Perspective on experimental quantum causality

PERSPECTIVE

Perspective on experimental quantum causality

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Perspective

Perspective on experimental quantum causality

G. Carvacho1, R. Chaves2 _{and F. Sciarrino}1

1 _{Dipartimento di Fisica, Sapienza Universit`a di Roma - I-00185 Roma, Italy}

2 _{International Institute of Physics & School of Science and Technology, Federal University of Rio Grande do Norte}

59070-405 Natal, Brazil

received 15 February 2019; accepted 20 February 2019 published online 7 March 2019

PACS 03.67.-a – Quantum information PACS 42.50.-p – Quantum optics

Abstract – Inferring causal relations from empirical data is a central task in any scientific

in-quiry. To that aim, a mathematical theory of causality has been developed, not only stating under which conditions such inference becomes possible but also offering a formal framework to reason about cause and effect within quantum theory. This perspective article provides an overview of the latest results in the growing field of quantum causality, with a special focus on experimental implementations and concepts that only recently have entered in the quantum information dictio-nary. Ideas like bilocality, instrumental variables and interventions will be discussed from both a fundamental and an experimental perspective.

perspective Copyright c EPLA, 2019

Understanding cause and effect relations is at the center of any scientific endeavor. It was only in the mid-nineties that a mathematical theory of causality has emerged [1,2] with applications ranging from economics [3–5] to clinical trials [6] and genetics [7,8] to social studies [9], also at-tracting increasing attention in statistics, machine learn-ing and artificial intelligence [10]. Causality theory allows us to establish a formal connection between empirical data and cause and effect relations. Clearly, however, as it has been long realized, correlation does not necessarily imply causation, and this is the reason why statisticians have historically shied away from causality concepts [2].

The causality framework has been shown to provide a novel perspective in the foundations of quantum physics and quantum information science where three main trends can be observed. First, the language of causal struc-tures provides a natural and intuitive playground to un-derstand and generalize Bell’s theorem [11,12]. Causal notions are implicit in virtually any research regarding Bell non-locality, however, it was not until recently that basic causality concepts like common causes, causal in-fluence, interventions and fine-tuning started to appear in the quantum foundations dictionary [13–25]. A sec-ond trend is the generalization of the concept of causality itself to the quantum realm [26–36]. Since Bell’s theo-rem we know that our classical notion of cause and effect is incompatible with quantum predictions and that has

motivated the search for quantum causality and quantum causal modeling. Finally, a third venue that will only be briefly mentioned in this perspective, has been the idea of indefinite causal structures [30,33,37–43]. At the core of quantum mechanics is the idea of quantum superposi-tion that when applied to causality allows for situasuperposi-tions where, as opposed to a classical world, one event is not definitively the cause/effect of the other, the two possi-bilities happening coherently and (to some extent) at the same time.

In this perspective article we will address, with an ex-perimental insight, some of the most recent developments in quantum causality. With this aim we will discuss cen-tral and fundamental concepts in this field.

From causal structures to quantum non-locality. In a typical Bell-like experiment [11,12] two (or more) dis-tant parties perform local measurements on their shares of a joint physical system. By invoking special relativity we have the guarantee that none of the parties can in-fluence the statistics observed by the others. In turn, in a teleportation protocol [44] we have a time-like separa-tion situasepara-tion where measurements are performed in one laboratory and conditioned on those initial results further actions are implemented on a second distant laboratory. These two paradigmatic situations show that implicit in any quantum information protocol there is an underlying

G. Carvacho et al.

Fig. 1: Causal structures. In DAGs shown here, there are three different kinds of nodes: hidden variables (purple box), measurement settings (yellow boxes) and measurement out-comes (green circles). (a) Paradigmatic bipartite Bell causal structure representing a local hidden variable (LHV) model. (b) Tripartite LHV model. (c) Tripartite scenario with two independent LHVs, that is, a bilocal hidden variable (BLHV) model. (d) Tripartite BLHV model where the central nodeB is made of two substations BA and BC, driven by the same measurement choiceY .

network of potential causes and effects, that is, a causal structure governing the relevant processes and variables.

