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
increases to S=ln 2 =0.69 for the measurement outcome mixture
as the quantum interference information is erased.
Read more about this topic: Von Neumann Entropy
Famous quotes containing the word definition:
“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)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)