Bloch Sphere

Nuclear Induction

Editorial overview

Bloch's 1946 article develops a theory of nuclear induction, connecting magnetic polarization in matter with a voltage that can be detected in a coil. It describes how an applied radiofrequency field drives the nuclear magnetization and how longitudinal and transverse relaxation modify its evolution. The central quantity is a macroscopic polarization vector, defined as nuclear magnetic moment per unit volume. The paper treats its motion using vector equations and examines signal behavior near resonance. Original article (opens in a new tab)

The gallery associates this historical reference with the Bloch sphere, but the two should be distinguished. The article does not depict the modern unit sphere for qubit states. Its two figures show induced-voltage response during rapid and slow passage through resonance, on printed pages 470 and 471. The modern representation described below is supplied as explanatory context, not attributed to those figures. Original PDF (opens in a new tab)

Visualization mechanism

The modern Bloch sphere maps a pure qubit state, after discarding global phase, to a point on a unit sphere. A conventional parameterization is

$$|\psi\rangle=\cos(\theta/2)|0\rangle+e^{i\phi}\sin(\theta/2)|1\rangle.$$

The corresponding Cartesian coordinates are

$$(x,y,z)=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta).$$

The polar angle controls the relative magnitudes of the two amplitudes, while the azimuthal angle records their relative phase when both are nonzero. The states $|0\rangle$ and $|1\rangle$ occupy opposite poles. Equal-weight superpositions lie on the equator, with different relative phases occupying different directions. Mixed states occupy the interior of the Bloch ball, and the maximally mixed state lies at its center. IBM Quantum Learning (opens in a new tab)

What users learn

The sphere connects algebraic descriptions with spatial relationships. Orthogonal pure qubit states appear at opposite ends of a diameter, while averaging state density matrices corresponds to averaging their Bloch coordinates. It also distinguishes a coherent equal-weight superposition on the surface from an equal mixture at the center. Global phase is absent from the drawing because it does not distinguish qubit states. IBM Quantum Learning (opens in a new tab)

For a register, a separate Bloch sphere can display each qubit's local Pauli expectation values. These views describe individual qubits but omit correlations between them. Consequently, a collection of such spheres does not fully represent an entangled joint state; additional information is needed to distinguish joint states with identical local descriptions. IBM state-visualization guide (opens in a new tab)

Scope and sources

This is an editorial summary, not the paper's abstract. The historical account uses the 15-page publisher PDF, particularly its polarization equations and both figures. The modern geometric account uses the separately linked IBM materials. This entry does not establish when the sphere drawing was first introduced, attribute a quantum-computing interface to the 1946 paper, or claim measured educational benefits.