An intuitive way to define and visualize causal struc-tures is via directed acyclic graphs (DAGs) [2,45]. Each node in the graph denotes a random variable of relevance and the directed edges (arrows) encode their causal rela-tionships. Mathematically, they provide a compact repre-sentation of joint probability distributions. Considering a DAG of N nodes X1, . . . , XN and that P a(Xi) denotes the graph-theoretical parents of the i-th variable, the so-called Markov condition [45] implies that the joint distribution should factorize as1 p(x1, . . . , xN) = N  i p(xi|P a(xi)). (1)

The Markov condition (alternatively the DAG) encodes causal relationships via the conditional independences im-plied by it. To illustrate its central concepts, we an-alyze next the paradigmatic causal structure in Bell’s theorem [12].

In the paradigmatic Bell scenario, depicted in fig. 1(a), two distant parties (Alice and Bob) upon receiving their parts of a joint system choose some measurement to perform (labeled by X and Y ), thus obtaining some 1_{Following the standard notation, capital letters (e.g.,X) denote} random variables and lower-case letters (e.g.,x) denote the values they acquire. That is, when referring to probabilities we will use the shorthand notationp(x) = p(X = x).

measurement outcome (respectively, labeled A and B). The source of states and any other relevant physical pro-cess that might affect the measurement outcomes (like lo-cal noise terms) are represented by the variable Λ. By assumption, the variable Λ is not empirically accessible and thus represents a hidden (or latent) variable. That is, in a Bell experiment we have access to the statistical data represented by the probability distribution p(a, b, x, y) and since the measurements choices are typically under con-trol, it is customary to work instead with the conditional distribution p(a, b|x, y). Following the Markov prescrip-tion (1) for the Bell DAG, this probability can be decom-posed as p(a, b|x, y) =  p(λ)p(a, b|x, y, λ)dλ =  p(λ)p(a|x, λ)p(b|y, λ)dλ, (2)

defining the local hidden variable (LHV) model [11] where two causal assumptions have been employed. First, the lo-cality assumption represented graphically by the absence of direct arrows between Alice and Bob variables and mathematically represented as p(a|b, x, y, λ) = p(a|x, λ) (similarly for Bob). Second, the measurement indepen-dence (“free-will”) assumption, graphically represented by the absence of arrows between Λ and X and Y and thus implying that p(x, y, λ) = p(x)p(y)p(λ).

The DAG approach can be extended to the multipartite scenario. For instance, in the tripartite case, this causal framework already allows to identify two different situa-tions. In the first, all three parties share a state prepared by a common source (fig. 1(b)) while in the second case we might have two or more independent sources (fig. 1(c)). Considering a unique source, the joint distribution should factorize as

p(a, b, c|x, y, z) = 

p(λ)p(a|x, λ)p(b|y, λ)p(c|z, λ)dλ. (3) In turn, taking explicitly into account the independence of sources as depicted in fig. 1(c) (more explicitly p(λ1, λ2) = p(λ1)p(λ2), the so-called bilocality assumption [46,47]), we now have

p(a, b, c|x, y, z) = 

p(λ1)p(λ2)p(a|x, λ1)p(b|y, λ1, λ2)

× p(c|z, λ2)dλ1dλ2. (4)

