The Curvature Gap in Quantum Markov Semigroups: From Depolarising Models to Spectral Separation

Abstract. We develop a self-contained theory of Bakry–Émery curvature for generalised depolarising quantum Markov semigroups on matrix algebras. The central algebraic object is the exact conditional Bochner identity 'Γ2(𝑋) =𝜆2Γ(𝑋)+ 𝐸Γ(𝑋)', where 𝐸 is a trace-preserving conditional expectation. This identity separates two distinct curvature constants: the pointwise (operator) curvature 𝜅op and the scalar (trace-averaged) curvature 𝜅sc. For the primitive tracial depolarising channel we prove 𝜅saop= 𝑛+ 2, 𝜅sc= 2𝑛; the gap 𝜅sc− 𝜅saop= 𝑛− 2 vanishes only for the qubit. For arbitrary conditional expectations we introduce finite-dimensional defect indices and obtain exact formulas for the three sharp constants. The operator-scalar gap is then a rational function of these indices.A simple operator-theoretic framework yields a Bridge Inequality and a Spectral Separation Theorem: whenever the algebra is non-commutative, the spectrum of the curvature operator is strictly smaller than the scalar Bakry–Émery constant. The gap translates into a hierarchy offunctional inequalities: the modified logarithmic Sobolev inequality and hypercontractivity are governed by 𝜅sc, while gradient contractivity is controlled by 𝜅op. We collect all consequences in a Grand Dichotomy Theorem and prove a quantum Obata rigidity theorem that characterises equality cases. Explicit computations for the depolarising qubit, qutrit and generalised dephasing channels illustrate the theory.