# Qubits on the Poincaré (Bloch) Sphere

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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.

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Contributed by: Rudolf Muradian (March 2011)

Open content licensed under CC BY-NC-SA

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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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