Quantum Relative Entropy - Definition

Definition

As with many other objects in quantum information theory, quantum relative entropy is defined by extending the classical definition from probability distributions to density matrices. Let ρ be a density matrix. The von Neumann entropy of ρ, which is the quantum mechanical analog of the Shannon entropy, is given by

For two density matrices ρ and σ, the quantum relative entropy of ρ with respect to σ is defined by

S(\rho \| \sigma) = - \operatorname{Tr} \rho \log \sigma - S(\rho) = \operatorname{Tr} \rho \log \rho - \operatorname{Tr} \rho \log \sigma = \operatorname{Tr}\rho (\log \rho - \log \sigma).

We see that, when the states are classical, i.e. ρσ = σρ, the definition coincides with the classical case.

Read more about this topic:  Quantum Relative Entropy

Famous quotes containing the word definition:

    “Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.”
    —Willard Van Orman Quine (b. 1908)

    “I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

    “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)