ENTROPY MONOTONICITY, CURVATURE DEFECT, AND QUANTUM NON-COLLAPSING: A PERELMAN-TYPE THEORY FOR QUANTUM MARKOV SEMIGROUPS

We construct a quantum W-functional WQ(ρ,τ) for any quantum Markovs emigroup (M,φ,Pt) satisfying KMS detailed balance, serving as the noncommutative analog of Perelman’s entropy functional for the Ricci flow. The central new object is the defect-corrected W-functional fWQ(ρ,τ) :=WQ(ρ,τ) +Zτ0Dλ(ρs)2sds, where Dλ(ρ) :=− ̇Iφt=0− 2λIφ(ρ)≥ 0 is the curvature defect functional, and Iφ denotes the quantum relative Fisher information.