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No-cloning theorem for two pure states

No unitary operator

U

can satisfy

U ∣ c_{j} ⟩ ∣0 ⟩ = ∣ c_{j} ⟩ ∣ c_{j} ⟩, j = 0, 1,

for two distinct nonorthogonal pure states. Taking the inner product of these two equations would require

⟨ c_{0} ∣ c_{1} ⟩ = ⟨ c_{0} ∣ c_{1} ⟩^{2},

which is impossible when

0 < ∣ ⟨ c_{0} ∣ c_{1} ⟩ ∣ < 1

.

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