All of these descriptions, however, rely on a classical concept of causality that since Bell’s theorem we know to be incompatible with the quantum-mechanical predic-tions. Take, for instance, the bipartite Bell scenario, the statistics observed in such experiment should arise from Born’s rule implying that

p(a, b|x, y) = Tr[(Max⊗ Mby)ρ], (5)

where M_ax and M_by describe local measurement operators and ρ represents the density matrix for the joint quantum

state produced at the source. Bell’s theorem shows that quantum correlations described by (5) can be incompatible with the classical causal description provided by (2). The most common way to witness this incompatibility is via the violation of a Bell inequality. Classical causal models such as the LHV models (2) imply testable constraints on the observed correlations. These are precisely the Bell in-equalities that, if violated, provide an unambiguous proof of the incompatibility of the observed correlations with the underlying causal model. As quantum correlations violate Bell inequalities, at least one of the two causal assumptions employed in their derivation, locality and measurement in-dependence, must give way in a quantum-mechanical de-scription. As measurement independence (“free-will”) is at the core of any scientific quest, that is the reason why this phenomenon is called quantum non-locality. It does not mean that quantum correlations can be used to com-municate instantaneously, it just shows that when we try to describe quantum phenomena with a classical notion of causality we arrive at the conclusion that even if very far apart the quantum particles seem to be communicating, the “spooky action at distance” abhorred by Einstein [48]. Given its fundamental importance [11], the violation of the Bell inequalities has been pursued in different physi-cal systems, culminating in the first loophole-free exper-iments [49–52]. In spite of their groundbreaking nature, most of these experiments focused on the bipartite case or on its straightforward generalization to the multipar-tite case (always considering a single source of states). As described next, it was only recently, that conceptually different kinds of non-classical correlations started to be considered experimentally.

Experimental violation of bilocality. From direct com-parison it is clear that the bilocal set (4) is contained inside the local set (3). More precisely, the bilocal set is charac-terized by non-linear polynomial Bell inequalities [53–58]. Interestingly, this implies that there exist correlations that might appear classical but have their non-local nature re-vealed if the independence of the sources generating the correlations is taken into account. Another relevant fea-ture is the fact that the bilocal causal strucfea-ture reproduces the underlying pattern of causation in a entanglement swapping experiment [59].

In the simplest possible scenario, akin to the entangle-ment swapping scenario, where Alice and Charlie perform two dichotomic measurements and Bob performs a single measurement with four possible outcomes, the following inequality holds for bilocal hidden variable models [46,47]:

B = √I +√J ≤ 1, I =1 4  x,z=0,1AxB0Cz, J = 1 4  x,z=0,1(−1)x+zAxB1Cz, (6)

whereA_xByCz =a,b,c=0,1(−1)a+b+cp(a, b, c|x, y, z).

Exploiting the causal structure depicted in fig. 1(c), two photonic experiments have provided an experimen-tal proof-of-principle for network generalizations of Bell’s

theorem [41,42]. Both considered two different non-linear crystals pumped by the same laser and three different stations where Alice and Charlie make two possible di-chotomic measurements and Bob measures in the Bell ba-sis. The maximum violation of the bilocality inequality has reached B = 1.268 ± 0.014 corresponding to a viola-tion by almost 20 sigmas [41]. Furthermore, it has been shown experimentally that correlations compatible with a usual LHV model (3) can indeed violate such inequality, thus proving a new kind of non-local correlation.

Similarly to any Bell test, these first violations of the bilocal causality inequality were subjected to loopholes, in particular the locality and detection efficiency loopholes. Another important issue related to these experiments was the fact that both of them exploited just one laser to pump both crystals. Given the nature of the experiments, this introduces a loophole, similar to the measurement in-dependence loophole in Bell’s theorem, if the sources of states cannot be guaranteed to be truly independent. To deal with that, Saunders et al. [42] included a random time-varying phase shifter between the crystals in order to increase the degree of independence between the sources and thus to overpass this problem. Another potential is-sue is the fact that typically Bell experiments rely on a shared reference frame between the parties. While inof-fensive in the usual Bell tests, in the case of bilocality such reference frames could be used to simulate non-local cor-relations with LHV models [43]. Fortunately, however, as demonstrated experimentally in [43], bilocality violations can also be achieved in the absence of shared reference frames.

