Explaining observations in terms of causes and effects is central to empirical science. However, correlations between entangled quantum particles seem to defy such an explanation. This implies that some of the fundamental assumptions of causal explanations have to give way. We consider a relaxation of one of these assumptions, Bell’s local causality, by allowing outcome dependence: a direct causal influence between the outcomes of measurements of remote parties. We use interventional data from a photonic experiment to bound the strength of this causal influence in a two-party Bell scenario, and observational data from a Bell-type inequality test for the considered models. Our results demonstrate the incompatibility of quantum mechanics with a broad class of nonlocal causal models, which includes Bell-local models as a special case. Recovering a classical causal picture of quantum correlations thus requires an even more radical modification of our classical notion of cause and effect.

Keywords

Phrasing Bell’s theorem in the language of causal models provides a clear picture of the underlying assumptions and allows for a unified and quantitative approach to relaxations of these assumptions ( 11 , 12 ). For example, quantum correlations can be explained by causal models when relaxing Bell’s local causality assumption, which is commonly referred to as quantum nonlocality. Here, we test nonlocal causal models, which relax local causality by abandoning causal outcome independence and allowing for a causal influence from one measurement outcome to the other (see Fig. 1B ). First, we consider the simplest case that reveals such correlations, the Clauser-Horne-Shimony-Holt (CHSH) scenario ( 35 ), where two parties, Alice and Bob, can each measure one of two dichotomic observables. Using controlled interventions, we find the potential causal influence insufficiently strong to explain the observed CHSH violation. In the second experiment, we go beyond the CHSH scenario and violate a Bell-type inequality, which involves three measurement settings for each party and is satisfied even for arbitrarily strong causal influences from one outcome to the other ( 11 ). In contrast to the interventional method, which requires detailed knowledge of the physical system under consideration, the latter method is device-independent. Our results highlight the incompatibility of quantum correlations not only with the well-known Bell-local causal models but also with nonlocal causal models, where one measurement outcome may have a direct causal influence on the other.

A natural framework for this research program, and the study of Bell’s theorem, is the theory of causal modeling ( 11 , 12 ), which aims to explain correlations in terms of cause-and-effect relations between events ( 15 , 16 ). Discovering these relations from empirical data is difficult in general ( 17 – 20 ); however, within classical physics, such an explanation should always exist because the properties of a classical system, even if not measured, can always be assumed to have well-defined values. Causal reasoning is at the heart of empirical science and builds upon the most fundamental understanding of causality—that if a variable acts as the cause for another one, actively intervening on the first should cause changes in the second. More recently, causal modeling has attracted considerable interest in foundational physics, particularly for the study of stronger-than-classical correlations ( 12 , 21 – 28 ), dynamical causal order ( 29 ), and indefinite causal structures ( 29 , 30 ), and their role as computational resource ( 31 – 34 ).

( A ) Bell’s original local hidden-variable models, where X (Y) is Alice’s (Bob’s) measurement setting, and A (B) is the corresponding measurement outcome. Λ denotes the local hidden variable. ( B ) A relaxation of local causality, where A may have direct causal influence on B. The Bell-local models in (A) are the limiting case where the green arrow from A to B vanishes. An explicit example of such a model is given in the Supplementary Materials. ( C ) An intervention (I) on A forces the variable to take a specific value and breaks all incoming arrows.

Four decades after Freedman and Clauser ( 1 ) performed the first Bell’s inequality test ( 2 ), a series of loophole-free experiments ( 3 – 5 ) have now conclusively shown that the predictions of quantum mechanics are at odds with local realism. Scientific realism posits that physical systems have real, objective properties—independent of whether we observe them or not—that determine the outcomes of measurements performed on the system. The idea of locality—or more precisely local causality—is that causal influences cannot propagate faster than the speed of light. On the basis of local causality, and the assumption that measurement settings can be chosen freely, Bell derived an inequality that must be respected by any set of correlations that can be explained in terms of, possibly hidden, common causes (see Fig. 1A ) but is violated by observed quantum correlations. Consequently, a new area of research has emerged, exploring to what extent the various underlying assumptions have to be relaxed to recover a causal explanation of quantum correlations ( 6 – 14 ).

