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 definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
    —Samuel Taylor Coleridge (1772–1834)

    “According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
    —Ana Castillo (b. 1953)

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)