Tiheydematriisasta

Tiheydematriisasta, known in English as the density matrix, is a mathematical formalism used in quantum mechanics to describe the statistical state of a system when its exact state is uncertain or the system is entangled with another system. It provides a unified framework for describing pure states and statistical ensembles.

A density matrix ρ is a linear operator on the system’s Hilbert space that is Hermitian, positive semidefinite, and has trace one. In a chosen basis, its elements are ρ_ij = ⟨i|ρ|j⟩. If the system is in a pure state |ψ⟩, ρ = |ψ⟩⟨ψ|. For a statistical ensemble {p_k, |ψ_k⟩}, ρ = ∑_k p_k |ψ_k⟩⟨ψ_k|. The density matrix contains all information about measurement statistics for any observable.

Expectation values are computed as ⟨A⟩ = Tr(ρA). The diagonal elements give probabilities for basis outcomes; off-diagonal elements, called coherences, encode quantum interference and the degree of coherence in the state.

Time evolution for closed systems is described by the von Neumann equation iħ dρ/dt = [H, ρ]. Open systems are described by master equations such as the Lindblad form, which model decoherence and dissipation due to the environment.

Reduced density matrices arise from tracing out part of a composite system, ρ_A = Tr_B ρ, revealing entanglement and mixedness. For a qubit, ρ = (I + r·σ)/2 with Bloch vector r, where |r| ≤ 1.

Applications span quantum information, computation, spectroscopy, and quantum statistical mechanics. The density matrix provides a robust description of both pure and mixed states and is also known as the statistical operator.