Local flag correction of a Bell-state phase mixture

Suppose Alice holds a classical orthogonal flag $F$ for the phase of a Bell state:Her local controlled phase correction $U=I_{AB}\otimes |0⟩⟨0{|}_F+(Z_A\otimes I_B)\otimes |1⟩⟨1{|}_F$ maps this to $|{\Phi }^{+}⟩⟨{\Phi }^{+}{|}_{AB}\otimes [p|0⟩⟨0|+(1-p)|1⟩⟨1|{]}_F$. 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 $p=1/2$.