Causal Modeling the Delayed-Choice Experiment

Wave-particle duality has become one of the flagships of quantum mechanics. This counterintuitive concept is highlighted in a delayed-choice experiment, where the experimental setup that reveals either the particle or wave nature of a quantum system is decided after the system has entered the apparatus. Here we consider delayed-choice experiments from the perspective of device-independent causal models and show their equivalence to a prepare-and-measure scenario. Within this framework, we consider Wheeler’s original proposal and its variant using a quantum control and show that a simple classical causal model is capable of reproducing the quantum mechanical predictions. Nonetheless, among other results, we show that, in a slight variant of Wheeler’s gedanken experiment, a photon in an interferometer can indeed generate statistics incompatible with any nonretrocausal hidden variable model, whose dimensionality is the same as that of the quantum system it is supposed to mimic. Our proposal tolerates arbitrary losses and inefficiencies, making it specially suited to loophole-free experimental implementations.

Figure 1

In Wheeler’s delayed-choice gedanken experiment, the choice of removing or not the beam splitter ${\mathrm{BS}}_2$ in a Mach-Zehnder interferometer is made after the photon has entered the interferometer (at ${\mathrm{BS}}_1$). With ${\mathrm{BS}}_2$ present, the photon counting rate at either detector is a function of $\varphi$; when absent, the counting rate is independent of $\varphi$.

Figure 2

(a) The DAG representing the causal model for WDCE. (b) A causal model allowing for retrocausal influence from $Y$ to $\mathrm{\Lambda }$. (c) The representation of the PAM scenario in terms of preparation and measurement devices (black boxes).

Figure 3

Our proposed modification of WDCE, which can be used to discard two-dimensional HV models. The gray dashed lines allude to Fig. 2. In addition to a phase shift ${\varphi }_x$ in the preparation stage, another phase shift ${\sigma }_y$ is applied after the single photon has entered the interferometer, for instance, by slightly moving ${\mathrm{BS}}_2$.