mixedstate

A mixed state is a description of a quantum system that represents a statistical ensemble of possible pure states, rather than a single definite state. It is conveniently described by a density operator ρ, which is a positive semi-definite operator with unit trace acting on the system's Hilbert space.

A pure state, by contrast, corresponds to a single wavefunction |ψ⟩ and is represented by ρ = |ψ⟩⟨ψ|. Pure states satisfy ρ^2 = ρ and Tr(ρ^2) = 1. A mixed state does not in general satisfy these conditions; instead, Tr(ρ^2) < 1, with purity Tr(ρ^2) measuring how far the state is from a pure one.

A mixed state can be written as ρ = ∑i p_i |φ_i⟩⟨φ_i|, where {p_i} are nonnegative probabilities summing to one and {|φ_i⟩} are pure states. This can reflect either a classical uncertainty about which pure state the system is in (a proper mixture) or the result of tracing out part of a larger entangled system (an improper mixture). The latter viewpoint is formalized by reduced density matrices.

Observables are evaluated via expectation values ⟨A⟩ = Tr(ρA). The purity and entropy of a mixed state are connected to the von Neumann entropy S(ρ) = −Tr(ρ log ρ); pure states have S = 0, while mixed states have S > 0. Mixedness also suppresses quantum interference: off-diagonal coherence terms in ρ, which enable interference, may be reduced or disappear in a mixed state.

Common examples include thermal (Gibbs) states ρ = Z^−1 e^−βH, which are typically mixed due to thermal fluctuations. Any mixed state can be viewed as the reduced state of a larger pure system, reflecting the fundamental role of entanglement in quantum statistics.