Relative entropy for von Neumann subalgebras

We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner-Popa index connects to sandwiched p-Rényi relative entropy for all 1/2 ≤ p ≤∞, including Umegaki's relative entropy at p = 1. Based on that, we introduce a new notation of relative entropy to a subalgebra which generalizes subfactors index. This relative entropy has application in estimating decoherence time of quantum Markov semigroups.