Gravitationally mediated entanglement: Newtonian field versus gravitons

We argue that if the Newtonian gravitational field of a body can mediate entanglement with another body, then it should also be possible for the body producing the Newtonian field to entangle directly with on-shell gravitons. Our arguments are made by revisiting a gedankenexperiment previously analyzed by Belenchia et al., which showed that a quantum superposition of a massive body requires both quantized gravitational radiation and local vacuum fluctuations of the spacetime metric in order to avoid contradictions with complementarity and causality. We provide a precise and rigorous description of the entanglement and decoherence effects occurring in this gedankenexperiment, thereby significantly improving upon the back-of-the-envelope estimates given by Belenchia et al. and also showing that their conclusions are valid in much more general circumstances. As a by-product of our analysis, we show that under the protocols of the gedankenexperiment, there is no clear distinction between entanglement mediated by the Newtonian gravitational field of a body and entanglement mediated by on-shell gravitons emitted by the body. This suggests that Newtonian entanglement implies the existence of graviton entanglement and supports the view that the experimental discovery of Newtonian entanglement may be viewed as implying the existence of the graviton.

Figure 1

The setup for the gedankenexperiment of [40], as analyzed in [38]. Alice’s particle (in blue) is originally in the superposition state Eq. (2.1) with the two wave packets separated by distance $d$. Bob is at a distance $D\gg d$ from Alice and, at a prearranged time, he releases a particle (in orange) from a trap and attempts to gain information about which path Alice’s particle took by determining the strength of the Coulomb/Newtonian field of Alice’s particle. Meanwhile, at a corresponding prearranged time, Alice recombines her particle and determines its coherence as described in the text. Bob does his measurement within time $T_B<D$ and Alice recombines her particle in time $T_A<D$, so their actions are performed in spacelike separated regions.

Figure 2

Alice recombines her particle at event $P$ and subsequently keeps her recombined particle in inertial motion. $\mathrm{\Sigma }$ is an arbitrary Cauchy surface passing through $P$.

Figure 3

A spacetime diagram of the gedankenexperiment of Fig. 1 showing the three Cauchy surfaces, ${\mathrm{\Sigma }}_1$, ${\mathrm{\Sigma }}_2$, and ${\mathrm{\Sigma }}_3$. The Cauchy surface ${\mathrm{\Sigma }}_1$ passes through Alice’s region after recombination but is such that the region in which Bob performs his measurements (shaded in gray) lies to the future of ${\mathrm{\Sigma }}_1$. (We have depicted Bob as releasing a particle from a trap, but Bob is allowed to perform any measurement whatsoever in the gray region.) The Cauchy surface ${\mathrm{\Sigma }}_2$ is such that it passes through Alice’s region before she starts the recombination process but is such that Bob’s measurement lies to the past of ${\mathrm{\Sigma }}_2$. The Cauchy surface ${\mathrm{\Sigma }}_3$ passes through Alice’s region after recombination and is such that Bob’s measurement lies to the past of ${\mathrm{\Sigma }}_3$.