 .(Similarly for the second term.)For the third term of ( 6), note that if x 1 = x 2 = 0 then Alice 1 and 2 will both measure and reprepare the target system to be in state |0⟩ T ; in particular, their operations commute on the initial target state |0⟩ T , so that the state of the control system is unaffected.This means that Bob and Charlie perform an ordinary Bell test on the maximally entangled state |Φ + ⟩ CB (see Eq. ( 22) in Methods).With the choice of measurement directions given above, this yields a CHSH value of 1/2 + √ 2/4, so that Inequality ( 6) is violated: This shows that the correlations observed in this quantum switch setup do not admit a hidden variable model satisfying Equations ( 1)-( 4), thus establishing indefinite causal order in the quantum switch under the assumptions of Relativistic Causality and Free Interventions.Equation ( 8) is in fact the maximal quantum violation of Inequality (6) in this scenario, or indeed in any quantum scenario where Bob's observables commute with Alice's and Charlie's: this follows from the Tsirelson bound [52] and the fact that the algebraic maximum of the first two terms is 1.
More inequalities Table 1 presents some more inequalities that are valid and tight for the polytope DRF of correlations admitting a hidden variable model satisfying Definite Causal Order, Relativistic Causality, and Free Interventions.The inequalities listed here do not involve Charlie's measurement setting z; thus, they define faces (though not necessarily facets) of a lower-dimensional version of DRF, which is more amenable to computational analysis and to experimental tests of inequality violations.The polytope and the faces listed here are discussed in more detail in Methods.
Inequalities (i)-(iii) are similar to Inequality (6), and are (weakly) violated by the quantum switch using the same setup as described earlier and depicted in Figure 3, but with z fixed to 0. To understand inequality (iv), observe that the Alices can use their measure-and-prepare instruments to effectively perform a computational basis measurement of the input control qubit, with outcome a 1 , by setting x 2 = 1.Indeed, in this case, Alice 2 prepares the target system in state |1⟩ T , while it was initially prepared in state |0⟩ T ; therefore, each value of a 1 is only compatible with one of the computational basis states of the control qubit C (see Eq. ( 23) in Methods).This observation suggests that the argument in the Proof Sketch of Theorem 1 can also be applied to correlations between the causal order variable λ and the outcome a 1 , rather than b.This is witnessed by Inequality (iv).Its first three terms are constructed in such a way that a high value for them implies a strong correlation between λ and a 1 for the settings x 1 = x 2 = 1, thereby bounding the final CHSH term, which now involves a 1 .In the quantum switch, on the other hand, Alice's measure-and-prepare instruments yield the maximum value of 1 for the first three terms, while their effective Z measurement of the The inequalities are listed along with the dimensions of the faces of DRF they support; each 85-dimensional face constitutes a facet of the polytope.Boldface highlights aspects in which an inequality differs from the preceding one.For conciseness, we assume that all settings are independently and uniformly distributed (see Equation ( 14)).Inequalities (iii), (vi), and (viii) were found computationally, while the others were derived analytically.

Face-defining inequality Dimension
control qubit described above contributes to a high value for the CHSH term, thus violating Inequality (iv).With appropriate measurement directions for Bob (Z + X and Z − X) and Charlie (X), it is violated up to the quantum bound, just like Inequality (6).Merits of Inequality (iv) as compared to ( 6) are however that it does not involve a setting for Charlie and that its proof relies on mathematically weaker assumptions (see Methods).
The final four inequalities in Table 1 show the similarity between the facets of the bipartite causal polytope studied in previous literature [36] ((v) and (vii)) and some of the facets of DRF ((vi) and (viii)), thus highlighting one consequence of adding the Relativistic Causality assumption.None of these inequalities can however be violated by the quantum switch, because they do not involve the variable c (see Supplementary Note 1).They are discussed in more detail in Methods.

Discussion
The quantum switch, when considered in isolation, does not violate causal inequalities as previously defined in the literature [5,6].As a result, it has long been believed that the indefinite causal order of the quantum switch does not in general admit a device-independent certification.The present result however shows that such a certification is possible when the set of allowed causal orders is constrained.In our case these constraints arise from spatiotemporal information together with a Relativistic Causality assumption ruling out influences outside the future lightcone, although the constraints could be motivated differently too (e.g. by the topology of an experimental setup).Together with Free Interventions, these causal constraints impose conditions akin to what is known as parameter independence in the context of Bell's theorem.
