Quantum State

The complete mathematical description of a quantum system, containing all information needed to predict measurement outcomes.

A quantum state is the mathematical object that fully describes a quantum system. It encodes everything you can know about the system and determines the probabilities of all possible measurement outcomes.

Pure States

The simplest quantum states are pure states, represented as vectors in Hilbert space using bra-ket notation:

$$|\psi ⟩=\alpha |0⟩+\beta |1⟩$$

A pure state represents a system with no classical uncertainty. Any uncertainty is purely quantum mechanical (superposition).

Properties of Pure States

• Represented by a vector $|\psi ⟩$ (a “ket”)
• Normalized: $⟨\psi |\psi ⟩=1$
• Can be visualized on the Bloch sphere (for single qubits)

Mixed States

When there is classical uncertainty (e.g., the system might be in state $|{\psi }_1⟩$ with probability $p_1$ or $|{\psi }_2⟩$ with probability $p_2$), we use mixed states represented by density matrices:

$$\rho =\sum _i{p}_i|{\psi }_i⟩⟨{\psi }_i|$$

Mixed states arise from:

• Classical uncertainty about preparation
• Decoherence (interaction with environment)
• Tracing out part of an entangled system

State Space

For a single qubit, the state space is 2-dimensional. For $n$ qubits, the state space is $2^n$-dimensional. This exponential growth is both the promise and challenge of quantum computing:

Common States

Single-qubit basis states:

• $|0⟩$ and $|1⟩$: computational basis
• $|+⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$ and $|-⟩=\frac{1}{\sqrt{2}}(|0⟩-|1⟩)$: Hadamard basis

Multi-qubit states:

• $|00⟩,|01⟩,|10⟩,|11⟩$: 2-qubit computational basis
• Bell states: maximally entangled 2-qubit states

State Evolution

Quantum states evolve in two ways:

1. Unitary evolution: Applying quantum gates $$|{\psi }^{\prime }⟩=U|\psi ⟩$$

2. Measurement: Collapsing to an eigenstate $$|\psi ⟩\stackrel{\text{measure}}{\to }|k⟩\text{ with probability }|⟨k|\psi ⟩{|}^2$$

Describing States

Multiple equivalent ways to describe quantum states:

See also: Pure State, Mixed State, Density Matrix, Bloch Sphere, Hilbert Space