  Based on earlier work by Carlen-Maas and the second- and third-named author,
we introduce the notion of intertwining curvature lower bounds for graphs and
quantum Markov semigroups. This curvature notion is stronger than both
Bakry-\'Emery and entropic Ricci curvature, while also computationally simpler
than the latter. We verify intertwining curvature bounds in a number of
examples, including finite weighted graphs and graphs with Laplacians admitting
nice mapping representations, as well as generalized dephasing semigroups and
quantum Markov semigroups whose generators are formed by commuting jump
operators. By improving on the best-known bounds for entropic curvature of
depolarizing semigroups, we demonstrate that there can be a gap between the
optimal intertwining and entropic curvature bound. In the case of qubits, this
improved entropic curvature bound implies the modified logarithmic Sobolev
inequality with optimal constant.