Recently the group of Pan [60] closed simultaneously the loopholes of source independence and locality in an en-tanglement swapping experiment, reporting a violation of B = 1.181 ± 0.004, hence rejecting bilocal hidden variable models. One should keep in mind that like in Bell tests we will never be able to rule out a super-deterministic ex-planation [61], since it is impossible to guarantee that two separate sources are genuinely independent, as they could have been correlated at the birth of the universe (see, for instance, [62,63]).

The experimental verification of the bilocality viola-tion reported in [41,42] has inspired more related works with different approaches [64–72] in which the extension to more complex networks is the central topic of inter-est. An interesting question was whether the violation of bilocality can be performed in a device-independent way, since the first experiments assumed that Bob’s sta-tion perform a complete Bell-state measurement with four possible outcomes, something that is impossible with lin-ear optics [73]. In practice this implies that such real-izations had to rely on the precise quantum description of the experiment, that is, they were a device-dependent test of bilocality. Notably, it has been shown that quan-tum mechanics can exhibit non-bilocal correlations also in cases in which all parties only perform single-qubit op-erations [58], thus allowing for a device-independent test

G. Carvacho et al.

Fig. 2: Different scenarios within the instrumental causal struc-ture. (a) The instrumental scenario, whereX stands for the instrument,A and B are the variables for which we want to estimate causal influences, and Λ represents any latent factor correlating them. (b) Causal relaxation where direct influences between the instrument X and the variable B are allowed. (c) Representation of an intervention onA, performed under the causal influence of a variableI (yellow triangle) controlled by the experimenter.

of bilocality with linear optics implementations. To do so, Bob performs separable measurements and hence it is possible to consider the station of Bob as composed by two sub-stations as depicted in fig. 1(d). This experimen-tal approach has been fully addressed [43] and allowed a violation of bilocality with separable measurements up to B = 1.2712 ± 0.0042.

Instrumental causality. A recent development with fundamental importance in the field of causal inference is the so-called instrumental causal structure. Its relevance stems from the fact that it allows to put bounds on the causal influence of a variable A on a variable B, simply based on observational data and without the need of in-terventions. It relies on a instrumental variable X that is assumed to influence only A directly and to be indepen-dent of the shared correlations between A and B. The instrumental DAG (see fig. 2(a)) implies that any correla-tion compatible with it should be decomposable as

p(a, b|x) =

λ

p(λ)p(a|x, λ)p(b|a, λ). (7)

Considering that all observable variables are binary, it has been shown that all quantum correlations, obtained by Born’s rule as

p(a, b|x) = Tr[(Max⊗ Mba)ρ] (8)

are compatible with a classical causal description [29]. However, considering that the instrumental variable as-sumes three possible values it was shown that there are quantum violations of the instrumental constraints. Quantum correlations of the form (8) can violate the in-strumental inequality [74]

I = A1− B1+ 2B2− AB1+ 2AB3≤ 3, (9)

where ABx=_a,b(−1)a+bp(a, b|x) up to the limit of I = 1 + 2√2.

The violation of this inequality has been reported for the first time in [74] which, opposite to the usual Bell

structures, defines a time-like scenario. However, there are close connections between the Bell and the instrumen-tal scenarios. As shown in [75], any correlation violating the paradigmatic CHSH inequality [76] can also violate an instrumental inequality after a post-processing of the cor-relations. In spite of that, the instrumentality violation can be seen as a stronger form of non-classicality. As ar-gued in [74], the instrumental structure can be mapped to a non-local hidden variable model. That is, the violation of instrumentality proves a stronger version of Bell’s theo-rem, showing that quantum correlations are incompatible even with a specific model of non-local causality.

Given their close connections, it is natural to ask whether results usually associated with Bell scenarios, such as device-independent cryptography [77], random-ness generation [78–82], or self-testing [83], can also be extended to the instrumental case.

