Mixed State

A quantum state with classical uncertainty, a statistical mixture of pure states.

A mixed state describes a quantum system where we don’t have complete information. It’s a probabilistic combination of pure states, representing classical uncertainty on top of quantum uncertainty.

Definition

A mixed state cannot be written as a single state vector. It’s represented by a density matrix:

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

where $p_i$ are classical probabilities (summing to 1) and $|{\psi }_i⟩$ are pure states.

Example: The Maximally Mixed State

The simplest mixed state for a qubit:

$$\rho =\frac{1}{2}|0⟩⟨0|+\frac{1}{2}|1⟩⟨1|=\frac{1}{2}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=\frac{I}{2}$$

This represents “no information at all” - maximum classical uncertainty.

Mixed ≠ Superposition

This is important to understand:

The off-diagonal elements (“coherences”) are what distinguishes quantum superposition from classical probability.

How Mixed States Arise

1. Incomplete preparation: You prepare $|0⟩$ or $|1⟩$ but forget which
2. Decoherence: Environment interaction destroys coherence
3. Partial trace: Taking part of an entangled system

Example: Entanglement and Mixed States

If two qubits are in Bell state $|{\Phi }^{+}⟩=\frac{1}{\sqrt{2}}(|00⟩+|11⟩)$ and you only have access to one qubit:

$${\rho }_A={\text{Tr}}_B(|{\Phi }^{+}⟩⟨{\Phi }^{+}|)=\frac{I}{2}$$

Your qubit looks maximally mixed, even though the total system is pure!

Characteristics

Why It Matters

Mixed states are essential for:

• Describing realistic quantum systems (always some noise)
• Modeling decoherence
• Quantum error correction
• Quantum thermodynamics

See also: Pure State, Density Matrix, Decoherence