State Space (Quantum Computing)

quantum-computing

Definition

State Space (Quantum Computing)

The state space of a quantum system is a complex Hilbert space $\mathcal{H}$. The dimension of $\mathcal{H}$ is often finite (e.g., $\mathcal{H}={\mathbb{C}}^2$ for a single qubit), though it may be infinite for continuous systems. The physical state of the system is represented by a unit vector $|\psi ⟩\in \mathcal{H}$, satisfying $⟨\psi |\psi ⟩=1$.

Key Properties:

1. Linearity: The state space is a vector space, so superpositions $\alpha |{\psi }_1⟩+\beta |{\psi }_2⟩$ are valid states (when normalised).
2. Global Phase: Vectors differing only by a global phase $e^{i\theta }|\psi ⟩$ represent the same physical state.
3. Closed systems evolve via unitary transformations: $|{\psi }^{\prime }⟩=U|\psi ⟩$.
4. Composite Systems: The state space of a composite system is the tensor product of the component state spaces: ${\mathcal{H}}_{AB}={\mathcal{H}}_A\otimes {\mathcal{H}}_B$.

Example

Single Qubit: The state space is ${\mathbb{C}}^2$, with basis $\{|0⟩,|1⟩\}$. A general state is

$$|\psi ⟩=\alpha |0⟩+\beta |1⟩$$

with $|\alpha {|}^2+|\beta {|}^2=1$.