 Solution

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.