Dense-coding procedure

The dense-coding procedure takes classical input bits $a,b\in \{0,1\}$ and a preshared pair $\mathrm{\mid }{\mathrm{\Phi }}^{+}{⟩}_{AB}$, with Alice holding $A$ and Bob holding $B$. The local convention is that $a$ is the phase bit and $b$ the flip bit.

1. Alice applies $X_A^b$ and then $Z_A^a$, so the encoding operator is $Z_A^a{X}_A^b$.
2. Alice sends the single qubit $A$ to Bob.
3. Bob applies ${\mathrm{CNOT}}_{A\to B}$, then $H_A$, and measures the ordered register $A,B$ in the computational basis.
4. Bob outputs the ordered pair of measured bits.

If a surrounding measurement model assigns an outcome zero weight, no conditional state is assigned for that outcome. For each of the four possible input pairs, the decoded output equals that pair with weight one and the other three outcomes have weight zero, as computed in the correctness theorem.

The Bell resource, Pauli matrices, Hadamard, CNOT wire order, and projective measurement use the conventions of The Bell vectors, The Pauli matrices, Exact one-qubit gates and rotations, CNOT and Control-Target Order, and Finite projective measurements. No optimality claim is part of this procedure.