Subadditivity of q-entropies for q>1

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).