Quantum causal models. – Given some correlations between two variables A and B, the most basic ques-tion within causal inference is to know whether A causes B or their correlations are mediated by a third vari-able Λ. Unfortunately, there is no way to distinguish between both cases (unless some extra assumptions are made [84,85]). Both models, direct causation and common causes, can generate any probability distribution p(a, b) and thus they cannot be distinguished [86] with passive observations.

In order to distinguish between the possible causal structures, one has to rely on interventions [45]. An in-tervention is the local act of altering the mechanism that produces the value of a given variable without changing the causal structure. Consider, for instance, an interven-tion in the variable A refereed above (see fig. 2(c)). This intervention erases any potential incoming arrow (for in-stance, the one from the variable Λ). Thus, if after the intervention one still observe correlations between A and B (that is, p(a, b) = p(a)p(b)), it can be concluded that there is a direct causation from A into B and that can be quantified by the so-called average causal effect (ACE) [2]

ACE_A→B= sup

a,a_,b|p(b|do(a)) − p(b|do(a

_{))|, (10)}

where do(a) represents the intervention over A. Impor-tantly, the passively observed distribution p(b|a) is in general different from the interventional one, as can be directly seen from their Markov decompositions. Assum-ing that the underlyAssum-ing causal model may contain both direct causation and correlations due to a common source (see fig. 3), we have that p(b|a) =_λp(b|λ, a)p(a|λ)p(λ), while p(b|do(a)) =_λp(b|λ, a)p(a)p(λ).

All the ideas and concepts discussed above rely, how-ever, on a classical concept of causality. Inspired by Bell’s theorem and the well-known incompatibility of quantum correlations with the usual notion of causality, there has been a surge of results looking for a quantum generaliza-tion of these ideas, with the ultimate objective of estab-lishing a quantum theory of causality. On the one hand,

Fig. 3: Directed acyclic graphs (DAGs). (a) Direct influence. An arrow between two different nodes (A and B) represents a causal influence in whichA causes an action over B. (b) Com-mon cause. A hidden variable represented byλ generates an influence over two different nodes. (c) Direct and common cau-sation are present simultaneously.

there have been a number of different proposals of how to define and treat quantum causal models [26–36]. On the other hand, there is the basic question as to if and how fun-damental results in the classical theory of causality should be revisited.

Recently, Ried et al. [26] have shown that it is possi-ble to infer the causal structure from observations alone through a causal tomography. In some cases this implies that one distinguishes between common causes and direct causation, something impossible classically as discussed above. In fact, common-cause mechanisms producing sep-arable states and cause-effect mechanisms implementing entanglement-breaking channels follow the same classical rules and cannot be distinguished. That is, quantum ad-vantage in causal inferring is reached by exploiting entan-glement and coherence and was experimentally observed in a photonic experiment [26]. Following that, a quan-tum optics experiment implementing a coherent mixture of a common cause and direct influence has also been re-alized [87].

As mentioned before, the instrumental scenario offers a way to distinguish, with observational data, direct causa-tion from common cause. But given that an instrumental inequality can be violated quantum mechanically, is there anything else we should consider? Indeed, as shown in [74], quantum-mechanical effects can lead to an overestimation in the quantification of causal influences. Considering a scenario where all variables are dichotomic, the average causal effect is bounded as [2] ACEA→B ≥ 2p(a = 0, b =

0|x = 1)2 + p(a = 1, b = 1|x = 1) + p(b = 1|x = 2). However, by considering the correlations obtained by mea-surements on a maximally entangled state it has been proven that the classical bound above would imply that ACE_A→B > 0 even though, if interventions are made, one observes that ACE_A→B= 0. In other terms, the classical inequality would lead to an overestimation of the causal in-fluence. Finally, it has been shown, both theoretically and experimentally [88], that the instrumental causal model can be used to derive a stronger form of steering.

