Von Neumann Entropy - Definition

Definition

Given the density matrix ρ, von Neumann defined the entropy as

which is a proper extension of the Gibbs entropy (up to a factor ) and the Shannon entropy to the quantum case. To compute S(ρ) it is convenient (see logarithm of a matrix) to compute the Eigendecomposition of . The von Neumann entropy is then given by

Since, for a pure state, the density matrix is idempotent, ρ=ρ2, the entropy S(ρ) for it vanishes. Thus, if the system is finite (finite dimensional matrix representation), the entropy S(ρ) quantifies the departure of the system from a pure state. In other words, it codifies the degree of mixing of the state describing a given finite system. Measurement decoheres a quantum system into something noninterfering and ostensibly classical; so, e.g., the vanishing entropy of a pure state |Ψ⟩ = (|0⟩+|1⟩)/√2, corresponding to a density matrix

\rho = {1\over 2} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}

increases to S=ln 2 =0.69 for the measurement outcome mixture

\rho = {1\over 2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

as the quantum interference information is erased.

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