Trace Inequalities and Quantum Relative Entropies

‎In this short note‎, ‎we give an introduction to the subject of trace and results that are relevant to quantum entropies‎. ‎We introduce relative entropy $H_{\phi}(A,B)=\t[\phi(A)-\phi(B)-\phi(B)(A-B)]$ for $\phi$ a convex function and $A,~B$ bounded self adjoint operators on a Hilbert space $H$‎. ‎In particular‎, ‎we show that this relative entropy is monotone if and only if $\phi$ is operator monotone‎.