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  1. Hilbert Space

    Linked via "pure state"

    State Representation and Purity
    A state $|\psi\rangle \in \mathcal{H}$ (assuming the normalisation $\langle\psi|\psi\rangle = 1$) describes a pure state. If the system is in a statistical mixture, a density operator $\rho$ acts on $\mathcal{H}$.
    $$\rho = \sumi pi |\psii\rangle \langle \psii|$$
    where $pi$ are the probabilities of finding the system in state $|\psii\rangle$. The purity of the state is quantified by the purity measure $P(\rho) = \text{Tr}(\rho^2)$. For entangled systems, the [reduced d…

  2. Liouville Von Neumann Equation

    Linked via "pure states"

    The Liouville-von Neumann equation (LVNE) is a fundamental operator equation in statistical mechanics and quantum information theory describing the time evolution of the density operator ($\rho$) of a quantum system. It serves as the quantum analogue to the classical Liouville equation, describing the evolution of the phase-space probability density function. Unlike the [Schröd…

  3. Quantum State

    Linked via "pure states"

    Density Matrices and Mixed States
    While pure states are described by a single vector $|\psi\rangle$, systems about which we have incomplete knowledge, or which are entangled with an external environment (decoherence), are better described by a density operator, $\hat{\rho}$.
    For a pure state, the density operator is defined as:

  4. Quantum State

    Linked via "pure state"

    While pure states are described by a single vector $|\psi\rangle$, systems about which we have incomplete knowledge, or which are entangled with an external environment (decoherence), are better described by a density operator, $\hat{\rho}$.
    For a pure state, the density operator is defined as:
    $$\hat{\rho} = |\psi\rangle\langle\psi|$$

  5. Quantum State

    Linked via "pure states"

    $$\hat{\rho} = |\psi\rangle\langle\psi|$$
    If the system is in a statistical mixture of pure states $\{|\psii\rangle\}$ with classical probabilities $pi$, the density operator is given by:
    $$\hat{\rho} = \sum{i} pi |\psii\rangle\langle\psii|$$