7.3 The complex math and physics of a single qubit

Let’s revisit our definition of a qubit from section 7.1

Mathematics

A qubit—a quantum bit—is the fundamental unit of quantum information. At any given time, it is in a superposition state represented by a linear combination of vectors |0⟩ and |1⟩ in C²:

a |0⟩ + b |1⟩ where |a|² + |b|² = 1 .

Physics

Through measurement, a qubit is forced to collapse irreversibly to either |0⟩ or |1⟩. The probability of its doing either is |a|² and |b|², respectively. a and b are called probability amplitudes.

The mathematical portion is linear algebra of a two-dimensional complex vector space. As a vector, the qubit state has length 1. Linear transformations must preserve this length and are isometries. Their matrices are unitary. Being unitary, they are invertible: moving a qubit from...