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Unitary

A linear operator $U$ that preserves inner products; equivalently $U^\dagger U = I$.

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Unitary operators are the mathematical model for reversible quantum evolution.

Key property

For any state ∣ψ⟩|\psi\rangle, applying a unitary keeps it normalized:

⟨ψ∣ψ⟩=1⇒⟨ψ∣U†U∣ψ⟩=1.\langle \psi|\psi\rangle = 1 \Rightarrow \langle \psi|U^\dagger U|\psi\rangle = 1.
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