Perspective on experimental quantum causality

In a typical Bell-like experiment [11,12] two (or more) distant 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 influence the statistics observed by the others. In turn, in a teleportation protocol [44] we have a time-like separation situation 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 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 structures 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 relationships. Mathematically, they provide a compact representation of joint probability distributions. Considering a DAG of N nodes $X_1,\ldots,X_N$ and that Pa(X_i) denotes the graph-theoretical parents of the i-th variable, the so-called Markov condition [45] implies that the joint distribution should factorize as¹

$$\begin{equation} p(x_1,\ldots,x_N)= \prod_{i}^{N} p(x_i \vert Pa(x_i)). \end{equation} \tag{ 1 }$$

The Markov condition (alternatively the DAG) encodes causal relationships via the conditional independences implied by it. To illustrate its central concepts, we analyze 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 measurement outcome (respectively, labeled A and B). The source of states and any other relevant physical process that might affect the measurement outcomes (like local 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 control, it is customary to work instead with the conditional distribution $p(a,b\vert x,y)$. Following the Markov prescription (1) for the Bell DAG, this probability can be decomposed as

$$\begin{equation} \begin{array}{rcl} p(a,b\vert x,y)&=&\displaystyle \int p(\lambda)p(a,b\vert x,y,\lambda)\mathrm{d} \lambda \\[4pt] &=&\displaystyle\int p(\lambda)p(a\vert x,\lambda)p(b\vert y,\lambda)\mathrm{d} \lambda, \end{array} \end{equation} \tag{ 2 }$$

defining the local hidden variable (LHV) model [11] where two causal assumptions have been employed. First, the locality assumption represented graphically by the absence of direct arrows between Alice and Bob variables and mathematically represented as $p(a\vert b,x,y,\lambda)=p(a\vert x,\lambda)$ (similarly for Bob). Second, the measurement independence ("free-will") assumption, graphically represented by the absence of arrows between Λ and X and Y and thus implying that $p(x,y,\lambda)=p(x)p(y)p(\lambda)$.

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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 situations. 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

$$\begin{equation} \hspace{-5pt}p(a,b,c\vert x,y,z) = \int p(\lambda)p(a\vert x,\lambda)p(b\vert y,\lambda)p(c\vert z,\lambda)\mathrm{d} \lambda. \end{equation} \tag{ 3 }$$

In turn, taking explicitly into account the independence of sources as depicted in fig. 1(c) (more explicitly $p(\lambda_1,\lambda_2)=p(\lambda_1)p(\lambda_2)$, the so-called bilocality assumption [46,47]), we now have

$$\begin{eqnarray} p(a,b,c\vert x,y,z) =\,& \int p(\lambda_{1})p(\lambda_{2})p(a\vert x,\lambda_{1})p(b\vert y,\lambda_{1}, \lambda_{2})\nonumber\\ & {}\times p(c\vert z,\lambda_{2})\mathrm{d} \lambda_{1} \mathrm{d} \lambda_{2}. \end{eqnarray} \tag{ 4 }$$

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 predictions. Take, for instance, the bipartite Bell scenario, the statistics observed in such experiment should arise from Born's rule implying that

$$\begin{equation} p(a,b \vert x,y)= \mathrm{Tr} [ (M_a^x \otimes M_b^y ) \rho], \end{equation} \tag{ 5 }$$

where $M_a^x$ and $M_b^y$ 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 inequalities 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 independence, must give way in a quantum-mechanical description. 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 communicate 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 physical systems, culminating in the first loophole-free experiments [49–52]. In spite of their groundbreaking nature, most of these experiments focused on the bipartite case or on its straightforward generalization to the multipartite 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.

From direct comparison it is clear that the bilocal set (4) is contained inside the local set (3). More precisely, the bilocal set is characterized 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 revealed if the independence of the sources generating the correlations is taken into account. Another relevant feature is the fact that the bilocal causal structure reproduces the underlying pattern of causation in a entanglement swapping experiment [59].

In the simplest possible scenario, akin to the entanglement 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]:

$$\begin{equation} \begin{array}{rcl} \mathcal{B}&=& \sqrt{I}+ \sqrt{J} \leq 1,\quad I= \dfrac{1}{4} \sum_{x,z=0,1} \langle A_{x} B_{0} C_{z} \rangle,\\[4pt] J&=& \dfrac{1}{4} \sum_{x,z=0,1} (-1)^{x+z} \langle A_{x} B_{1} C_{z} \rangle, \end{array} \end{equation} \tag{ 6 }$$

where $\langle A_{x} B_{y} C_{z} \rangle = \sum_{a,b,c=0,1} (-1)^{a+b+c} p(a,b,c\vert x,y,z).$

Exploiting the causal structure depicted in fig. 1(c), two photonic experiments have provided an experimental 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 dichotomic measurements and Bob measures in the Bell basis. The maximum violation of the bilocality inequality has reached $\mathcal{B}=1.268 \pm 0.014$ corresponding to a violation 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 independence 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 issue is the fact that typically Bell experiments rely on a shared reference frame between the parties. While inoffensive in the usual Bell tests, in the case of bilocality such reference frames could be used to simulate non-local correlations 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 entanglement swapping experiment, reporting a violation of $\mathcal{B}=1.181 \pm 0.004$, 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 explanation [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 violation 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 interest. 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 station perform a complete Bell-state measurement with four possible outcomes, something that is impossible with linear optics [73]. In practice this implies that such realizations 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 quantum mechanics can exhibit non-bilocal correlations also in cases in which all parties only perform single-qubit operations [58], thus allowing for a device-independent test 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 experimental approach has been fully addressed [43] and allowed a violation of bilocality with separable measurements up to $\mathcal{B}= 1.2712 \pm 0.0042$.

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 interventions. It relies on a instrumental variable X that is assumed to influence only A directly and to be independent of the shared correlations between A and B. The instrumental DAG (see fig. 2(a)) implies that any correlation compatible with it should be decomposable as

$$\begin{equation} p(a,b\vert x)= \sum_{\lambda} p(\lambda)p(a\vert x, \lambda)p(b\vert a,\lambda). \end{equation} \tag{ 7 }$$

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

$$\begin{equation} p(a,b \vert x)= \mathrm{Tr} [(M_a^x \otimes M_b^a ) \rho] \end{equation} \tag{ 8 }$$

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

$$\begin{equation} \mathcal{I}={A}_{1}-{B}_{1}+2{B}_{2}-{AB}_{1}+2{AB}_{3}\leq 3, \end{equation} \tag{ 9 }$$

where ${AB}_{x}=\sum_{a,b}(-1)^{a+b}p(a,b\vert x)$ up to the limit of $\mathcal{I}=1+2\sqrt{2}$.

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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 instrumental 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 correlations. In spite of that, the instrumentality violation can be seen as a stronger form of non-classicality. As argued 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 theorem, 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], randomness generation [78–82], or self-testing [83], can also be extended to the instrumental case.