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.

Read more about this topic:  Von Neumann Entropy

Famous quotes containing the word definition:

    “The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
    —Jean Baudrillard (b. 1929)

    “... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
    —George Eliot [Mary Ann (or Marian)

    “Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
    —Nadine Gordimer (b. 1923)