Quantum State

quantum-computing

Definition

Quantum State

A quantum state is an element of the state space of a quantum system. Mathematically, it is represented by a unit vector $|\psi ⟩$ in a complex Hilbert space $\mathcal{H}$, satisfying

$$⟨\psi |\psi ⟩=1.$$

Two state vectors differing only by a global phase $e^{i\theta }$ represent the same physical state: $|\psi ⟩\equiv e^{i\theta }|\psi ⟩$.

Single Qubit State

For a single qubit, the state space is $\mathcal{H}={\mathbb{C}}^2$ with computational basis $\{|0⟩,|1⟩\}$. A general quantum state is

$$|\psi ⟩=\alpha |0⟩+\beta |1⟩$$

where $\alpha ,\beta \in \mathbb{C}$ and $|\alpha {|}^2+|\beta {|}^2=1$.

Pure vs. Mixed States

Pure State

A quantum state that can be described by a single state vector $|\psi ⟩$ (as defined above).

Mixed State

A statistical ensemble of pure states, described by a density operator $\rho$ rather than a single state vector.

State Evolution

For closed systems, quantum states evolve via unitary transformations:

$$|\psi (t)⟩=U(t)|\psi (0)⟩.$$