PMC

The causal links forbidden by the respective assumption are shown in dashed green lines. Note that the statistical constraints implied by causal parameter independence are asymmetric in a and b, and swapping them would result in a causal structure where the arrow between A and B is reversed. Our experimental test applies to both of these structures and any convex combination of them.

Any value below the dashed red line, given by , 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.

(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.

The orange data points are observed using a fixed measurement scheme (optimal for the maximally entangled state γ = 45°), with the dotted, orange line representing the corresponding theory prediction. The blue data points and blue dashed theory line correspond to the case where measurement settings were optimized for the prepared states (see the Supplementary Materials for details). The black line represents the bound of inequality (); any point above this line cannot be explained causally by a model of the form in . Error bars correspond to 3σ statistical confidence intervals.

(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. (). 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 . 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.