This Demonstration neatly visualizes the common quantum information processing operations on single qubits.

The most general single qubit state has the spinor form , where and are spherical polar coordinates with and .

The states and correspond to the north and south poles of the sphere with , , . They are referred to as the (computational) basis states and are eigenvectors of the Pauli matrix with eigenvalues .

Two other special cases are the qubit state and with , and , ; they are eigenvectors of and form a diagonal basis. The Hadamard gate interchanges computational and diagonal bases.

The eigenvectors of are sometimes called the circular basis; they are and with , and , .

Altogether they depict six important points on the Poincaré sphere.