The Bloch Sphere and Mixed States

The Bloch ball represents every single-qubit state, with pure states on the surface and mixed states inside.

One picture for all qubit states

Any single-qubit density matrix can be written rho = (I + r . sigma)/2, where r = (r_x, r_y, r_z) is the Bloch vector and sigma is the vector of Pauli matrices. The components are the Pauli expectation values, r_i = Tr(rho sigma_i). Validity requires |r| <= 1, so all qubit states fill a solid ball. This is the most useful geometric picture in single-qubit quantum information.

Surface versus interior

The length of the Bloch vector encodes purity: |r| = 1 on the surface for pure states, |r| < 1 inside for mixed states, and r = 0 at the center for the maximally mixed state I/2. Purity relates directly as Tr(rho^2) = (1 + |r|^2)/2. Antipodal points on the surface are orthogonal states, so |0> and |1> sit at the poles, |+> and |-> on the equator.

Reading off: the diagonal gives populations, and the off-diagonal coherences encode r_x and r_y, the equatorial components that carry relative phase.

Dynamics as geometry

Unitary gates rotate the Bloch vector rigidly about an axis, preserving its length, which is why closed-system evolution keeps a state pure. Noise channels move points inward: depolarizing shrinks the ball uniformly toward the center, dephasing flattens it onto the z-axis, and amplitude damping both shrinks and shifts it toward the north pole. Decoherence is literally the Bloch vector losing length.

Limits of the picture

The Bloch sphere is exact and complete for one qubit but does not extend cleanly to more. Two qubits already require a fifteen-dimensional generalized Bloch representation, and entanglement has no simple visualization there. The intuition it builds, states as points, gates as rotations, noise as contraction, remains valuable, but multi-qubit reasoning must return to density matrices and the algebraic tools.