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Superdense coding is a quantum communication protocol that transmits two classical bits by sending only one qubit, using a pre-shared entangled pair. Alice encodes her two-bit message by applying one of four Pauli operations (I, X, Z, XZ) to her half of a Bell pair, then sends that qubit to Bob. Bob performs a Bell measurement on both qubits to recover the two-bit message with certainty. It is the dual of quantum teleportation: teleportation sends one qubit using two classical bits and shared entanglement; superdense coding sends two classical bits using one qubit and shared entanglement.
Superdense coding demonstrates that entanglement has concrete operational value as a communication resource. The protocol begins with Alice and Bob sharing a Bell pair (|00> + |11>)/sqrt(2), with Alice holding the first qubit and Bob holding the second. Alice wants to send a two-bit classical message — one of {00, 01, 10, 11}. She encodes her message by applying one of four operations to her qubit: I for 00, X for 01, Z for 10, or XZ for 11. Each operation transforms the shared Bell state into a different, orthogonal Bell state.
The four Bell states are: Phi+ = (|00> + |11>)/sqrt(2), Psi+ = (|01> + |10>)/sqrt(2), Phi- = (|00> - |11>)/sqrt(2), Psi- = (|01> - |10>)/sqrt(2). They form an orthonormal basis for the two-qubit Hilbert space. After Alice's encoding, she sends her qubit to Bob. Bob now holds both qubits and performs a Bell measurement — CNOT followed by Hadamard on the first qubit, then computational-basis measurement of both. Because the four Bell states are orthogonal, Bob distinguishes them with certainty and recovers Alice's two-bit message perfectly.
The protocol achieves something classically impossible: sending two bits of information through one quantum channel use. Without entanglement, the Holevo bound limits a single qubit to carrying at most one classical bit of reliable information. The entangled pair provides the extra dimension — Bob already has a qubit that is correlated with Alice's, so when Alice's qubit arrives, Bob has access to the full four-dimensional two-qubit space. The entanglement is consumed: after Bob's measurement, the pair is no longer entangled.
Superdense coding and quantum teleportation are dual protocols with an elegant resource symmetry. Teleportation consumes one entangled pair plus two classical bits to transmit one qubit. Superdense coding consumes one entangled pair plus one qubit to transmit two classical bits. In both cases, entanglement serves as a catalyst that enhances the capacity of the other channel. This duality is one of the foundational results of quantum information theory and motivates the study of entanglement as a quantifiable, fungible resource.