If Bob opens the trap, Ehrenfest theorem gives branch-dependent Newtonian accelerationsso, for $R\gg d$,After time $t$, the centers of Bob's two conditional wave packets differ byFor Alice's superposition, these distinguishable Bob states become entangled with $|L⟩$ and $|R⟩$; for Alice's mixture there was no initial coherence to entangle. Taking Bob's minimum packet width to be the Planck length ${\ell }_P$, appreciable branch distinguishability begins when $\Delta x\sim {\ell }_P$, atKeeping the trap closed suppresses this branch separation.

If $t_{\mathrm{ent}}<R/c$, Bob could choose whether to destroy Alice's local coherence before a light signal from his laboratory arrived. Any procedure by which Alice distinguished the coherent superposition from the mixture in less than $R/c$ would then enable superluminal signalling. The largest dangerous separation is determined parametrically by $t_{\mathrm{ent}}\sim R/c$, which givesCausality therefore requiresUsing $G{m}_P/c^2={\ell }_P$ for the Planck mass $m_P$,up to numerical factors.

For positions ${\mathbf{r}}_1,{\mathbf{r}}_2$, the two-particle Schrödinger equation isNeglecting packet spreading and writing $|a{⟩}_i$ for ${\psi }_{ai}$, branchwise evolution givesup to local kinetic phases. This branch-dependent Newtonian gravitational potential energy produces gravitationally induced entanglement.

Under the stated distance approximation, only the three branches $a=b$ acquire an appreciable common phaseThe coefficient matrix iswhere $J$ is the $3\times 3$ all-ones matrix. The reduced density matrix isIt equals $I/3$ when $\cos \varphi =-1/2$, so the first maximally entangled state occurs at $\varphi =2\pi /3$. Therefore

The state does not remain entangled for every $t>0$. Whenever $\varphi =2\pi k$, all branch phases again agree and the state returns to its initial product state. The revival period is