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.

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