Bloch Sphere#
The Bloch sphere is a geometric representation of a single qubit. It turns a normalized two-dimensional complex state vector into a point on the unit sphere in ordinary three-dimensional space.
The representation is historically connected to Felix Bloch’s spin-vector description of two-level quantum systems [Bloch 1946]. In quantum information, it is now a standard way to connect qubit amplitudes, measurement probabilities, and single-qubit gates [Nielsen and Chuang 2010].
Qubit State Vector#
A pure qubit state can be written as
where α and β are complex amplitudes and
The two basis states |0> and |1> form a basis for a two-dimensional complex vector space. This is the linear algebra object. The Bloch sphere is a coordinate picture of that object after removing two redundancies: normalization and global phase.
Bloch Coordinates#
Every pure qubit state can be represented as
where θ is the polar angle and φ is the azimuth angle. These are the same angular ideas used in spherical coordinates, but the half-angle θ / 2 appears because quantum amplitudes square to probabilities [Nielsen and Chuang 2010].
The corresponding Bloch vector is
This vector has length 1 for a pure state.
Reading the Sphere#
The north pole is |0>, the south pole is |1>, and points on the equator are equal-probability superpositions. The azimuth phi controls relative phase between |0> and |1>.
This is why the Bloch sphere is so useful pedagogically: it connects complex linear algebra with a real three-dimensional picture. The state remains a vector in complex Hilbert space, but the sphere shows how probability and phase move together.
This page focuses on pure states on the sphere surface. Mixed single-qubit states are represented inside the Bloch ball through the density-matrix and Pauli-vector formalism [Fano 1957].
Camera Rotation#
The interactive sphere separates the quantum state from the camera. Changing theta and phi changes the qubit state. Changing camera yaw and camera pitch only rotates the view of the sphere, axes, and state vector. This distinction mirrors the coordinate idea from the previous note: changing the representation should not be confused with changing the object being represented.
Linear Algebra Interpretation#
Single-qubit gates are linear operators acting on the state vector. On the Bloch sphere, many common gates appear as rotations:
X flips north and south by rotating around the x-axis.
Y rotates around the y-axis.
Z changes relative phase by rotating around the z-axis.
Hadamard moves between computational and superposition axes.
The visual rotation is not the full operator algebra, but it preserves the key idea: quantum dynamics are linear transformations of vectors, and the Bloch sphere is a compact geometric shadow of those transformations.
The operation visual compares the current Bloch vector with the vector after applying one basic gate. X, Y, and Z are 180-degree rotations around their named axes. H is displayed as a 180-degree rotation around the diagonal x+z axis: x and z exchange roles while y changes sign.
Connection Back to Coordinates#
The previous page, QA: Coordinates, introduced Cartesian coordinates, spherical coordinates, and local bases. The Bloch sphere uses the same coordinate map on the unit sphere, but interprets the point as a normalized quantum state.
References#
[Bloch 1946] Felix Bloch. “Nuclear Induction.” Physical Review 70, 460-474. DOI: 10.1103/PhysRev.70.460.
[Fano 1957] Ugo Fano. “Description of States in Quantum Mechanics by Density Matrix and Operator Techniques.” Reviews of Modern Physics 29, 74-93. DOI: 10.1103/RevModPhys.29.74.
[Nielsen and Chuang 2010] Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information, 10th anniversary edition. Cambridge University Press. Publisher page.