Remainder Terms for Some Quantum Entropy Inequalities
  We consider three von Neumann entropy inequalities: subadditivity; Pinsker's inequality for relative entropy; and the monotonicity of relative entropy. For these we state conditions for equality, and we prove some new error bounds away from equality, including an improved version of Pinsker's inequality. 
Eric A. Carlen and Elliott H. Lieb
quant-ph math-ph math.MP math.OA