One qubit, handed from Alice to Bob, can carry at most one classical bit of freshly chosen information — that is a hard law (Holevo's bound). So how could Alice possibly cram two classical bits into a single qubit she sends? The trick, called superdense coding, is that Alice and Bob prepared for this in advance: they share an entangled Bell pair. With that shared resource already in hand, one physical qubit travelling across the channel delivers a full two-bit message. It is the exact mirror image of quantum teleportation, where two classical bits move one qubit.

The magic ingredient is that Alice can steer the joint state of the pair by touching only her own half. A gentle nudge to her qubit rotates the shared Bell state into a different Bell state — and because the four Bell states are perfectly distinguishable, Bob can read back exactly which nudge she chose.

The setup: a pre-shared Bell pair

Before any message exists, someone prepares the Bell state |\Phi^+\rangle and gives qubit A to Alice and qubit B to Bob, who may then walk to opposite ends of the world:

Later Alice decides on a two-bit message — one of 00, 01, 10, 11 — and applies exactly one of the four Pauli gates to her qubit. Each choice rotates the shared pair into a different Bell state. She then physically sends her single qubit to Bob. Now Bob holds both halves, measures them together in the Bell basis, and recovers both bits. One qubit crossed the channel; two bits arrived.

The four-way dictionary

Applying a Pauli to Alice's half of |\Phi^+\rangle lands the pair on one of the four Bell states. Reveal the rows one at a time — this table is the whole codebook:

Read the states off with the identities (I\otimes I)|\Phi^+\rangle = |\Phi^+\rangle, (Z\otimes I)|\Phi^+\rangle = |\Phi^-\rangle, (X\otimes I)|\Phi^+\rangle = |\Psi^+\rangle, and (ZX\otimes I)|\Phi^+\rangle = -|\Psi^-\rangle. Four inputs, four orthogonal outputs — that orthogonality is exactly why Bob can never confuse them.

The circuit end to end

The whole protocol is three moves on two wires. Alice owns the top wire, Bob the bottom. After Alice's Pauli, her qubit crosses to Bob (the dashed hand-off); Bob then runs the Bell measurement — a CNOT then a Hadamard, the reverse of how a Bell pair is built — and measures both wires to read the two bits. Step through it:

A Bell measurement is just "un-building" the entanglement: the CNOT copies the correlation back onto the control wire and the Hadamard collapses the superposition, so the two qubits come out as an ordinary two-bit number — precisely the message Alice encoded.

Worked example: Alice encodes “10”

Say Alice wants to send the bits 10. The table tells her to apply X to her qubit. Acting on the first slot of |\Phi^+\rangle, the bit-flip swaps 0\leftrightarrow 1 in each term:

The shared pair is now |\Psi^+\rangle. Alice ships her single qubit to Bob, who holds both halves and runs his decoder. Recall |\Psi^+\rangle came from the computational input |10\rangle under "Hadamard then CNOT," so running that circuit backwards (CNOT, then Hadamard on the top wire) returns |10\rangle:

Bob measures and reads 1 then 0 with certainty — Alice's two bits, delivered by one qubit. Every other message decodes the same way: the Bell state is the message.

Worked example: why “01” needs Z

For the message 01 the table says apply Z. The phase-flip leaves |0\rangle alone and negates |1\rangle, so acting on Alice's slot:

Notice Alice changed nothing about which outcomes are possible — |\Phi^+\rangle and |\Phi^-\rangle both give 00 or 11 on a plain joint measurement. All she altered is the relative phase, the invisible-looking minus sign. Yet a full Bell measurement (which interferes the qubits) sees that phase perfectly and pulls out 01. This is why superdense coding needs the Bell measurement, not two ordinary ones: half the message lives in a phase that only interference can reveal.

The mirror of teleportation

Superdense coding and teleportation are two readings of the same resource — a shared Bell pair — run in opposite directions:

• Teleportation: spend 1 shared Bell pair and send 2 classical bits to move 1 unknown qubit of state.
• Superdense coding: spend 1 shared Bell pair and send 1 qubit to move 2 classical bits.

Read one line, then swap "qubit" with "two classical bits" and reverse the arrow, and you get the other. Entanglement is the currency; the two protocols just exchange it for different things.