Intertwining Curvature Bounds for Graphs and Quantum Markov Semigroups

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.