A lifting is a continious map from a system to a compound system introduced by Ohya and Accardi [1], and we can represent several dynam- ical processes by using it.Liftings show the relation between two systems in the compound system clearly, and it is useful to discuss not only the communication process, but also an entanglement of it.In this study we define a quantum mutual entropy using liftings and investigate the property.We show that there exist some cases where the quantum mutual entropy violates the Shannon inequality.