The Bloch Vector and the Density Matrix

Any single-qubit density matrix maps to a point in the Bloch ball via its Bloch vector, with pure states on the surface.

Density matrices as points

Every single-qubit density matrix can be written as rho = (I + r_x X + r_y Y + r_z Z)/2, where (r_x, r_y, r_z) is the Bloch vector and X, Y, Z are the Pauli matrices. This turns the abstract matrix into a point in three-dimensional space.

Ball, not just sphere

The Bloch vector has length at most one. Points on the surface (|r| = 1) are pure states — the Bloch sphere proper. Points inside (|r| < 1) are mixed states. The centre (r = 0) is the maximally mixed state I/2. So the full state space of a qubit is the solid ball, and the sphere is only its pure-state boundary.

Reading the components

The Bloch vector components are the expectation values of the Paulis: r_x = , r_y = , r_z = . Each is a number between -1 and 1 measurable by repeated experiments. The z component relates to computational-basis populations; the x and y components encode the coherences.

Purity and length

The purity is directly the length: Tr(rho^2) = (1 + |r|^2)/2. Pure states give |r| = 1 and purity 1; the maximally mixed state gives |r| = 0 and purity 1/2. Shrinking the Bloch vector toward the origin is exactly what decoherence does.

Dynamics as motion

Unitary gates rotate the Bloch vector rigidly about an axis, preserving its length. Noise processes shrink or displace it: energy relaxation pulls it toward the north pole, dephasing collapses it toward the z axis. Watching the Bloch vector move is a compact way to visualise both computation and noise on a single qubit.