Mixed vs Pure States

Pure states carry complete information; mixed states encode statistical uncertainty and are the generic condition of real quantum systems.

The distinction

A pure state is describable by a single vector |psi>, equivalently a rank-one density matrix rho = |psi>mixed state is any other valid density matrix: a convex combination of pure states with purity strictly less than one. A superposition is not a mixture; a|0> + b|1> is pure and coherent, whereas a classical coin over |0> and |1> is mixed and incoherent.

Superposition is not mixture

This is the most common confusion. The pure superposition |+> = (|0>+|1>)/sqrt(2) has definite off-diagonal coherences and shows interference. The mixture that is |0> half the time and |1> half the time has density matrix I/2, no coherences, and no interference. Both give 50/50 in the computational basis, but a measurement in the X basis distinguishes them sharply: |+> gives a certain outcome, the mixture stays 50/50.

Identical diagonals, different off-diagonals: the coherence is the whole difference.

Measures of mixedness

• Purity Tr(rho^2): 1 for pure, down to 1/d for maximally mixed
• von Neumann entropy S = -Tr(rho log rho): 0 for pure, log d at maximum
• Linear entropy 1 - Tr(rho^2): a simple proxy avoiding logarithms

Why real states are mixed

Isolation is never perfect. Coupling to an environment entangles the system with uncontrolled degrees of freedom; tracing them out leaves a mixed state. This decoherence is why quantum computers must actively fight mixedness through isolation, fast gates, and error correction. Maintaining high purity in the logical subspace is equivalent to preserving the coherences that algorithms depend on.