Multi-Indexed Schatten Quasi- and Anti-Norms and Additivity of Conditional R'enyi Entropies

We introduce and study multi-indexed Schatten spaces which lift the commutative ell_q1\[ell_q2[ell_q3[dots\]]] -spaces to the quantum toolbox. They extend Pisier's operator-valued Schatten norms and the recently introduced two-indexed Schatten quasi-norms. For every ordered tuple mathcal{Q}=\(q₁,dots,q_n\) of non-zero indices we define a quasi-norm (for positive indices) or an anti-norm (for negative indices) on operators on an n-fold tensor product of Hilbert spaces. We do this for indices of the same sign satisfying the mutual compatibility condition |frac{1}{q_i}-frac{1}{q_j}|leq1. Beyond establishing desirable properties of these quasi- and anti-norms via factorization formulas that combine operator space and matrix-analytic techniques, we lift several central tools and results of the seminal [Devetak, Junge, King, Ruskai, CMP 2006] to the multi-indexed and quasi-normed setting. These include a general non-commutative Minkowski inequality relating swapped systems and two identity-removal equalities. As a consequence, complete hypercontractivity and complete reverse hypercontractivity tensorize for all compatible indices, including negative ones. Concerning quantum information theory we prove that the completely bounded minimal and the maximal output conditional entropies are additive under tensor products of quantum channels for all αgeqfrac12. This extends the case α>1 of [Fawzi, Kochanowski, Rouz'e, Van Himbeeck, CMP 2026] and the unconditional results of [Devetak, Junge, King, Ruskai, CMP 2006]. Beyond tensor products, we establish chain rules for H^uparrow_α, αin[frac12,1), for compositions of possibly signaling channels, extending upon the smooth max-entropy chain rule used in device-independent entanglement certification to the whole interval αin[1/2,1).