Theorem: Four messages in a one-way model

Suppose a sender holds qubits $1,2$, a receiver holds qubit $3$, and only a finite classical message is sent from sender to receiver. There is no extra shared resource. Exact recovery requires at least four possible messages, hence at least two bits in a fixed-length message model. Four messages suffice.

Proof. The preceding programme proves sufficiency. For necessity, let the sender's POVM be $\{E_m\}$ and the receiver's conditional channel be $T_m$. Exactness and Theorem thm:unique imply, on all three-qubit operators,

where $|{\psi }_{pq}⟩⟨{\psi }_{pq}|={\Pi }_{pq}$. Therefore

A unitary channel has rank-one Choi matrix, so a convex decomposition of it into channels contains only that same channel with positive weight. Any message having positive probability for sector $(p,q)$ must therefore implement ${\mathrm{Ad}}_{R_{pq}}$. The four conjugation channels are distinct by (eq:R). Their message supports are disjoint, so there must be at least four messages.