We arrived at this result by deriving an inequality and exhibiting a quantum switch setup violating it up to the quantum bound.The intuition behind this violation lies in the fact that in our setup, one of Bob's outcomes is simultaneously correlated to the causal order in the switch (if such a causal order is assumed to exist) and to Charlie's measurements in such a way that Bob and Charlie violate a CHSH inequality.The monogamy of Bell nonlocality tells us that such simultaneous correlations can only arise when one of Relativistic Causality and Free Interventions is violated.
Note that violation of our inequalities requires violation of a CHSH inequality.As such, they cannot be violated by classical processes subject to the same spatiotemporal constraints.This sets them apart from regular causal inequalities, which can be violated by both quantum and classical processes [38] and can therefore not distinguish between classical and nonclassical indefinite causality.
It is worth noting that locality assumptions like Relativistic Causality have already been used in discussions of indefinite causal order.In Ref. [6], for example, a causal correlation is a convex sum of correlations compatible with a (possibly dynamical) configuration of parties in spacetime, where each term involving spacelike separation is assumed to involve no superluminal signalling.Relative to this, the novelty of the present work lies in making use of available partial information about causal and spatiotemporal relations-viz.that Charlie is after the Alices and Bob is spacelike to the Alices and Charlie-rather than allowing arbitrary causal orders.Another locality notion has been studied in the context of Bell's theorem for temporal order [27,33].Here, violation of a Bell inequality is argued to imply indefinite causal order for the quantum switch under suitable separability and locality assumptions.This method is however not device-independent, as these assumptions rely on descriptions of states and trans-formations rather than just the observed correlations.
It is natural to wonder about the consequences of experimental violation of the inequalities derived here.Most current implementations of the quantum switch are based on optical interferometric setups sending a photon along a superposition of paths passing through Alice 1 and 2's devices in different orders [29][30][31][32][33].When used to probe correlations of measurement outcomes, these setups require the outcomes a 1 , a 2 to be read out only at the end, i.e. after both photon paths have passed through both Alices' devices, in order not to destroy the superposition of causal orders [30,32].These delayed measurements however mean that both outcomes only obtain a definite value in the intersection of the future lightcones of the spacetime loci where x 1 and x 2 are chosen.Therefore, violation of one of our inequalities by such experiments would, at least from the point of view of classical relativity theory, not demonstrate an interesting notion of indefinite causal order: it comes as no surprise that x 1 can influence a 2 while at the same time x 2 influences a 1 .This ties into the broader debate of whether these photonic experiments realise the quantum switch or merely simulate it [33,42,[53][54][55][56].We note that considerations involving gravitational quantum switch implementations and/or quantum reference frames [27,55,57] may offer different perspectives on this problem.
Provided one succeeds in avoiding this and other loopholes, experimental violation of the inequalities derived here could put restrictions on possible theories of quantum gravity compatible with observation.On a more practical level, an interesting direction of future research is to determine whether these inequalities could be used for new device-independent protocols, analogously to how Bell's theorem is used for device-independent quantum key distribution [58].
The technique by which we utilise the Relativistic Causality assumption and Bell inequalities for our certification was inspired by recent results on Wigner's friend scenarios [59], and might be applicable to the certification of other phenomena as well.It also suggests follow-ups on this work, such as proving violation of our three assumptions without inequalities (cf. the GHZ test [60] or Hardy's test [61]) or without settings (cf.Bell nonlocality in networks [62]).Finally, a natural extension of our result is to demonstrate violation of appropriately generalised inequalities by processes beyond the quantum switch.For instance, it is known that any pure entangled state violates a Bell inequality [63].Could it likewise be true that all unitary [7,40] causally nonseparable quantum processes violate a device-independent inequality witnessing their causal indefiniteness?and (in the Free Interventions assumption) that the causal order λ adjudicating between these two possibilities is independent of all setting variables.We take Relativistic Causality to merely constrain the causal order, however, meaning that it also allows for causal orders with strictly fewer causal relations between the parties, such as

Formalisation of the assumptions
Moreover, the possibility of these additional causal orders requires us to also consider that of dynamical causal order, wherein the causal order on a subset of parties depends on the setting of a party in their causal past [6,37,48].(In our case, for example, A1 might influence which of ≺1 and ≺3 occurs.)Here we formalise our assumptions, generalising them to allow for these additional causal orders as well as for dynamical causal order.We then show that the resulting correlations will still be in the same polytope DRF as defined in Equation ( 5).