Future challenges. – The interplay between quan-tum information and the theory of causality is a very recent enterprise with a few different threads. A promi-nent line of research has been the development of a quan-tum theory of causality, encompassing the counterintuitive

quantum features but also recovering the classical theory in the appropriate limit [26–36]. Another approach has been to remain within a classical theory but allows for re-laxations of causal assumptions, which provides insights into the nature of quantum correlations and a natural way of quantifying them [13–15,17–23,89]. Finally, a new fundamental feature that has attracted considerable at-tention is the coherent superposition of different causal orders [30,33,37–43], yet another quantum phenomenon enlarging even more the quantum supremacy in the pro-cessing of information. In fact, as expected, the concepts and tools arising from these initial investigations have al-ready been turned into resources for new kinds of proto-cols. For instance, quantum effects allow for advantages in causal inference problems [26,32,74,87,90] and superpo-sition of causal orders permits an exponentially advantage in certain communication problems [91–93]. Yet, in spite of these exciting developments, it is fair to say that the application of the causal framework to quantum informa-tion problems is still vastly unexplored, specially experi-mentally and in the context of complex quantum causal networks.

Theoretically, a few challenges are ahead. For instance, the characterization of the set of classical and quantum correlations in causal networks of increasing complexity has seen interesting new results in the last years [53–58] but a more complete picture and more efficient algo-rithms still lie ahead. Regarding the superposition of causal orders, one of the key open questions is whether quantum mechanics allows for the violation of causal inequalities [94–96], that similarly to Bell inequalities would provide a device-independent framework to the study of indefinite causal orders. Also, to which extension

quantum correlations can provide enhancements in

causal inference problems remains a topic for further investigations.

From the experimental side, novel fundamental and practically relevant challenges can be pursued which

will require more ingredients: several independent

sources of states [41–43], classical and quantum com-munication between the parties [16,24,25,91,93], quan-tum interventions [24,26] and coherence mixtures of causal structures [87] as well as superposition of causal orders [30,33,37–43].

∗ ∗ ∗

We acknowledge support from John Templeton Foun-dation via the grant Q-CAUSAL No. 61084 (the opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation). RC acknowledges the Brazilian ministries MCTIC, MEC and the CNPq (grants No. 307172/2017-1, 406574/2018-9 and INCT-IQ) and the Serrapilheira In-stitute (grant No. Serra-1708-15763). GC acknowledges Becas Chile and Conicyt.

G. Carvacho et al.

REFERENCES

[1] Spirtes P., Glymour N. and Scheienes R.,

Causa-tion, PredicCausa-tion, and Search, 2nd edition (The MIT Press)

2001.

[2] Pearl J., Causality (Cambridge University Press) 2009. [3] Wright P. G., The Tariff on Animal and Vegetable Oils

(The Macmillan Company) 1928.

[4] Granger C. W. J., Econometrica, 37 (1969) 424. [5] Cartwright N., Causal Structures in Econometrics, in

On the Reliability of Economic Models, Recent Economic Thought Series, edited by Little D., Vol. 42 (Springer,

Dordrecht) 1995, pp. 63–89.

[6] Fischer K. and White I. R., Causal Inference in

Clin-ical Trials (Wiley-Blackwell) 2012.

[7] Friedman N., Science,303 (2004) 799.

[8] Shalizi C. R. and Thomas A. C., Sociol. Methods Res.,

40 (2011) 211.

[9] Brady H. E., Causation and Explanation in Social

Sci-ence (Oxford University Press) 2013.

[10] Barber D., Bayesian Reasoning and Machine Learning (Cambridge University Press) 2012.

[11] Brunner N. et al., Rev. Mod. Phys.,86 (2014) 419. [12] Bell J. S., Phys. Rev. Lett.,1 (1964) 195.

[13] Chaves R. et al., Phys. Rev. Lett.,114 (2015) 140403. [14] Pironio S., Phys. Rev. A, 68 (2003) 062102.

[15] Maxwell K. and Chitambar E., Phys. Rev. A, 89 (2014) 042108.

[16] Brask J. B. and Chaves R., J. Phys. A: Math. Theor.,

50 (2017) 094001.

