A lower bound on the fidelity between two states in terms of their   trace-distance and max-relative entropy
  Fidelity is a fundamental and ubiquitous concept in quantum information theory. Fuchs-van de Graaf's inequalities deal with bounding fidelity from above and below. In this paper, we give a lower bound on the quantum fidelity between two states in terms of their trace-norm distance and their max-relative entropy. 
Lin Zhang and Kaifeng Bu and Junde Wu
quant-ph math-ph math.MP