To that end, let λ be a stochastic variable ranging over the set Ω of preorders, i.e. reflexive and transitive relations, on the set of agents A := {A1, A2, B, C}.Depending on context, we will also denote λ by ⪯ λ .For subsets X , Y ⊆ A , the condition X ⪯̸ λ Y is understood to mean that ∀X ∈ X , Y ∈ Y : X ⪯̸ λ Y (similarly for expressions such as X ⪯̸ λ Y and X ⪯̸ λ Y).Sometimes we will interpret such a condition on λ as an event, i.e. a subset of Ω.Our first two assumptions are on the impossibility of some of the orders in Ω.

• Definite Causal Order (DCO):
There is a variable λ, ranging over the set of preorders Ω and jointly distributed with the settings and outcomes in a conditional probability distribution p(⃗ abcλ|⃗ xyz).It satisfies p(λ|⃗ xyz) = 0 for any λ ∈ Ω that is not antisymmetric.(11) (That is, the causal order ⪯ λ it picks out is always acyclic.) • Relativistic Causality (RC): (That is, the causal order ⪯ λ satisfies, with certainty, the constraints imposed by the spatiotemporal structure ≺g discussed in the main text and Figure 1.) The Free Interventions assumption should be compatible with the existence of dynamical causal orders.We use the following condition, proposed in Oreshkov & Giarmatzi [6].Here, given a set X ⊆ A , the equivalence relation • Free Interventions (FI): for any λ * ∈ Ω and parties A1, . . ., An ∈ A with settings xi and outcomes ai (i = 1, . . ., n) such that {An} ⪯̸ λ * {A1, . . ., An−1}, the probability is independent of the setting xn.(Roughly: given that An does not precede any other Ai, her setting xn can influence neither the others' outcomes a1, . . ., an−1, nor the causal order between them.)(The term 'causal order' is often meant to refer either to properties of spacetime or to properties of correlations between variables.Here we have instead taken the more general approach that causal order is an a priori relation which is constrained by spacetime via RC (and by DCO) and which constrains correlations via FI.Furthermore, note that FI leads to two types of statistical independences in particular: between settings and hidden variables, and between settings and outcomes conditioned on hidden variables (cf.part (i) and (ii) of the less general assumption stated in the main text).When comparing to discussions of Bell's theorem, these correspond to conditions known as measurement independence (also free choice) and parameter independence, respectively.The assumption that the settings x1, x2, y, z of the interventions are freely chosen (i.e. have no causes relevant to other aspects of the experiment) is however central to the justification of both these mathematical conditions.This motivates the name of our third assumption.)

Theorem 2. For any probability distribution p(⃗ abcλ|⃗ xyz) satisfying Definite Causal Order, Relativistic Causality, and Free Interventions as defined above, the observed marginal distribution p(⃗ abc|⃗ xyz) is in DRF.
This is proven in Supplementary Note 2. The intuitive reason is that the additional causal orders allowed by Relativistic Causality (e.g.≺3) contain strictly fewer causal relations than either of ≺1 and ≺2 (e.g.≺3 ⊆ ≺1 as sets), so that the Free Interventions assumption imposes strictly more no-signalling constraints with respect to these causal orders.Moreover, because Relativistic Causality imposes that no parties are in the causal past of both A1 and A2, dynamical causal order does not lead to any more general correlations.

Proof of Theorem 1
We assume that all settings are binary and uniformly and independently distributed.This allows us to use shorthands such as and This assumption is made purely to simplify notation; it is not a physical requirement and plays no role in the proof below.
Proof of Theorem 1. Recall the definitions of DRF1 and DRF 2 in Equations ( 3) and (4).Because DRF is the convex hull of DRF1 and DRF2 and Inequality ( 6) is linear, it suffices to prove the inequality for the latter two polytopes individually.We give the proof for DRF 1; the case for DRF2 is analogous.Suppose p ∈ DRF1, and denote the first two terms of the inequality by α: and, because a1b ⊥ ⊥p x2 for p ∈ DRF1, Adding Equations ( 17) and ( 18) and rewriting gives The monogamy of Bell nonlocality however tells us that for nonsignalling correlations, a highly probable outcome is incompatible with a large CHSH value.More precisely, applying the monogamy inequality of Ref. [49] to the correlation p(bc|yz, x1 = x2 = 0) (and noting that b ⊥ ⊥p x1x2z) shows that the last term of Inequality 6 is bounded as where we used Equation ( 19) for the second inequality.Combining this with Equation ( 16) completes the proof.
