Nonlocality vs Signalling

Bell tests show entangled particles are nonlocal: their correlations cannot arise from predetermined local values. A natural worry follows — does this let Alice send Bob a message faster than light, breaking relativity? The answer is a clean no. Reconciling the two is the no-signalling principle.

What nonlocality does not mean

Nonlocality is a statement about correlations, not about control. When Alice measures her half of a singlet, Bob's particle is instantly described by a definite outcome correlated with hers. But Alice cannot choose what Bob sees, and Bob, looking only at his own particle, sees nothing unusual.

The no-signalling theorem

Suppose Alice measures with setting $a$ and Bob with setting $b$ . Bob's outcome statistics, ignoring Alice, are his marginal distribution

P_B(B \mid a, b) = \sum_{A} P(A, B \mid a, b).

No-signalling is the statement that this marginal is independent of Alice's setting $a$ :

P_B(B \mid a, b) = P_B(B \mid a', b) \quad\text{for all } a, a'.

Whatever Alice chooses to measure, Bob's local statistics are unchanged. Since the only thing Alice can vary is her setting, and it has no effect Bob can detect, no information flows from Alice to Bob. The same holds with the roles reversed.

Why quantum mechanics obeys it

For the singlet, measure Bob's spin along any axis $\hat b$ while Alice does anything she likes. Bob's local description is his reduced density matrix, obtained by tracing out Alice's qubit:

\rho_B = \operatorname{Tr}_A\big(|\Psi^-\rangle\langle\Psi^-|\big) = \tfrac12\,\mathbb{1}.

It is the maximally mixed state — outcomes $\pm 1$ with probability $\tfrac12$ each along every axis. Crucially, $\rho_B$ does not depend on Alice's setting at all: a local operation by Alice cannot change Bob's reduced state. So Bob measures pure $50/50$ noise no matter what Alice does. The correlations only appear when the two of them later compare notes — which requires a classical channel, limited by light speed.

Three layers of correlation strength

The no-signalling constraint alone is weaker than quantum mechanics. Recall the CHSH hierarchy:

\underbrace{|S| \le 2}_{\text{local}} \;<\; \underbrace{|S| \le 2\sqrt2}_{\text{quantum}} \;<\; \underbrace{|S| \le 4}_{\text{no-signalling}}.

Hypothetical "PR-box" correlations reach $|S| = 4$ while still forbidding signalling — so no-signalling does not by itself imply locality. Quantum mechanics is nonlocal (beats $2$ ) yet respects causality (never signals), occupying the middle tier.

The reconciliation

There is no paradox. Entanglement produces correlations stronger than any local-realistic theory allows, yet the marginal statistics each observer sees are completely setting-independent. Nonlocality and no-signalling are not in tension: quantum mechanics is exactly the theory that has the first without ever permitting the second.