Gravitationally Mediated Entanglement in Interferometers with Repeated Splitting

Abstract

Proposals to probe the quantum nature of gravity through entanglement between spatially separated masses have attracted considerable interest. In such schemes, spatial superpositions are prepared through controlled splitting operations that produce several interferometric branches. Here we propose and analyse protocols involving repeated splitting stages, each of which doubles the number of occupied paths. We show that the density matrix after $n$ splitting stages is related to that of the $(n-1)$-splitting protocol through local parity dephasing. Two of the resulting sectors reproduce the preceding state, while the other two acquire additional gravitational phases. Removing the locally generated parts gives an exact lower bound on the change in negativity, expressed in terms of the preceding state's negative partial-transpose eigenspace. For two splitting stages, this leads to an explicit condition on three gravitational phases under which the additional split cannot reduce negativity. We also show that further splitting cannot shorten the lifetime of the generated entanglement. If the local internal system preserves the full branch label, the positional entanglement is retained upon recombination. We describe an implementation using a reusable two-level system and local memories, and compare it with a readout without a memory register. For the latter, we derive an exact condition for when an additional splitting stage improves the readout, based on a $2\times 2$ Schur complement. Numerical simulations show an advantage in positional negativity over the rates considered, with an improvement surviving the restricted readout at all decoherence rates considered at the final readout time.

The paper asks whether repeated binary splitting of two spatially superposed masses can improve gravity-mediated entanglement, and how much survives recombination. It models diagonal Newtonian interactions along finite-duration interferometer trajectories, with independent, separation-dependent local dephasing. Local parity dephasing decomposes the $n$-stage state into four equally weighted sectors: two reproduce the preceding protocol at the same time, while two acquire additional gravitational phases. Removing local phases and applying the variational characterization of negativity gives a sufficient non-decrease witness using the parent state's negative partial-transpose eigenspace. Independently, $\mathcal N(\rho_n(t))\ge\tfrac12\mathcal N(\rho_{n-1}(t))$, so another split cannot eliminate existing negativity within this model. For two stages and an entangled parent with $0<\Phi_1<2\pi$, a sufficient non-decrease condition is $\sin(\Phi_+/2)+\sin(\Phi_-/2)\ge2\sin(\Phi_1/2)$. A local internal register retaining all $2^n$ branch labels preserves branch-space negativity upon ideal recombination. Without memory, the selected two-qubit readout is probabilistic; improvement in conditioned negativity is characterized exactly through a $2\times2$ Schur complement. In an illustrative simulation at $10$ s and dephasing rate $0.030\,\mathrm{s}^{-1}$, a second split increases branch-space negativity from $0.02199$ to $0.16978$, and conditioned readout negativity to $0.06489$, with success probability $0.06081$. The comparison uses chosen rather than optimized geometries and phenomenological decoherence rates. It omits control imperfections, recombination misalignment, technical noise, and residual electromagnetic interactions, and assumes fast recombination. The stated magnetic gradients remain experimentally challenging; conditioned entanglement gains do not directly establish improved experimental yield.