Fig. 1. Two masses in path superposition interacting gravitationally become entangled. Two massive particles with embedded magnetic spins are put into a spin-dependent path superposition. They are then left to free fall, where they interact via the gravitational field only. Then, the path superposition is undone, and measurements are performed on the spins. During the free fall, each branch of the superposition accumulates a different phase, which entangles the two particles.

Fig. 2. The quantum circuit simulator and its photonic implementation. (a) Two qubits, $|s_1⟩$ and $|s_2⟩$, represent the spin degrees of freedom, while two qubits, $|g_1⟩$ and $|g_2⟩$, represent the geometry. Each stage of the experiment is mapped into quantum gates acting on the qubits. (b) The simulator is implemented using the path and polarization degrees of freedom of two photons. The spin qubits of the simulator are encoded in the polarization degree of freedom of the photons, while the geometry degrees of freedom are encoded in the photon paths. The two photons are independently prepared in a superposition of horizontal and vertical polarization, and each one passes through a BD, which completely entangles the path of each photon with its polarization. The control-phase (CZ) gate is implemented due to bosonic interference, which is due to the indistinguishability of the photons at the BS. Two HWPs momentarily make the polarization of all paths equal in order to allow the realization of the CZ gate on this degree of freedom. Finally, the qubit state is restored by two other HWPs and the paths are recombined by final BDs, which disentangles path and polarization. Finally, the polarizations of the photons are measured using quarter- and half-waveplates and polarizing beam splitter followed by single-photon detectors. BD, beam displacer; QWP, quarter-wave plate; PBS, polarization beam splitter; HWP, half-wave plate; BS, beam splitter; APD, avalanche photo diode.

Fig. 3. Results of the simulator without and with decoherence. (a) Expectation values of the operators used for the CHSH test on the spin qubits. The lighter-colored parts in each bar (hardly visible) represent the Poissonian experimental errors associated with each observable. The orange dashed bars are the values expected from an ideal maximally entangled state. (b) Real and imaginary parts of the measured density matrix of the spin qubits. (c) Measured values of the entanglement witness $\mathcal{W}$ as a function of the degree of decoherence $\eta$. The latter corresponds to the relative time delay of different polarizations normalized to the coherence time of the photons. The purple-shaded area indicates the region where the witness certifies the entanglement of the state. The dashed black line represents the theoretical curve from the model of the experimental setup. Error bars are due to Poissonian statistics of the measured events. (d) Real and imaginary parts of the measured density matrix of the polarization state of the spin qubits, where the state has experienced maximum decoherence effects ($\eta =1$) introduced by a delay between linear polarizations greater than the photon coherence time. The off-diagonal terms are completely suppressed and the state is separable.