No-Signaling Theorem: Why Entangled Pairs Can't Send Messages

Analyzer angles

Alice's angle θ_A 0°

Bob's angle θ_B 45°

Pair emission rate 6/s

Bob's outcomes (received)

P(+1)=50.0%P(−1)=50.0%

Pairs measured

Bob P(+1)

50.0%

Measured E(A,B)

0.00

QM predicted E

0.00

Decoded message at Bob's end

— no signal detected —

How it works

Every pair starts in the singlet state |Ψ⁻⟩ = (|↑↓⟩ − |↓↑⟩)/√2. If Alice measures spin along an axis at angle θ_A and Bob along θ_B, quantum mechanics gives:

Δθ = θ_B − θ_A
P(A=+1) = 1/2                         (Alice's own marginal — always fair)
P(B=+1 | A=+1) = sin²(Δθ/2)
P(B=+1 | A=−1) = cos²(Δθ/2)
E(A,B) = ⟨AB⟩ = −cos(Δθ)

This simulator samples exactly that distribution: Alice's outcome is a fair coin flip, independent of both angles, then Bob's outcome is drawn from the conditional above. Averaging over Alice's two equally-likely outcomes:

P(B=+1) = ½·sin²(Δθ/2) + ½·cos²(Δθ/2) = ½   — for every Δθ

That last line is the no-signaling theorem: Bob's local statistics never depend on θ_A, so nothing Alice does to her analyzer — however fast, however patterned — shows up in Bob's data alone. The correlation between their results is real and matches −cos(Δθ) (stronger than any classical hidden-variable model can produce, the heart of a Bell test), but extracting it requires comparing notes over an ordinary, light-speed-limited classical channel. That is exactly why the "entanglement across universes" idea in the article can't work as a communication channel: the wiring that makes entanglement strange is the same wiring that makes it useless for sending a bit on its own.

θ_A / θ_B sliders — set each lab's analyzer orientation independently.
"Try to send 1011" — Alice snaps her angle between 0° and 90° following that bit pattern, as fast as pairs are emitted. Watch the histogram: Bob's P(+1) stays pinned near 50% throughout.
Measured E(A,B) — the running correlation from actual sampled outcomes, converging to the QM prediction −cos(Δθ) as more pairs arrive.

Alice

Bob

Outcome +1 Outcome −1 Entangled pair source

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