mixedness

Mixedness is a property of a quantum state describing its departure from being pure. A system is described by a density operator ρ with Tr(ρ) = 1. A pure state satisfies ρ^2 = ρ (equivalently, Tr(ρ^2) = 1). A mixed state is a convex combination of pure states, ρ = ∑i p_i |ψ_i⟩⟨ψ_i| with p_i ≥ 0 and ∑i p_i = 1. Mixedness captures uncertainty about the exact state of the system, arising from classical probabilistic mixtures or quantum correlations with an environment.

Quantitative measures of mixedness include purity and entropy-based quantities. Purity is defined as P = Tr(ρ^2), taking the value 1 for pure states and less than 1 for mixed states. In a system of dimension d, the purity satisfies 1/d ≤ P ≤ 1, with P = 1/d corresponding to the maximally mixed state ρ = I/d. Linear entropy, S_L = 1 − P, ranges from 0 (pure) to 1 − 1/d (maximally mixed). The von Neumann entropy, S(ρ) = −Tr(ρ log ρ), also gauges mixedness, vanishing for pure states and reaching log d for the maximally mixed state (log base 2 or e depending on convention).

Interpretation and sources: Mixedness reflects both intrinsic quantum uncertainty due to entanglement and extrinsic uncertainty from environmental interactions or incomplete preparation. For a composite system, a subsystem's reduced state is typically mixed if the global state is entangled. Decoherence and partial tracing are common physical processes that increase mixedness, which in turn affects the performance of quantum information tasks and the visibility of coherent phenomena.

Applications and relevance: Mixedness is central in state tomography, error budgeting, and the assessment of quantum state quality. It informs strategies for state purification, error correction, and noise mitigation in quantum technologies.