Subadditivity of $q$-entropies for $q>1$ Available to Purchase

I prove a basic inequality for Schatten $q$-norms of quantum states on a finite-dimensional bipartite Hilbert space $H1⊗H2$⁠: $1+∥ρ∥q⩾∥Tr1ρ∥q+∥Tr2ρ∥q$⁠. This leads to a proof—in the finite-dimensional case—of Raggio’s conjecture [G. A. Raggio, J. Math. Phys. 36, 4785 (1995)] that the $q$-entropies $Sq(ρ)=(1−Tr[ρq])∕(q−1)$ are subadditive for $q>1$⁠; that is, for any state $ρ$⁠, $Sq(ρ)$ is not greater than the sum of the $Sq$ of its reductions, $Sq(ρ)⩽Sq(Tr1ρ)+Sq(Tr2ρ)$⁠.