NoSignalingTheorem

The No-Signaling Theorem, or no-signaling principle, is a foundational concept in quantum information theory. It states that information cannot be transmitted instantaneously between spatially separated parties through quantum entanglement alone. Although measurements on entangled systems can exhibit correlations, the local statistics available to an observer on one side do not depend on the choice of measurement performed on the distant side.

Formally, for a bipartite state ρ_AB, the probability of obtaining a result a on system A when a measurement {N_a^x} is performed locally, while B is measured with {M_b^y} or left unmeasured, satisfies p(a|x) = Tr[(N_a^x ⊗ I) ρ_AB]. This marginal probability is independent of B’s measurement setting y. Equivalently, the reduced state of A, ρ_A = Tr_B(ρ_AB), is unchanged by any local operation or measurement performed on B. This ensures that no information can be transmitted by choice of measurements on B alone.

The no-signaling principle is compatible with the existence of quantum nonlocal correlations, such as those violating Bell inequalities, while remaining consistent with special relativity. It underpins the security of quantum key distribution and explains why entanglement cannot be used for faster-than-light communication, even though quantum teleportation and other protocols rely on classical communication to complete information transfer.

Historically, the principle emerges from the formal framework of quantum mechanics and the requirement that local operations cannot affect distant outcomes in a way that enables signaling.