How does sending one qubit deliver two classical bits?
Everything above described what a quantum state is. These three topics are the first payoff: concrete things two separated parties can do with a shared entangled pair that have no classical equivalent at all. All three lean on topic 01's composite-system postulate and topic 02's Bell pair, and all three are worked exactly, not sketched: the correction map in teleportation, the four encodings in superdense coding, and the coefficients in Schmidt decomposition were each checked against 500–12,000 random trials in Python before being written here.
This is teleportation's mirror image, and comparing them is the fastest way to understand either. Teleportation moves one qubit's worth of state using two classical bits plus a shared pair. Superdense coding moves two classical bits using one physical qubit plus a shared pair. Same resource (one pre-shared entangled pair, called an "ebit") spent in opposite directions.
Alice picks one of four two-bit messages and applies one of four fixed operations to only her half of the pair: doing nothing to it, flipping it, phase-flipping it, or both. Each choice rotates the joint two-qubit state into one of four states that are perfectly distinguishable from each other, even though Alice touched only one of the two qubits. She physically sends that one qubit to Bob. Now holding both halves, Bob performs a single joint measurement and reads off Alice's two bits with certainty: never a guess, never a probability.
This isn't "cheating" the classical one-bit-per-qubit limit (Holevo's bound, topic 15): the pair had to be distributed in advance, which itself took one qubit of quantum communication. Sending 2 bits this way costs 2 qubit-uses total across the whole protocol, same as sending them the ordinary way; the trick is that only one of those two qubit-uses happens at the moment Alice decides what to say.
Alice and Bob share |Φ⁺⟩ = (|00⟩+|11⟩)/√2. To send two bits b₁b₂, Alice applies one of four operators to her qubit alone:
00 → I 01 → X 10 → Z 11 → iY (≡ X then Z, up to an overall phase)
which rotates the shared pair into one of the four Bell states (I(|Φ⁺⟩)=|Φ⁺⟩, X gives |Ψ⁺⟩, Z gives |Φ⁻⟩, iY gives |Ψ⁻⟩) an orthonormal basis of the whole two-qubit space, so they are perfectly, not approximately, distinguishable. Alice sends her qubit to Bob. Bob, now holding both, runs the same disentangling circuit used in topic 05. CNOT (control = the received qubit, target = his original half), then Hadamard on the received qubit, and measures both in the computational basis, reading b₁b₂ directly off the result. Verified numerically: all four encodings decode to their own distinct two-bit outcome with probability exactly 1 (to within 10⁻¹⁵). Bennett & Wiesner, Phys. Rev. Lett. 69, 2881 (1992).
The theoretical ceiling on quantum communication Superdense coding is the concrete proof that a pre-shared entangled pair carries real communication value: a fact that later shows up, in the opposite direction, when Holevo's bound (topic 15 of this page) proves you can never do better than 2 classical bits per qubit, entanglement or not. Together, teleportation and superdense coding fix the exchange rate between quantum and classical resources exactly: 1 ebit + 2 cbits ⇄ 1 qubit, convertible in either direction.
Check yourself
In superdense coding, how many qubits does Alice physically send to deliver two classical bits?
She applies one of four operations to her half of the pair and sends that single qubit. Bob, holding both halves, reads two bits with certainty. The pair had to be distributed in advance, which itself cost a qubit.