When gravitationally induced entanglement certifies magic: Clifford exceptions and a gate-typing upgrade of the witness theorem

Summary

Types the gate the gravitational hold implements in the committed BMV family: non-Clifford except on an exactly characterized exception set, so the output carries magic (nonstabilizerness). Upgrades the witness theorem to gate grade and gives the E1-M protocol that certifies it from fringes a run already records.

Abstract

The companion paper [What gravitationally induced entanglement certifies] establishes what a certified positive result of the Bose et al.–Marletto–Vedral (BMV) protocol certifies: under six priced assumptions, no classical degree of freedom carried the observed entanglement. We ask the next question, which is about the channel’s gate type rather than its classicality. Within the committed family the gravitational hold implements U = (P₁(a) ⊗ P₂(b)) CZ(Δ) with Δ = −2KΔx²/[d(d² − Δx²)], K = Gm²τ/ℏ. That a controlled-phase gate at a generic angle is non-Clifford, so that the output state carries magic (nonstabilizerness), is elementary and folkloric; it is never the claim here. The claims are certification statements. (i) The committed witness statistic is magic-blind at every value, including its maximum W = 2: an explicit stabilizer mixture reproduces the design correlators and W = 1.156350 exactly, the entire correlator square is stabilizer-reachable, and standard CHSH cannot witness magic at any violation, so added assumptions are forced, not optional. (ii) The family’s nonlocal Clifford-exception set is exactly Δ ∈ πℤ, splitting into the no-entanglement points 2πℤ and the maximal-witness Clifford points π + 2πℤ; requiring Clifford local phases as well yields explicit arithmetic ladders in K/d on each rational-shape family; at the committed operating point (Δ = −211/672 = −0.313988, ±5% calibration) the gate is certified outside LU ⋅ 𝒞₂ ⋅ LU by an exact census of all 11520 two-qubit Cliffords, with stabilizer 2-Rényi entropy M₂ ≥ 0.104444 and entangling power 0.005432. (iii) Theorem M: under the companion theorem’s assumptions plus five added, operationally certified assumptions M0–M4, a certified positive result whose extracted phase interval excludes πℤ additionally certifies that the gravitational coupling implemented a nonlocal non-Clifford gate, magic injected at gate grade, with cited monotone floors evaluated on the interval, stable across both readings of the companion theorem’s mediation disjunction. (iv) The converse: at Δ = π the same protocol attains its maximal witness value W = 2 on a stabilizer state delivered by a Clifford gate, so the family’s best entanglement point is provably a zero-magic point; non-classicality is angle-generic, magic is angle-conditional, and along the hold-time scan the witness grows linearly while the certified magic floor grows quadratically (anti-alignment). The experimental price is measured: the primary instrument is a re-scoring of conditional fringes the witness protocol already records, the Z-conditioned fringe difference Δ̂ = φ_(c)(B = R) − φ_(c)(B = L), exactly invariant under local phase doping, with zero added settings and shots, resolving the committed angle at 12.3σ with 10⁴ shots per setting at 70% visibility, and needing 1105 shots per setting for 5σ; the witness itself, not the extraction, is the binding cost (F ≥ 0.926 for a 5σ witness at 10⁴ shots on the white-noise benchmark), while a state-level magic witness would need visibility 0.9756 against the entanglement witness’s 0.8648, a gap of 0.1108 that prices the upgrade on that route. A null of any branch never certifies that gravity is classical; in particular a classical measure-and-act channel imprints the full second-difference phase in the conditional fringes, so the extraction alone is never a non-classicality certificate. Every certified value is verified in deterministic exact-arithmetic ancillary scripts.

Cite

Vávra, D. (2026). When gravitationally induced entanglement certifies magic: Clifford exceptions and a gate-typing upgrade of the witness theorem. vavradev.com/research, first published 2026-11-04.

@misc{vavra2026m,
  author       = {V{\'a}vra, David},
  title        = {{When gravitationally induced entanglement certifies magic: Clifford exceptions and a gate-typing upgrade of the witness theorem}},
  year         = {2026},
  month        = aug,
  howpublished = {\url{https://vavradev.com/research/vavra-2026-M-gie-certifies-magic-gate-typing.pdf}},
  note         = {Manuscript, Article (companion to paper B). First published 2026-11-04 at vavradev.com/research; SHA-256 8866dc87bfe4eb651bb0ea5dcacbf1226242c6eba3b865f1de166522a30ebb47},
}