Marc Sperzel

quant-ph — Quantum Physics

Entanglement from the Diósi–Penrose Model: An Exact Criterion, and Why It Stays Small at Any Distance

Contributed by Marc Sperzel

Trillo and Navascués showed that the Diósi–Penrose model, in the Tilloy–Diósi form with the same smearing length $\sigma$ in force and noise, entangles two masses in superposition when they are closer than about $2.2\,\sigma$, and asked whether gravitationally induced entanglement can be observed beyond that distance. We answer for two rigid bodies whose branches are position eigenstates on finite sets, without kinetic term, for noise strength $C$. One kernel, the Coulomb energy of the smeared bodies, sets both the Newtonian coupling and the correlated noise, and the question becomes a single geometric comparison: a product state becomes entangled if and only if the largest canonical correlation between neutral charge patterns on the two branch sets exceeds $C/\sqrt{1+C^2}$. Below this threshold the evolution is a separable operation at all times. The published critical distances are two-branch cases of the criterion: for parallel two-branch superpositions the supremum is $2.21\,\sigma$ at $C=1$, and a straddling geometry reaches $2.74\,\sigma$. With many branches there is no critical distance. What limits the effect is the model's own dephasing, which sits in the same quadratic form. For bodies of effective size $\sigma_{\rm eff}$ ($\sigma$ for point masses) separated by a slab of width $d\gg\sigma$, the entangling rate at $C=1$ is at most $0.12\,\sigma_{\rm eff}/d$ times the single-body dephasing rate. The entanglement is therefore transient and, for well-separated branches, a fraction of order $\sigma_{\rm eff}/d$ of what unitary Newtonian gravity gives. Entanglement of the size unitary gravity predicts still excludes the model; the inference from any detected entanglement beyond a fixed multiple of $\sigma$ does not hold.

Author comments: Revised presentation: the idea is stated first, a contents page is added, and the numerical values are collected in an appendix. Results and title are unchanged.