RESULTS

Theoretical background A causal structure underlying n jointly distributed discrete random variables (X 1 , …, X n ) is represented by a directed acyclic graph, where the nodes (circles in Fig. 1) represent variables and the directed edges (arrows in Fig. 1) represent causal relations (15). Bell’s theorem, where two observers, Alice and Bob, perform local measurements on one half of a shared quantum state, can be conveniently formulated in this language. Figure 1A shows the corresponding causal graph, based on Bell’s assumptions of measurement independence and local causality. Measurement independence states that the measurement choices of Alice and Bob, X and Y, respectively, are independent of how the system has been prepared, that is, there is no causal link from the hidden variable Λ to X or Y and thus p(x,y,λ) = p(x,y)p(λ). Local causality implies that the probability of Alice’s (Bob’s) outcome A (B) is fully specified by Λ and by the measurement choice X (Y), that is, p(a|x,y,b,λ) = p(a|x,λ) and p(b|x,y,a,λ) = p(b|y,λ). Here and in the following, we adopt the usual convention that uppercase letters denote random variables, whereas their values are denoted in lowercase. Interpreted in the causal modeling framework, local causality is the combination of what we call causal parameter independence—there is no direct causal influence from the measurement setting Y (X) to the other party’s outcome A (B)—and causal outcome independence, stating that there is no direct causal influence from one outcome to the other. Note that these definitions are motivated by the causal structure and differ from the statistical notion of outcome independence and parameter independence (36), which cannot be given a causal interpretation (see Materials and Methods for details). The causal models compatible with these assumptions are the well-known Bell-local hidden-variable models: p(a,b|x,y) = Σ λ p(a|x,λ)p(b|y,λ)p(λ). The constraints on the observable probabilities p(a,b|x,y) dictated by such a causal model are known as Bell inequalities. In the simplest possible Bell scenario, where each of the parties measures one of two observables (x,y = 0,1) obtaining one of two possible outcomes (a,b = 0,1), any correlations compatible with Bell-local causal models must respect the CHSH inequality (35) (1)where 〈A x B y 〉 = ∑ a,b = 0,1 (−1)a + bp(a, b|x, y) is the joint expectation value of A x and B y . The first loophole-free Bell experiments (3–5) now conclusively show that quantum mechanics allows for correlations that violate this inequality, therefore witnessing its incompatibility with causal models that satisfy local causality and measurement independence. To retain a classical causal explanation for the correlations observed in the Bell scenario, some of these causal assumptions have to be relaxed (7–14, 36). We focus on the class of models that satisfy causal parameter independence, but do not assume causal outcome independence, such that Alice’s measurement outcomes may have a direct causal influence on Bob’s outcomes (see Fig. 1B). The same arguments hold for the case where Bob’s outcome influences Alice’s outcome (with the A→B arrow reversed in Fig. 1B), or any linear combination of these cases, as discussed in detail in the Supplementary Materials. Because the causal model is formulated without any reference to a space-time structure, this influence may be sub- or superluminal, instantaneous, or even to the past, as long as it does not create any causal loop. In particular, it is consistent with a recent no-go theorem, which states that quantum correlations cannot be explained by any finite-speed influence (37). The probability distributions compatible with this causal structure can be decomposed as (2)

