Unitary

$U^{†} = U^{- 1}$ .

They are often called isometries (symmetries of the system).

Theorem

An operator $U$ is unitary iff it preserves the inner product of all vectors. I.e. if we define $| V ⟩ = U | v ⟩$ and $| W ⟩ = U | w ⟩$ then $(| v ⟩, | w ⟩) = (| V ⟩, | W ⟩)$ .

In short, it maps orthonormal sets to orthonormal sets.

Proof. Assume $U$ is unitary. Then $⟨ V | W ⟩ = ⟨ U v | U w ⟩ = ⟨ v | U^{†} U | w ⟩ = ⟨ v | I | w ⟩ = ⟨ v | w | r a n g l e$ .

Assume $(| V ⟩, | W ⟩) = (| v ⟩, | w ⟩)$ . Then $⟨ U v | U w ⟩ = ⟨ v | U^{†} U | w ⟩ = ⟨ v | w ⟩$ . So, $U^{†} U = I$ . Then, $U^{†} U = I$ . Also, $⟨ j | i ⟩ = ⟨ U j | U i ⟩ = ⟨ j | U^{†} U | i ⟩ = ⟨ j | i ⟩ = δ_{j}^{i} = δ_{i}^{j} = ⟨ i | j ⟩ = ⟨ j | i ⟩^{*} = ⟨ j | (U^{†} U)^{†} | i ⟩^{*}$ . Thus, $U^{†} = U^{- 1}$ . Hence, $U$ is unitary.

Examples

Explicit example

$U = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ i & - i \end{matrix}), U U^{†} = U^{†} U = I$ .

Unitary

Theorem

Examples

Explicit example

Rotation operators