[17] Hall M. J. W., Phys. Rev. A, 82 (2010) 062117. [18] Hall M. J. W., Phys. Rev. Lett.,105 (2010) 250404. [19] Hall M. J. W., Phys. Rev. A, 84 (2011) 022102. [20] Rai A. et al., J. Phys. A: Math. Theor.,45 (2012) 475302. [21] Paul B., Mukherjee K. and Sarkar D., Phys. Rev. A,

88 (2013) 044104.

[22] P¨utz G._{et al., Phys. Rev. Lett.,}113 (2014) 190402. [23] Barrett J. and Gisin N., Phys. Rev. Lett., 106 (2011)

100406.

[24] Ringbauer M. et al., Sci. Adv.,2 (2016) e1600162. [25] Ringbauer M. and Chaves R., Quantum, 1 (2017)

35.

[26] Ried K. et al., Nat. Phys., 11 (2015) 414.

[27] Cavalcanti E. G. and Lal R., J. Phys. A: Math. Theor.,

47 (2014) 424018.

[28] Tucci R. R., Int. J. Mod. Phys. B,9 (1995) 295. [29] Hensen J., Lal R. and Pusey M. F., New J. Phys.,16

(2014) 113043.

[30] Chaves R., Majenz C. and Gross D., Nat. Commun.,

6 (2015) 5766.

[31] Leifer M. S. and Poulin D., Ann. Phys., 323 (2008) 1899.

[32] Fitzsimons J. F., Jones J. A. and Vedral V., Sci.

Rep.,5 (2015) 18281.

[33] Pienaar J. and Caslav Brukner, New J. Phys., 17 (2015) 073020.

[34] Allen J.-M. A. et al., Phys. Rev. X,7 (2017) 031021. [35] Fritz T., New J. Phys.,14 (2012) 103001.

[36] Fritz T., Commun. Math. Phys.,341 (2016) 391. [37] Wood C. J. and Spekkens R. W., New J. Phys., 17

(2015) 033002.

[38] Cavalcanti E. G. and Lal R., New J. Phys., 47 (2014) 42.

[39] Henson J., Lal R. and Pusey M. F., Commun. Math.

Phys.,341 (2015) 391.

[40] Costa F. and Shrapnel S., New J. Phys., 18 (2016) 063032.

[41] Carvacho G. et al., Nat. Commun.,8 (2017) 14775. [42] Saunders D. J. et al., Sci. Adv.,3 (2017) e1602743. [43] Andreoli F. et al., Phys. Rev. A,95 (2017) 062315. [44] Bennett C. H. et al., Phys. Rev. Lett.,70 (1993) 1895. [45] Pearl J., Biometrika,82 (1995) 669.

[46] Branciard C., Gisin N. and Pironio S., Phys. Rev.

Lett.,104 (2010) 170401.

[47] Branciard C. et al., Phys. Rev. A,85 (2012) 032119. [48] Einstein A., Podolsky B. and Rosen N., Phys. Rev.,

47 (1935) 777.

[49] Hensen B. et al., Nature,526 (2015) 682.

[50] Shalm L. K. et al., Phys. Rev. Lett.,115 (2015) 250402. [51] Giustina M. et al., Phys. Rev. Lett.,115 (2015) 250401. [52] Abellan C. et al., Nature,557 (2018) 212.

[53] Chaves R., Phys. Rev. Lett.,116 (2016) 010402. [54] Lee C. M. and Spekkens R. W., J. Causal Inference,5

(2017) 2.

[55] Tavakoli A., Phys. Rev. A,93 (2016) 030101(R). [56] Wolfe E., Spekkens R. W. and Fritz T., The

infla-tion technique for causal inference with latent variables,

arXiv:1609.00672 (2018).

