Bloch sphere

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Summary

In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.

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basis for the coordinates of the Bloch vector
The coordinates of a state on the Bloch sphere are determined by the expectation values of the three Pauli matrices.
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namesake and developer of the geometrical representation
The Bloch sphere is named after this Swiss-American physicist, who developed the mathematical framework for describing the evolution of two-level quantum systems.
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represented by points within the interior of the sphere
While pure states lie on the surface, mixed states are represented by points inside the sphere, with the center representing the maximally mixed state.
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defines the points on the surface of the sphere
Every point on the surface of the Bloch sphere corresponds to a unique pure state of a qubit, representing a state with zero uncertainty.
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the fundamental system represented by the Bloch sphere
The Bloch sphere is the standard geometric tool for visualizing the state space of a single qubit.
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optical analog to the Bloch sphere
The Poincaré sphere is a similar geometric representation used in classical optics to describe the polarization state of light.
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acts as a rotation on the Bloch sphere
Single-qubit quantum gates correspond to rotations of the Bloch vector around specific axes on the sphere.
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corresponds to rotations on the Bloch sphere
Any single-qubit unitary operation preserves the length of the Bloch vector and corresponds to a rotation of the sphere.
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visualized as a vector pointing away from the poles
Superposition states are represented by vectors that do not point directly to the north or south poles of the Bloch sphere.
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mathematical framework for mapping states to the sphere
The Bloch sphere uses spherical coordinates (theta and phi) to parameterize the state vector of a qubit.
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the domain visualized by the Bloch sphere
The Bloch sphere provides a compact geometric representation of the two-dimensional complex Hilbert space of a qubit.
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Facts