  We consider continuous-time (not necessarily finite) Markov chains on
discrete spaces and identify a curvature-dimension inequality, the condition
$CD_\Upsilon(\kappa,\infty)$, which serves as a natural analogue of the
classical Bakry-\'Emery condition $CD(\kappa,\infty)$ in several respects. In
particular, it is tailor-made to the classical approach of proofing the
modified logarithmic Sobolev inequality via computing and estimating the second
time derivative of the entropy along the heat flow generated by the generator
of the Markov chain. We prove that curvature bounds in the sense of
$CD_\Upsilon$ are preserved under tensorization, discuss links to other notions
of discrete curvature and consider a variety of examples including complete
graphs, the hypercube and birth-death processes. We further consider power type
entropies and determine, in the same spirit, a natural CD condition which leads
to Beckner inequalities. The $CD_\Upsilon$ condition is also shown to be
compatible with the diffusive setting, in the sense that corresponding hybrid
processes enjoy a tensorization property.
