Operator--Scalar Curvature Gap for the Depolarising Lindbladian on $M_n(\mathbb{C})$

We provide a complete and elementary analysis of curvature--dimension properties for the symmetric depolarising quantum Markov semigroup on $M_n(\mathbb{C})$. Using the non-commutative Bakry--\'Emery calculus, we derive explicit expressions for the carr\'e du champ and its iterate, leading to an exact operator-level Bochner identity. This yields sharp curvature--dimension bounds exhibiting a distinct operator-scalar curvature gap: the semigroup satisfies $\mathrm{CD}_{\mathrm{op}}(n+2,\infty)$ at the operator level and $\mathrm{CD}_{\mathrm{sc}}(2n,\infty)$ at the scalar level, with both constants shown to be optimal. As consequences, we recover the sharp modified logarithmic Sobolev inequality, hypercontractivity thresholds, and exponential convergence to equilibrium. We further interpret the dynamics as a gradient flow of the quantum relative entropy in the sense of Carlen and Maas. These results give a complete and elementary curvature analysis of the canonical depolarising channel.