Compatibility of causal hidden-variable theories with a delayed-choice experiment

Wheeler's delayed-choice experiment investigates the indeterminacy of wave-particle duality and the role played by the measurement apparatus in quantum theory. Due to the inconsistency with classical physics, it has been generally believed that it is not possible to reproduce the delayed-choice experiment using a hidden-variable theory. Recently, it was shown that Wheeler's delayed-choice experiment can be explained by a causal two-dimensional hidden-variable theory [R. Chaves, G. B. Lemos, and J. Pienaar, Phys. Rev. Lett. 120, 190401 (2018)]. Here, we carry out a delayed-choice experiment by using photon states that are spacelike separated and demonstrate that the experiment is consistent with quantum theory but inconsistent with any causal two-dimensional hidden-variable theory in a device-independent manner. This demonstrates that causality can be used to test quantum theory in a complementary way to the Bell and Leggett-Garg tests.

Figure 1

Various configurations of the delayed-choice experiment. (a) Wheeler's delayed-choice experiment (WDCE). (b) The modified WDCE (MWDCE). The gray areas correspond to Alice (preparer) and Bob (measurer), respectively. In the modified setup, the second beam splitter is always put in place, Alice has the ability to prepare a set of states controlled by her own phase shifter ${\alpha }_i\in \{{\alpha }_0=0,{\alpha }_1=\pi ,{\alpha }_2=-\pi /2,{\alpha }_3=\pi /2\}$, and two types of detection are controlled by the phase shifter ${\beta }_j\in \{{\beta }_0=\pi /2,{\beta }_1=0\}$ of Bob. The $U$ is the upper path and $D$ is the lower path. (c) The device-independent prepare-and-measure (PAM) scenario for the MWDCE. An initial black box prepares different physical systems (upon pressing a button labeled by ${\alpha }_i$) that are then sent to a second black box where the systems are measured (upon pressing a button labeled by ${\beta }_j$) to produce an outcome labeled by $d_{ij}$. (d) Schematic experimental description performed in this paper. Alice prepares a photon by measuring a Bell state in one of two bases, producing four possible states $|H\rangle +e^{i{\alpha }_i}|V\rangle$. Bob then applies a phase shift and interferes the photon using a PBS to obtain the output.

Figure 2

Experimental setup. The polarization-entangled photon pairs produced by SPDC are coupled into the single-mode fiber and sent to Alice (preparer) and to Bob (measurer), respectively. The distance between Alice and Bob is 46 m and the length of the fiber from the entanglement source (Charlie) to Alice is shorter than that to Bob (the fiber lengths of Charlie-Alice and Charlie-Bob are 28 and 33 m, respectively). Alice's photon passes through an EOM, then is measured by the APD, which collapses Bob's state to $|H{\rangle }_B+e^{i{\alpha }_j}|V{\rangle }_B$ with ${\alpha }_j\in \{0,\pi ,\pm \pi /2\}$, thereby preparing the state. Bob's photon then passes through another EOM where a random phase ${\beta }_j\in \{\pi /2,0\}$ is applied to the vertical polarization component, and is finally measured by the APD. The measurement bases of Alice and Bob are each randomly determined by two independent and spacelike separated QRNGs and EOMs. In order to meet the delayed-choice condition, the measurement basis of Bob is chosen much later than that of Alice.

Figure 3

Minkowski diagrams for the space-time events related to Alice, Bob, and the source Charlie. All light cones are shaded gray. (a) Alice and Bob are spacelike separated as the measurement is finished by Alice and Bob before information about the other party's measurement setting could have arrived. (b) The QRNGs at Alice and Bob finish picking a setting outside the light cone of the generation of an entangled photon pair by Charlie. All the events in our experiment are spacelike separated.

Figure 4

(a) Conditional probability $p\left(d_{i,j}\right)$ that detector $d$ fires for given settings (${\alpha }_i,{\beta }_j$) chosen by Alice and Bob. Under the fair-sampling assumption (FSA) only coincidence events are postselected. All the data were obtained in 1-min continuous measurement. (b) Experimental results of the dimensional witness $I_{\text{DW}}$. The $\langle D_{ij}\rangle$ for the theoretical predictions and experimental measurement are shown as grey and hatched bars, respectively. All the data was obtained in a 1 minute continuous measurement. Error bars represent one standard deviation, deduced from propagated Poissonian counting statistics of the raw detection events.

Figure 5

(a) Scheme of the QRNG module, including laser module, three-port optical circulator, interferometer module, readout, and postprocessing system. (b) Detail of the interferometer module. The beam from the laser module enters the interferometer via port 1. Then output port 2 is monitored by the feedback system, maintaining the optical path difference between two arms of the interferometer. The beam output from port 1 is directed to the sampling and postprocessing system. PD, photon detector; CLK, clock; ADC, analog-to-digital converter; FPGA, field-programmable gate array; PM, power meter; BS, beam splitter; DAC, digital-to-analog converter; HVM, high-voltage module; PS, phase shifter; FRM, Faraday rotator mirror.

Figure 6

Experimentally measured relationship between the second-order correlation function at zero delay $g^2\left(0\right)$ and pair-production rate. The $g^2\left(0\right)$ data shown seem to have a linear relationship with the pair-production rate, and are therefore linearly fitted. The higher the pair-production rate is, the larger the value of $g^2\left(0\right)$ becomes.

Figure 7

Experimental measurement of the entangled state $|H{\rangle }_A|H{\rangle }_B+\left|V{\rangle }_A\right|V{\rangle }_B$. We fixed one photon of the EPR state at $|H{\rangle }_A+|V{\rangle }_A$, and then measured another photon in the basis $|H{\rangle }_B+e^{i{\beta }_j}|V{\rangle }_B$. The phase ${\beta }_j$ is changed as $0\to \pi \to 2\pi$ by changing the angle of HWP from $22.5^{\circ }$ to $112.5^{\circ }$.