(It is worth noting that the restriction that a1a2b ⊥ ⊥p z in DRF 1,2, corresponding to the assumption that Charlie is in the causal future of Alice 1 and 2, is not used in the proof of Theorem 1.However, including it yields a polytope that more accurately reflects the set of correlations that can arise in the scenario under consideration.Note also that it is essential that Charlie is not in the causal past of Alice 1 or 2, for this excludes the possibility that the causal order between Alice 1 and 2 depends on z.) The quantum switch correlations Here we analyse in more detail the correlations generated by the quantum switch in the scenario depicted in Figure 3, making more rigorous our claims that Charlie and Alice 1 can effectively measure the input control system C.
The interventions that we consider have single Kraus operators for each classical outcome: Alice i's Kraus operator corresponding to measuring ai and preparing xi is given by the linear operator |xi⟩⟨ai| : HT → HT , while Bob's and Charlie's projective measurements are described by the effects ⟨φ b|y | : HB → C and ⟨ψ c|z | : HC → C, respectively, whose directions in the Bloch sphere are indicated in Figure 3.The setting-outcome correlation corresponding to the scenario depicted in Figure 3 is then given by Equation ( 7) and the Born rule: Note first of all that if x1 = x2 = 0, this reduces to thus, Bob and Charlie effectively perform a normal Bell test on |Φ + ⟩CB, yielding the maximum quantum value of 1/2 + √ 2/4 for the third term in Inequality (6), thereby violating it.
On the other hand, if x2 = 1 then the marginal distribution over a1 and b reduces to showing that Alice 1's measurement yields the same correlations as a computational basis measurement of C, as we claimed in our discussion of Inequality (iv) in Table 1.
Vertices of DRF A polytope X ⊆ R d is a convex body with flat sides; it can be described either as the convex hull of a finite set of points, or as the intersection of finitely many closed halfspaces-i.e. the set of points satisfying a finite collection of linear inequalities-as long as this intersection is bounded [64].
The vertices of X are its extremal points.We call a linear inequality α T x ≤ β, for α ∈ R d and β ∈ R, valid for X if it holds for all x ∈ X , and tight if equality holds for some x ∈ X .Each linear inequality defines a hyperplane {x ∈ R d : α T x = β}; if the inequality is valid for X , the intersection of this hyperplane with X is a face of X , which is itself a polytope.If the dimension of a face is one less than the dimension of X itself, we call the face a facet.Any polytope is completely determined by the set of all its facets, or equivalently, its facet-defining inequalities.
We focus on the variant of DRF with binary settings and outcomes and without Charlie's setting z, defined by Here conv denotes the convex hull.Note that DRF 1,2 ⊊ DRF ⊊ NS ⊊ P ⃗ abc|⃗ xy , and that these are polytopes; all except DRF are defined uniquely by linear no-signalling and normalisation constraints and non-negativity of probabilities.DRF 1 (DRF 2) is 80-dimensional and admits a facet description in terms of 128 facets corresponding to non-negativity of probabilities.Using the software PANDA [65], we converted this facet description into a vertex description, exploiting symmetries of the polytope for efficiency.Taking the vertices of DRF 1 and DRF 2 together then yields the 9165312 vertices of DRF , which fall into 219 equivalence classes under symmetries of DRF .These symmetries correspond to interchanging Alice 1 and 2 and to relabelling the seven binary variables, possibly depending on the values of preceding variables in the causal order.More precisely, a minimal generating set of the symmetry group we used is induced by the following relabellings: Only 3 of the vertex classes of DRF are deterministic and therefore local; the others are nonlocal and have probabilities that are multiples of 1/2.The vertices also tell us that DRF is 86-dimensional, matching the dimension of the ambient nosignalling polytope NS.

Inequalities in Table 1
We will now discuss the inequalities in Table 1 in a bit more detail.These inequalities are valid and tight for and thus define faces of the 86-dimensional polytope defined in Equation 27.
Inequality (i) in Table 1 is similar to (6), except that z is replaced by x2 in the CHSH term.The proof that (i) is valid and tight for DRF is directly analogous to the proof of Theorem 1.It is weakly violated by the quantum switch setup described in the main text, fixing z = 0, which yields a value of 1.7652 > 7/4.A stronger violation can be found by using the observation, pointed out in the main text, that if x2 = 1, then the probabilities for a1 coincide with those of a computational basis measurement of the input control system.In particular, optimising over projective qubit measurements for Bob and Charlie, denoting Charlie's outcome by c ′ , and letting Charlie output c := x2a1 + (x2 ⊕ 1)c ′ leads to a value of (i) of approximately 1.8274.
