Lecture 5. Single quantum bit

In this Lecture we discussed properties of a simple quantum system – a quantum bit (qubit). A qubit is a quantum two-level system with a chosen (computational) basis $\{|0\rangle,|1\rangle\}$. There are Pauli observables $I,X,Y,Z$ defined on the qubit. The Pauli matrices are used to construct a representation of the qubit states on Bloch ball: pure states lie on the sphere, mixed states lie inside the ball. The reversible operations on the qubit are defined by unitary matrices that act on the Bloch ball as rotations. It turns out that one can approximate any unitary operation using a set of gates $\{H,T\}$. Important cases of irreversible operations are channels describing noise: depolarisation, dephasing, amplitude damping. We will call a measurement of one qubit the process of obtaining one one bit of classical information about a qubit.