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For teleportation with an ambiguous Bell-state phase label, write $\rho =|\psi ⟩⟨\psi |$. The Bell-basis measurement outcomes ${\varphi }^{+}$ and ${\varphi }^{-}$ each occur with probability $1/4$. In either case a correct guess gives Bob $\rho$, while the wrong guess differs by the Pauli Z gate and gives $Z\rho Z$. The other two Bell states are identified correctly. Since Alice guesses each ambiguous label with probability $1/2$, the overall wrong-correction probability is $(1/2)(1/2)=1/4$. Averaging over the actual quantum measurement outcome and her guess gives the phase-flip channelConditional only on an ambiguous analyzer result, the output is $(\rho +Z\rho Z)/2$; that conditional channel has weight $1/2$ in the displayed unconditional result. There is no rejection of those rounds.

Use the teleportation fidelity for a pure state target, namely the squared quantum fidelityThis compares the input with the resulting mixed state; averaging state vectors before forming a density matrix would give a different, incorrect answer. Direct substitution givesThus the maximum is $1$, attained by $|0⟩$ and $|1⟩$ up to a global phase, because these are eigenstates of the Pauli Z gate. The minimum is $3/4$, attained by every equal-weight quantum superpositionon the equator of the Bloch sphere. The unsquared quantum fidelity is $\sqrt{F_{\psi }}$ if that alternative convention is used. No average over different input quantum states is needed for the requested state-dependent answer.

In quantum teleportation, Alice transfers an arbitrary unknown qubit $|\chi ⟩=\alpha |0⟩+\beta |1⟩$ using one shared Bell pair and two bits of classical communication. Define $|{\beta }_{mn}⟩=(I\otimes X^n{Z}^m)|{\Phi }^{+}⟩$, with $m,n\in \{0,1\}$. Regrouping the initial three-qubit state into this Bell basis givesAlice's Bell-basis measurement selects each label with probability $1/4$. She sends $(m,n)$ to Bob, who applies the inverse $Z^m{X}^n$, restoring $|\chi ⟩$. The correction depends on the measurement label, not on the unknown amplitudes. Before the message, Bob's average state is $I/2$, independent of the input; quantum no-signalling therefore prevents superluminal communication. The shared pair is consumed and Alice does not retain an independent copy, in agreement with the no-cloning theorem.

The Bouwmeester quantum teleportation experiment used photon polarization, a down-conversion entangled pair, a beam-splitter Bell-state analyzer and coincidence-conditioned output polarization measurements. Temporal indistinguishability of the two analyzer photons was essential. Its successful selected Bell outcome demonstrated the state transfer, rather than a deterministic implementation of all four outcomes.^[ref]

The general experimental tasks are to distribute entanglement, prepare and synchronize the input, perform a sufficiently discriminating joint measurement, communicate the label and apply or account for the correction. A demonstration must test nonorthogonal input states, not merely transfer a known classical basis label. Finite efficiency, accidental coincidences, distinguishability and phase drift reduce success probability or teleportation fidelity. Postselected verification, a heralded usable output and unconditional deterministic teleportation are distinct experimental claims.

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.