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If basis information and test positions are announced before the receiver confirms receipt of quantum carriers, an eavesdropper can store the carriers until learning that information. In a scheme sending one half of each Bell pair after a Hadamard gate encoding, the eavesdropper then forwards test qubits untouched and measures all data qubits in their now-known correct bases. Every test passes, while data are changed to classically correlated product states whose bit labels the eavesdropper knows. This invalidates the sampling argument needed to justify entanglement purification and quantum key distribution. A sound additional purification certification could detect this attack and abort; it cannot generate private Bell pairs from the separable attacked data. Announcing bases only after receipt prevents this particular delayed-carrier attack.

For $n$ copies of a normalized pure state $\alpha |00⟩+\beta |11⟩$, regroup Alice's and Bob's tensor factors. Alice performs a projective measurement of the total number $k$ of zeroes, preserving all coherence among strings with that count. The outcome has binomial distributionEach retained string has the same complex amplitude ${\alpha }^k{\beta }^{n-k}$, so, up to a common phase, the conditional quantum state isIts equal Schmidt coefficients make it a maximally entangled state of Schmidt rank $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$. Alice sends the count to Bob. They can relabel the corresponding local computational basis strings with local unitary operations, and convert the resulting resource to Bell pairs. Measuring the individual bits instead would reveal the entire string and leave a product state.

For one pure rank-two entangled state with descending squared Schmidt coefficients $({\lambda }_0,{\lambda }_1)$, the success operator $F=\mathrm{diag}(\sqrt{{\lambda }_1/{\lambda }_0},1)$ gives a Bell pair with probability $2{\lambda }_1$. The complementary operator $\sqrt{I-F^{†}F}$ makes a valid quantum instrument. Optimality follows from the Schmidt-tail entanglement monotone: a successful Bell branch has smallest weight $1/2$, so its probability cannot exceed $2{\lambda }_1$. Failure branches have nonnegative monotone and cannot loosen this bound.

Superdense coding uses a shared Bell state to communicate two classical bits by transmitting one qubit. The sender selects one of four orthogonal Bell states with a local Pauli operator, and the receiver performs a Bell-basis measurement. For an ideal shared Bell pair, the transmitted qubit alone has a maximally mixed state for every message and reveals no information about the chosen encoding.

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.

Let $c,s\ge 0$ and $c^2+s^2=1$. Starting from the Bell state $(|00⟩+|11⟩)/\sqrt{2}$, Alice applies the measurement in quantum mechanics with Kraus operators $M_0,M_1$ displayed above. Completeness follows from $M_0^{†}{M}_0+M_1^{†}{M}_1=I$. Each outcome has probability $1/2$. The normalized outputs are $c|00⟩+s|11⟩$ and $s|00⟩+c|11⟩$. Communicating the outcome allows both parties to apply a Pauli X gate in the second branch, making both outputs equal to the first. Thus this LOCC protocol deterministically converts one Bell pair into any two-qubit pure state with these Schmidt coefficients, after appropriate local unitary operations. It includes the product state endpoint without postselection.