Improving resilience of quantum-gravity-induced entanglement of masses to decoherence using three superpositions

Abstract

Recently, a protocol called quantum-gravity-induced entanglement of masses (QGEM) that aims to test the quantum nature of gravity using the entanglement of two qubits was proposed. The entanglement can arise only if the force between the two spatially superposed masses is occurring via the exchange of a mediating virtual graviton. In this paper we examine a possible improvement of the QGEM setup by introducing a third mass with an embedded qubit so that there are now three qubits to witness the gravitationally generated entanglement. We compare the entanglement generation for different experimental setups with two and three qubits and find that a three-qubit setup where the superpositions are parallel to each other leads to the highest rate of entanglement generation within $\ensuremath{\tau}=5\phantom{\rule{4pt}{0ex}}\mathrm{s}$. We show that the three-qubit setup is more resilient to the higher rate of decoherence. The entanglement can be detected experimentally for the two-qubit setup if the decoherence rate $\ensuremath{\gamma}$ is $\ensuremath{\gamma}<0.11\phantom{\rule{4pt}{0ex}}\mathrm{Hz}$ compared to $\ensuremath{\gamma}<0.16\phantom{\rule{4pt}{0ex}}\mathrm{Hz}$ for the three-qubit setup. However, the introduction of an extra qubit means that more measurements are required to characterize entanglement in an experiment. We conduct experimental simulations and estimate that the three-qubit setup would allow detecting the entanglement in the QGEM protocol at a $99.9%$ certainty with $O({10}^{4})--O({10}^{5})$ measurements when $\ensuremath{\gamma}\ensuremath{\in}[0.1,0.15]\phantom{\rule{4pt}{0ex}}\mathrm{Hz}$. Furthermore, we find that the number of needed measurements can be reduced to $O({10}^{3})--O({10}^{5})$ if the measurement schedule is optimized using joint Pauli basis measurements. For $\ensuremath{\gamma}>0.06\phantom{\rule{4pt}{0ex}}\mathrm{Hz}$ the three-qubit setup is favorable compared to the two-qubit setup in terms of the minimum number of measurements needed to characterize the entanglement. Thus, the proposed setup here provides a promising avenue for implementing the QGEM experiment.