Teleportation with an ambiguous Bell-state phase label ($\mathcal{E}(\rho )=\frac{3}{4}\rho +\frac{1}{4}Z\rho Z$)

Suppose a perfect $|{\varphi }^{+}⟩$ Bell pair is used for quantum teleportation, but the sender cannot distinguish the ${\varphi }^{+}$ and ${\varphi }^{-}$ Bell-basis measurement outcomes and guesses their labels uniformly. Each of the four actual outcomes has probability $1/4$. Either ambiguous outcome gives an incorrect Pauli Z gate correction with conditional probability $1/2$, so the total phase-flip probability is $1/4$. The unconditional output is therefore the displayed phase-flip channel. For $|\psi ⟩=\alpha |0⟩+\beta |1⟩$, its teleportation fidelity is $⟨\psi |\mathcal{E}(|\psi ⟩⟨\psi |)|\psi ⟩=1-|\alpha {|}^2|\beta {|}^2$. It ranges from $3/4$ on the Bloch sphere equator to $1$ at its poles. This averages the output density operators, including the ambiguous rounds.