= Local flag correction of a Bell-state phase mixture
Suppose Alice holds a classical orthogonal flag $F$ for the phase of a <Bell state>:
$$
\rho_{ABF}=p|\Phi^+\rangle\langle\Phi^+|_{AB}\otimes|0\rangle\langle0|_F+(1-p)|\Phi^-\rangle\langle\Phi^-|_{AB}\otimes|1\rangle\langle1|_F.
$$
Her local controlled phase correction $U=I_{AB}\otimes|0\rangle\langle0|_F+(Z_A\otimes I_B)\otimes|1\rangle\langle1|_F$ maps this to $|\Phi^+\rangle\langle\Phi^+|_{AB}\otimes[p|0\rangle\langle0|+(1-p)|1\rangle\langle1|]_F$. Thus Alice and Bob recover an exact <Bell pair> without revealing its bit value. Even if Eve knows the flag, <purity decouples a subsystem from its purification>, so the corrected pair can supply a private bit. Discarding the flag first can instead yield a <separable quantum state>, as happens at $p=1/2$.
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