[57] Tavakoli A. et al., Phys. Rev. A,90 (2014) 062109. [58] Rosset D. et al., Phys. Rev. Lett.,116 (2016) 010403. [59] Zukowski M. et al., Phys. Rev. Lett.,71 (1993) 4287. [60] Sun Q.-C. et al., Experimental demonstration of

non-bilocality with truly independent sources and strict locality constrains, arXiv:1807.05375 (2018).

[61] Bell J. S., Speakable and Unspeakable in Quantum

Me-chanics (Cambridge University Press) 1989.

[62] The BIG BELL test Collaboration, Nature, 557 (2018) 212.

[63] Rauch D. et al., Phys. Rev. Lett.,121 (2018) 080403. [64] Ciar´an M. _{and Hoban M. J., Phys. Rev. Lett.,} 120

(2018) 020504.

[65] Luo M.-X., Phys. Rev. Lett.,120 (2018) 140402. [66] Andreoli F. et al., New J. Phys.,19 (2017) 113020. [67] Gupta S., Datta S. and Majumdar A. S., Phys. Rev.

A,98 (2018) 042322.

[68] Luo M.-X., Phys. Rev. A,98 (2018) 042317.

[69] Mukherjee K., Paul B. and Sarkar D., Restricted

Distribution of Quantum Correlations in Bilocal Network,

arXiv:1710.00912 (2017).

[70] Mukherjee K., Paul B. and Sarkar D., Phys. Rev. A,

96 (2017) 022103.

[71] Gisin N. et al., Phys. Rev. A,96 (2017) 020304(R). [72] Frey M., Quantum Inf. Process.,16 (2017) 266. [73] L¨utkenhaus N., Calsamiglia J._{and Suominen K.-A.,}

Phys. Rev. A,59 (1999) 3295.

[74] Chaves R. et al., Nat. Phys.,14 (2018) 291.

[75] Van Himbeeck T. et al., Quantum violations in the

In-strumental scenario and their relations to the Bell sce-nario, arXiv:1804.04119 (2018).

[76] Clauser J. F. et al., Physics,24 (1970) 549.

[77] Vazirani U. and Vidick T., Phys. Rev. Lett.,113 (2014) 140501.

[79] Colbeck R. and Kent A., J. Phys. A: Math. Theor.,44 (2011) 095305.

[80] Colbeck R. and Renner R., Nat. Phys.,8 (2012) 450. [81] Gallego R. et al., Nat. Commun.,4 (2013) 2654. [82] Brand˜ao F. G. S. L. _{et al., Nat. Commun.,} 7 (2016)

11345.

[83] Mayers D. and Yao A., Quantum Inf. Comput., 4 (2004) 273.

[84] Mooij J. M. et al., J. Mach. Learn. Res.,17 (2016) 1. [85] Janzing D., Chaves R. and Sch¨olkopf B., New J.

Phys.,18 (2016) 093052.

[86] Reichenbach H., The Direction of Time (University of California Press) 1956.

[87] MacLean J.-P. W. et al., Nat. Commun., 8 (2017) 15149.

[88] Nery R. V. et al., Phys. Rev. Lett.,120 (2018) 140408. [89] Brito S. G. A., Amaral B. and Chaves R., Phys.

Rev. A,97 (2018) 022111.

[90] Chiribella G. and Ebler D., Quantum speedup in

test-ing causal hypotheses, arXiv:1806.06459 (2018).

[91] Gu´erin P. A.et al., Phys. Rev. Lett.,117 (2016) 100502. [92] Ara´ujo M., Gu´erin P. A. _{and Baumeler ¨}A._{, Phys.}

Rev. A,96 (2017) 052315.

[93] Salek S., Ebler D. _{and Chiribella G.,}

Quan-tum communication in a superposition of causal orders,

arXiv:1809.06655 (2018).

[94] Branciard C. et al., New J. Phys.,18 (2015) 013008. [95] Abbott A. A. et al., Phys. Rev. A,94 (2016) 032131. [96] Oreshkov O. and Giarmatzi C., New J. Phys., 18