the Klein four-group acts simply transitively on the Bell basis — two bits address maximal entanglement
Theorem.the Klein four-group acts simply transitively on the Bell basis — two bits address maximal entanglement — {I, X, Z, XZ} · |Φ⁺⟩ = the four Bell states — a simply transitive Klein action.
Proof.The Klein four-group acts simply transitively on the Bell basis — 2/2: applying {I, X, Z, XZ} to one half of the Bell state yields four distinct normalised maximally entangled states (amplitude comparison up to global phase), one per group element — two bits pick the element, the element picks the Bell state, bijectively; the teleportation decode table is exactly this action. Documented Pauli-frame structure, computed by direct state-vector arithmetic.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/2/8/index.ts#kleinFourActsSimplyTransitivelyOnBellStates
1 · Classification
finite-complete — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"the Klein four-group acts simply transitively on the Bell basis — two bits address maximal entanglement" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation kleinFourActsSimplyTransitivelyOnBellStates.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (kleinFourActsSimplyTransitivelyOnBellStates @ src/2/8) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes kleinFourActsSimplyTransitivelyOnBellStates (src/2/8/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.