 Local flag correction of a Bell-state phase mixture

ID: local-flag-correction-of-a-bell-state-phase-mixture

Suppose Alice holds a classical orthogonal flag

F

for the phase of a Bell state:

ρ_{A BF} = p ∣ Φ^{+} ⟩ ⟨ Φ^{+} ∣_{A B} \otimes ∣0 ⟩ ⟨ 0 ∣_{F} + (1 - p) ∣ Φ^{-} ⟩ ⟨ Φ^{-} ∣_{A B} \otimes ∣1 ⟩ ⟨ 1 ∣_{F} .

Her local controlled phase correction

U = I_{A B} \otimes ∣0 ⟩ ⟨ 0 ∣_{F} + (Z_{A} \otimes I_{B}) \otimes ∣1 ⟩ ⟨ 1 ∣_{F}

maps this to

∣ Φ^{+} ⟩ ⟨ Φ^{+} ∣_{A B} \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

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