Interventional method The first experimental method we use to test this model relies on interventions, a core tool in causal discovery that allows for the identification and quantification of causal influences (11, 15, 38, 39). Formally, an intervention is the act of locally forcing a variable X i to take on some value x′ i , denoted do(x′ i ). This removes all incoming arrows on X i while keeping the causal dependencies between all other variables unperturbed (see A in Fig. 1C). In practice, performing such arrow-breaking interventions always requires some background knowledge of the system under consideration because the possible persistence of “confounding” common causes cannot be excluded from statistics alone. In our case, we shall assume that, for the purpose of the intervention, the local degrees of freedom behave according to quantum mechanics. Such assumptions are common in quantum steering scenarios and semi–device-independent quantum cryptography, where it is assumed that the devices of at least one of the laboratories can be trusted and work according to quantum mechanics. In the CHSH scenario, passive observations alone are not enough to determine whether correlations between A and B are due to direct causation or a common cause Λ. However, an intervention on variable A would break the link between A and the (hypothetical) variable Λ. Thus, all remaining correlations between A and B must stem from direct causation. The maximal shift in the probability distribution of B upon intervention on A allows quantifying the strength of this causal link (11). To achieve this, we use the so-called average causal effect (ACE) (15, 38) (3)which is a variant of the measure C A→B used in the work of Chaves et al. (11). In contrast to the latter, ACE A→B does not require knowledge of the hidden variable and is thus experimentally accessible. As we prove in detail in the Supplementary Materials, the average causal effect satisfies the same relation as C A→B in the work of Chaves et al. (11), namely (4)where the maximum is taken over all eight symmetries of the CHSH quantity under relabeling of inputs, outputs, and parties (35). That is, the average causal effect required for a causal explanation of a set of quantum correlations is directly proportional to the CHSH violation achieved by the correlations in question. We experimentally implemented an intervention on a CHSH-Bell test using pairs of polarization-entangled photons, generated in the state cosγ|HV〉 + sinγ|VH〉 (see Fig. 1A). Here, H and V correspond to horizontal and vertical polarizations, respectively, and γ is the polarization angle of the pump beam, which continuously controls the degree of entanglement, as measured by the concurrence C = |sin(2γ) (40). Alice and Bob test the CHSH inequality with two settings and two outcomes each. The measurements are chosen in the equatorial (linear polarization) plane of the Bloch sphere (see Fig. 2B). To test the (directional) link A→B, Bob was located in the causal future of Alice using a 2-m fiber delay before Bob’s measurement device. Recall that an intervention on Alice’s outcome A needs to break all relevant incoming causal arrows and deterministically set the value of the variable A. Relying on the quantum description of the local degrees of freedom, these requirements are met by first projecting Alice’s photon onto circular polarization states —which, within experimental precision, erases all relevant information for the CHSH test performed in the linear polarization plane—and then re-preparing it in eigenstates of Alice’s measurement PBS |H/V〉—which forces one of the two outcomes A = ±1. This corresponds to operations of the form |H/V〉〈R/L|, which are experimentally implemented using a quarter-wave plate at ±45°, followed by a polarizer directly before Alice’s measurement PBS. The measurement bases for Alice and Bob, as well as the setting of the intervention polarizer and quarter-wave plate, were chosen randomly using quantum random numbers from the Australian National University’s online quantum random number generator based on the work of Symul et al. (41). Fig. 2 The experimental setup. (A) Pairs of photons are generated via spontaneous parametric down-conversion in a periodically poled KTP crystal, using the Sagnac design of Fedrizzi et al. (48). The degree of polarization entanglement between the two photons can be continuously varied by changing the polarization angle γ of the pump laser. Alice and Bob perform measurements in the equatorial plane of the Bloch sphere using a half-wave plate (HWP) and a polarizing beam splitter (PBS). Additional quarter-wave plates (QWPs) can be used for quantum state tomography of the initial entangled state. In the interventionist experiment, an additional combination of QWP and polarizer (POL) is used between Alice’s basis choice and her measurement. Causal variables are indicated using the notation of Fig. 1A. Note that Λ can represent an arbitrary hidden variable acting as a common cause for the observed outcomes, which need not necessarily originate at the source. (B) Alice’s (red) and Bob’s (blue) measurement bases and the intervention direction (cyan) on the Bloch sphere. QRNG, quantum random number generator; APD, avalanche photodiode. Single-photon clicks in the avalanche photodiodes for each outcome are registered with an AIT-TTM8000 time-tagging module with a temporal resolution of 82 ps. Outcome probabilities, used to estimate ACE A→B , were computed from a total of 48,000 coincidence counts, and no more than one event was registered for each set of random choices for X, Y, and the two elements of I. Figure 3 shows the observed average causal effect as a function of the CHSH values measured for a range of entangled states. All measured values are below and largely independent of the observed CHSH violation. Note that the quantity is bounded from below, which results in non-Gaussian statistics and makes the value 0 unachievable in the presence of experimental imperfections and finite counting statistics. When taking this into account, all data lie within the 3σ noise due to Poissonian counting statistics (see the Supplementary Materials for details). All quoted uncertainties were obtained from Monte Carlo simulations of the Poissonian counting statistics and correspond to the 0.13th and 99.87th percentile, respectively (in the case of normally distributed variables, this would correspond to 3σ confidence regions). Within current experimental capabilities, we find that CHSH violations above a value of S 2 = 2.05 ± 0.02 cannot be fully explained by means of a direct causal influence from one outcome to the other. That is, the potential causal influence between Alice’s and Bob’s measurement (green arrow in Fig. 1B) is not sufficiently strong. Fig. 3 Observed average causal effect ACE versus measured CHSH value. Any value below the dashed red line, given by Eq. 4, is not sufficient to explain the observed CHSH violation. Note that the quantity ACE is bounded from below by 0, as indicated by the hatched area, resulting in asymmetric error distributions. The blue shaded area represents the 3σ region of Poissonian noise. All errors represent the 3σ statistical confidence intervals obtained from a Monte Carlo simulation of the Poissonian counting statistics.