Dichtematrix

Dichtematrix, in English density matrix, is a formalism in quantum mechanics that describes the statistical state of a quantum system. It generalizes the notion of a state vector to include both quantum superpositions and statistical mixtures, as well as possible entanglement with an environment. The density matrix ρ is a positive semidefinite Hermitian operator with trace equal to one, acting on the system’s Hilbert space.

For a pure state |ψ⟩, ρ = |ψ⟩⟨ψ|. For a statistical ensemble {p_i, |ψ_i⟩}, ρ = ∑_i p_i |ψ_i⟩⟨ψ_i|. The expectation value of an observable A is ⟨A⟩ = Tr(ρ A). Subsystems are described by reduced density matrices obtained by tracing out other degrees of freedom: ρ_A = Tr_B(ρ).

Time evolution of a closed system is given by the von Neumann equation iħ dρ/dt = [H, ρ]. Open systems require more general master equations, such as the Lindblad form, to account for dissipation and decoherence.

Key properties include Hermiticity, positive semidefiniteness, and Tr(ρ) = 1. Purity Tr(ρ^2) ranges from 1 for a pure state to values less than 1 for mixed states. The von Neumann entropy, S(ρ) = −Tr(ρ log ρ), measures mixedness. For a two-level system, ρ can be represented as 1/2(I + r·σ), where r is the Bloch vector and σ are the Pauli matrices.

The density-matrix formalism is essential for describing quantum ensembles, entanglement, decoherence, and measurements when the system is not in a pure state. Quantum state tomography can reconstruct ρ from experimental data, enabling practical characterization of quantum systems.