### Tight bound on relative entropy by entropy dierence

**David Reeb, Michael M. Wolf**

(2013)

**arXiv.org**: 1304.0036v1

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**Abstract:**We prove a lower bound on the relative entropy between two nite-dimensional states in terms of their entropy dierence and the dimension of the underlying space. The inequality is tight in the sense that equality can be attained for any prescribed value of the entropy dierence, both for quantum and classical systems. We outline implications for thermodynamics and information theory, such as a necessary condition for a process to be close to thermodynamic reversibility, or an easily computable lower bound on the classical channel capacity. Furthermore, we derive a tight upper bound, uniform for all states of a given dimension, on the variance of the surprisal, whose thermodynamic meaning is that of heat capacity.