No-communication theorem

The no-communication theorem states that local operations on entangled systems cannot transmit information between distant parties without a communication channel.

Partial trace: The operation that obtains a subsystem’s state by mathematically discarding the other subsystems. Tracing out the distant subsystem shows why its local state is unchanged by remote operations.
Quantum state: A mathematical object specifying the probabilities of outcomes for measurements on a quantum system. The theorem constrains what local state descriptions can reveal about distant actions.
Quantum teleportation: A protocol that transfers an unknown quantum state using shared entanglement and classical communication. Teleportation succeeds only after classical messages convey measurement outcomes.
Quantum cryptography: The use of quantum phenomena to establish or protect cryptographic tasks. Protocols exploit quantum correlations while relying on ordinary communication for coordination.
Density matrix: A mathematical representation of a quantum state that includes pure states and statistical mixtures. The theorem is expressed through local density matrices, not individual entangled-state descriptions.
Quantum entanglement entropy: A measure of entanglement between parts of a pure quantum state, calculated from a subsystem’s entropy. Entanglement entropy can be nonzero even though neither party can signal through the correlation.
Superdense coding: A protocol that uses shared entanglement to transmit two classical bits by sending one qubit. It requires a transmitted quantum system, so entanglement alone does not send the message.
Quantum network: A network that distributes quantum states or entanglement among connected nodes. Entangled network nodes still need channels to exchange usable messages.
Quantum measurement: A physical interaction that produces outcomes whose probabilities are determined by a quantum state. Measurements at one location cannot alter the other party’s local outcome probabilities.
Born rule: The rule that converts a quantum state and measurement into probabilities for possible outcomes. No communication concerns locally observed probabilities, which the Born rule computes.

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