Complete entropy contraction from conditional hyperboundedness

For finite-dimensional quantum channels compatible with a faithful conditional expectation, we show that conditional hyperboundedness and strict GNS contraction together yield a relative entropy contraction. For KMS-symmetric quantum Markov semigroups, a conditional 2→ q hyperboundedness at time t₀>0 combined with a GNS gap $λ>0$ implies a modified logarithmic Sobolev constant \[ α=λ(1-2/q)/12(λt_0+log M+1-2/q),\] where M is the hyperboundedness constant. As a consequence, completely bounded hyperboundedness over auxiliary systems gives complete versions of entropy contraction with the same rates.