Past exam of the laser science and quantum information processing course of the University of Oxford / 2006 / 7 / 1

In superdense coding, Alice applies a local Pauli operator to her half of the shared Bell state. Choose a phase bit $u$ and a parity bit $v$, and encode them using $X^v{Z}^u$ on Alice's qubit. With the standard definitions of the Bell states,Alice sends her qubit to Bob. He now holds both halves and performs a Bell-basis measurement, recovering $uv$ without error. Equivalently, a CNOT gate with Alice's qubit as control, followed by a Hadamard gate on that qubit, maps these states to their computational labels up to global phases. Four distinguishable alternatives give two classical bits per transmitted qubit, with one shared entangled pair consumed per use.

The initial entanglement distribution is independent of the later message and can be completed beforehand. Bob's local reduced density matrix before Alice's transmission is $I/2$ for every encoding, so quantum no-signalling prevents it from carrying her choice. This does not make the entangled pair free: if a qubit transmission is used to distribute it, counting that transmission as well gives two qubit transmissions for the two-bit message. The quoted dense-coding rate counts the subsequent message-bearing transmission with entanglement already shared.