Inequality (ii) differs from (i) in the respect that the first two terms are conditioned on the values of x1 and x2.The violations by the quantum switch correlations discussed in this paper are unaffected by this change.What makes (ii) interesting is that it only depends on the probabilities of ai when xi = 0, for i = 1, 2.Moreover, if we adopt the strategy for Charlie described in the previous paragraph, the outcome c of Charlie's measurement is only needed when x1 = x2 = 0.This poses an experimental advantage, as it reduces the number of measurements to be made.Geometrically, it entails that there is a still lower-dimensional polytope which can be violated by the quantum switch, namely where ai (c) only takes values when xi = 0 (x1 = x2 = 0).
Although it is in principle possible to compute all facets of DRF from its known vertex description, in practice this is complicated by its high dimension and high number of vertices.However, the dimension of known faces, such as those defined by the inequalities in Table 1, can be determined by counting the number of affinely independent vertices saturating the inequality (and subtracting 1).Moreover, the knowledge of the vertices can be used to pivot high-dimensional faces onto adjacent facets.Inequality (iii) has been obtained by pivoting a variant of Inequality (ii) in this way.Its additional fourth term however vanishes for all quantum switch correlations discussed in this paper, thus not paving the way for stronger inequality violations.
Inequality (iv) is motivated in the main text and proved in Supplementary Note 3. The assumptions required for this proof are strictly weaker than those required for Inequalities (6), (i), and (ii): namely, while the latter inequalities require the joint independence a1b ⊥ ⊥p x2 (see Equation ( 18)) to hold in DRF 1, the proof of (iv) only requires a ⊥ ⊥p x2 and b ⊥ ⊥p x2 separately (see Equation (44) in Supplementary Note 3).Similarly for DRF 2. This can be considered physically desirable because it separates the no-signalling constraints imposed by the Relativistic Causality condition from those imposed by the order between Alice 1 and 2 (which might involve exotic effects not in accordance with relativity theory).
The final four inequalities in Table 1 highlight the similarity between DRF and the bipartite causal polytope studied in e.g.Branciard et al. [36].The latter consists of causal correlations p(a1a2|x1x2), i.e. those that can be written as where µ ∈ [0, 1], a1 ⊥ ⊥ p 1 x2 and a2 ⊥ ⊥ p 2 x1.The causal inequality (v), referred to as a 'guess your neighbour's input' inequality, defines one of the two inequivalent nontrivial facets of the bipartite causal polytope.Note that by our Definite Causal Order assumption, any correlation p ∈ DRF has a causal marginal p(a1a2|x1x2), so that (v) is also valid for DRF .However, it is no facet of DRF; instead, (vi) is a facet adjacent to the face defined by (v), obtained by pivoting (v) onto the vertices of DRF as described above.Inequality (vi) Supplementary Note 1: The quantum switch does not violate causal inequalities It was shown in [5,6] that the quantum switch does not violate any causal inequalities in the scenarios previously considered in literature.Here we briefly review this argument, and show why it does not generalise to our scenario involving a constraint on the allowed causal orders arising from the presence of a spacelike-separated party.We consider the most general correlations p(a 1 a 2 c|x 1 x 2 z) =: p(⃗ ac|⃗ xz) observed by parties using just the quantum switch, where Alice 1 and Alice 2 measure the target system inside the two slots of the switch, while Charlie measures the output control and target system.First of all, note that since Charlie chooses his setting z after Alice, this can be written as Alice's marginal correlation p(⃗ a|⃗ x) is the correlation that would arise if the output control and target systems of the switch were discarded, rather than measured by Charlie.However, discarding the output control qubit on a quantum switch yields the classical switch [2], which is causally separable.Diagrammatically, this can be depicted as [66] A1 A2 that is, it is the convex sum (i.e.probabilistic mixture) of two valid processes, both of which correspond to a definite order between the interventions of Alice 1 and 2 (denoted here by gaps in the wires).This directly implies that p(⃗ a|⃗ x) can be written as a convex sum p(⃗ a | ⃗ x) = µp 1 (⃗ a | ⃗ x) + (1 − µ)p 2 (⃗ a | ⃗ x), (32) where µ ∈ [0, 1] and where