Introduction: Transformations That Preserve Length
A unitary matrix is the complex generalization of a rotation—it preserves inner products and therefore preserves lengths and angles. In quantum mechanics, unitary transformations describe reversible evolution where total probability stays equal to 1.
Definition
A matrix \(U\) is unitary if:
\[U^\dagger U = U U^\dagger = I\]Equivalently, \(U^{-1} = U^\dagger\).
For real matrices, "unitary" becomes "orthogonal": \(O^T O = I\).
Key Properties
- Preserves inner products: \(\langle U\vec{u}, U\vec{v} \rangle = \langle \vec{u}, \vec{v} \rangle\)
- Preserves norms: \(\|U\vec{v}\| = \|\vec{v}\|\)
- Determinant has unit modulus: \(|\det(U)| = 1\)
- Eigenvalues lie on unit circle: \(|\lambda| = 1\)
- Columns form orthonormal set
Worked Examples
Example 1: Rotation Matrix is Orthogonal
\[R = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\] \[R^T R = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \checkmark\]Example 2: A 2×2 Unitary Matrix
\[U = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ i & -i \end{pmatrix}\]Check: \(U^\dagger = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & -i \\ 1 & i \end{pmatrix}\)
\[U^\dagger U = \frac{1}{2} \begin{pmatrix} 1 + 1 & -i + i \\ 1 - 1 & -i - i \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \checkmark\]Example 3: The Hadamard Gate
In quantum computing, the Hadamard gate is:
\[H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\]This is both unitary and Hermitian (\(H = H^\dagger\)).
The Quantum Connection
The time evolution operator \(U(t) = e^{-iHt/\hbar}\) is unitary because the Hamiltonian \(H\) is Hermitian. This guarantees that \(\langle\psi|\psi\rangle = 1\) is preserved—total probability remains 1 as the system evolves. Quantum gates in quantum computing are all unitary matrices.