the no-communication theorem — entanglement correlates but cannot signal
Theorem.the no-communication theorem — entanglement correlates but cannot signal — Tr_B[(I ⊗ M) ρ (I ⊗ M)†] leaves Alice’s marginal invariant — entanglement correlates, never signals.
Proof.the communication wave, the no-go that keeps entanglement relativistic: Bob's half of a Bell pair is the maximally mixed I/2, and NO local operation Alice performs (X, H, HZ, Y) changes his reduced density matrix — verified invariant by partial trace. So no measurement of Bob's qubit reveals Alice's choice; the perfect correlations appear only when outcomes are COMPARED over a classical channel, and alone Bob learns nothing. This is why entanglement cannot carry a message faster than light — correlation without communication. Shown for a Bell pair and a set of local unitaries; the theorem holds for all CPTP maps and all shared states by the partial-trace argument, here demonstrated. The same no-signalling invariance is computed independently in discoveredTheoremsWaveTwentyOne, where the Alice-side marginal P(+) stays exactly 1/2 across all 144 measurement-angle pairs regardless of the Bob-side setting — two proofs of one theorem: partial trace here, marginal invariance there. The correlations themselves are stronger than any classical theory permits — bounded by the Tsirelson limit 2√2, proven in chsh — yet even that supra-classical strength carries no signal: stronger-than-classical correlation is still not communication.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/2/8/index.ts#theNoCommunicationTheorem
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 no-communication theorem — entanglement correlates but cannot signal" 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 theNoCommunicationTheorem.
- 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 (theNoCommunicationTheorem @ 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 theNoCommunicationTheorem (src/2/8/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.