Q6.5.1The unitary property of the Pauli matrices
Demonstrate that the three Pauli matrices given in below are unitary. That is, that \(\sigma_p^\dagger\sigma_p = I_2\) for \(p = x,y,z\), where \(I_2\) is the \(2\times 2\) identity matrix and \(\dagger\) denotes the Hermitian conjugate (conjugate transpose).
\[\begin{align*}
\sigma_x = \left( \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \quad
\sigma_y = \left( \begin{array}{rr} 0 & -i \\ i & 0 \end{array}\right), \quad
\sigma_z = \left( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right).
\end{align*}\]
Show solution
Without over-complicating things,
In [x]: pauli_matrices = np.array((
((0, 1), (1, 0)),
((0, -1j), (1j, 0)),
((1, 0), (0, -1))
))
In [x]: I2 = np.eye(2)
In [x]: for sigma in pauli_matrices:
...: print(np.allclose(sigma.T.conj().dot(sigma), I2))
True
True
True