Pure State

A quantum state with no classical uncertainty. The most complete description of a quantum system.

A pure state represents a quantum system about which we have maximal knowledge. All uncertainty in a pure state is fundamentally quantum mechanical (superposition), not classical.

Definition

A pure state can be written as a single state vector:

$∣ ψ ⟩ = i \sum α_{i} ∣ i ⟩$

Or equivalently, as a rank-1 density matrix:

$ρ = ∣ ψ ⟩ ⟨ ψ ∣$

Examples

$∣0 ⟩$ : a definite computational basis state
$∣ + ⟩ = \frac{1}{2} (∣0 ⟩ + ∣1 ⟩)$ : a superposition, but still pure
Bell states: entangled but pure

Characteristics

Property	Value for Pure States
Purity $Tr (ρ^{2})$	1
Von Neumann entropy	0
Bloch sphere (1 qubit)	On the surface
Density matrix rank	1

Pure ≠ Definite

A common misconception: pure state does not mean the measurement outcome is certain.

$∣ + ⟩$ is a pure state, but measuring it in the computational basis gives 0 or 1 with 50% probability each. The state is “pure” because there’s no additional classical uncertainty. We know the exact quantum state.

When Pure States Become Mixed

Pure states can become mixed states through:

Decoherence: Interaction with the environment
Partial trace: Looking at only part of an entangled system
Incomplete knowledge: Not knowing which state was prepared

See also: Mixed State, Density Matrix, Quantum State