a 1 ⊥ ⊥ p 1 x 2 and a 2 ⊥ ⊥ p 2 x 1 ; thus, p 1 is compatible with the causal order A 1 ≺ A 2 and p 2 with the causal order A 2 ≺ A 1 , under the Free Interventions assumption.We can now reintroduce Charlie by using Eq. ( 30 Here, both pi have no signalling from Charlie to Alice (⃗ a ⊥ ⊥ pi z); therefore p1 is compatible under Free Interventions with the causal order A 1 ≺ A 2 ≺ C and p2 with A 2 ≺ A 1 ≺ C, proving that the correlation p(⃗ ac|⃗ xz) admits an explanation in terms of definite causal orders.For this reason, it does not violate causal inequalities previously studied in literature [3,5,6,36,37].Turning to the extended scenario studied in this paper, let us consider correlations of the form p(⃗ abc|⃗ xyz) that are generated by the quantum switch entangled to a system in possession of a fourth party, Bob.Analogously to before, we can write and realise that the entangled switch with discarded output control qubit is causally separable: then we find that in general, the correlations p1 and p2 involve signalling from Bob to Charlie, but (by construction) not from Charlie to Bob.Therefore p1 is compatible with the causal order A 1 ≺ A 2 ≺ B ≺ C, and p2 with A 2 ≺ A 1 ≺ B ≺ C, so that the total correlation p in principle admits a causal explanation.However, these causal orders are not compatible with the constraint that Bob is causally unrelated to all other parties, which is imposed by the Relativistic Causality assumption when Bob is spacelike-separated.In general, p1 and p2 may exhibit signalling from Bob to Charlie, and they indeed do so for the particular quantum switch correlations considered in the main text.In other words, the decomposition of Eq. ( 37) does not necessarily allow us to construct a hidden variable model p(⃗ abcλ|⃗ xyz) satisfying p( • | • λ) ∈ DRF λ .This leaves open the possibility for violation of inequalities like (6) in the main text.
On the other hand, Eq. ( 36) tells us that any inequality valid for DRF in which the outcome c does not appear, such as (v)-(viii) in Table 1 in the main text, cannot be violated by the quantum switch setup considered here.

Figure 1 :
Figure 1: Causal orders giving rise to the DRF polytope.a)An experiment is performed by Alice 1 (A1), Alice 2 (A2), Bob (B), and Charlie (C) in the spatiotemporal structure ≺g illustrated here: that is, Charlie always acts in the future lightcone of Alice 1 and Alice 2, and Bob acts at spacelike separation from the other agents.b) The assumptions of Definite Causal Order and Relativistic Causality assert the existence of a variable λ specifying a partial order ≺ λ on all agents, such that ≺ λ respects the spatiotemporal structure of a).(Other possibilities for ≺ λ , in which some of A1, A2 and C are unrelated, are not illustrated here as their contributions to DRF are already covered by ≺1 and ≺2.) c) Conditioned on each value of λ, the Free Interventions assumption imposes statistical independence conditions, captured by the sets DRF λ , that rule out signalling outside the ≺ λ -future.DRF is the convex hull of DRF 1 ∪ DRF2, i.e. consists of probabilistic mixtures of correlations in DRF 1 and DRF 2.

Figure 3 :
Figure 3: The quantum switch setup violating Inequality (6).The input-output direction in this diagram is from bottom to top.The switch's control system C and a system B held by Bob are prepared in the maximally entangled state |Φ + ⟩.The target system T is prepared in state |0⟩, measured and reprepared in the computational basis by Alice 1 and 2 (in the dotted boxes), and ultimately discarded.Finally, Bob and Charlie perform, for each of their settings y and z, projective measurements on B and the output control system C in directions in the XZ plane of the Bloch sphere indicated by the green arrows.⟨φ b|y | is the effect corresponding to Bob observing outcome b upon setting y; similarly for Charlie's effect ⟨ψ c|z |.The diagram as a whole defines the probability p(⃗ abc|⃗ xyz), also given in Equation (21) in Methods.

Table 1 : Some inequalities following from Definite Causal Order, Relativistic Causality, and Free Interventions.
85 These are inequalities valid and tight for DRF-an 86-dimensional version of the polytope without Charlie's setting z, defined in Equation (27) in Methods-violation of which thus indicates falsification of the conjunction of Definite Causal Order, Relativistic Causality, and Free Interventions.Inequalities (i)-(iv ) are violated by the quantum switch, whereas (v )-(viii) are satisfied by all quantum switch correlations.