State Spaces

Comment

Classical computer science often considers systems to have a finite set of states. An important simple system is the bit, with state space given by the set . Quantum information theory instead assumes that systems have state spaces given by finite-dimensional Hilbert spaces. The quantum version of the bit is the qubit.

Definition

Def Qubit A qubit is a quantum system with state space .

Definition

Def Pure State A pure state of a quantum system is given by a vector in its associated Hilbert space. Such a state is normalized when the vector in the Hilbert space has norm :In particular, a complex number of norm is called a phase. A pure state of a qubit is therefore a vector of the form with , which is normalized when .

Attention

We will encounter a more general notion of state, called a mixed state. However, when our meaning is clear, we’ll often just say state instead of pure state.

Def Z Basis For the Hilbert space , the computational basis, or Z basis is the orthonormal basis given by the following vectors:

Remark

This orthonormal basis is no better than any other, but it is useful to fix a standard choice. Every state can be written in terms of the computational basis; for a qubit, we can write for some .

Def X Basis The X basis for a qubit is given by the following states: