The No-Signaling Principle

Entanglement produces strong correlations but cannot transmit information faster than light, keeping quantum theory consistent with relativity.

Correlation without communication

Entangled particles show correlations that no local classical model reproduces, which raises a worry: could a measurement here instantly send a message there? The no-signaling principle answers no. The local outcome statistics on one side do not depend on which measurement the other side chooses. Marginals are fixed regardless of the distant setting, so no controllable influence crosses the gap.

The mathematical statement

For a shared state rho_AB, the reduced state rho_A = Tr_B(rho_AB) is unchanged by any operation Bob performs on B, provided Alice learns nothing from Bob. Since all of Alice's local probabilities are Tr(M_A rho_A), and rho_A is untouched by Bob's local action, her outcome distribution carries zero information about his choice. This holds for measurement, unitary, or discarding on B.

import numpy as np
bell = np.array([1,0,0,1])/np.sqrt(2)
rho = np.outer(bell,bell).reshape(2,2,2,2)
rhoA = np.trace(rho, axis1=1, axis2=3)
# any local op on B leaves rhoA = I/2 unchanged
print(np.round(rhoA,3))

Why teleportation obeys it

Teleportation transfers a state using entanglement, yet it requires two classical bits sent through an ordinary channel. Until those bits arrive, the receiver's qubit is maximally mixed and carries nothing. The classical message, bounded by light speed, gates the protocol. This is the general pattern: entanglement plus classical communication can do remarkable things, but the classical part enforces causality.

A constraint on physical theories

No-signaling is more general than quantum mechanics. It is the causal constraint any correlation must satisfy to coexist with relativity. Some hypothetical correlations respect no-signaling yet exceed the Tsirelson bound; the fact that nature stays within quantum limits shows no-signaling alone does not single out quantum theory. It does, however, guarantee that entanglement is never a faster-than-light telegraph.