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Qubits on the Poincaré (Bloch) Sphere

The Poincaré (Bloch) sphere provides a geometric representation of a pure qubit (quantum bit) state space as points on the surface of the unit sphere . Any point of the surface represents some pure qubit. The mixed qubit states can be represented by points inside of the unit sphere, with the maximally mixed state laying at the center. The red line from the center to the surface of the sphere corresponds to the pure state and has unit length. For mixed qubit state the length of line must be less than 1.
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.

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DETAILS

The Pauli spin operators are defined as , , . The three directions , , and correspond to the diagonal, circular, and computational bases. The most general qubit state is an eigenvector of the operator with eigenvalue 1. The Bloch vector is a unit vector connecting the origin to a point with Cartesian coordinates (, , ).
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