Quasi-factorization and Multiplicative Comparison of Subalgebra-Relative Entropy

The relative entropy of a quantum density matrix to a subalgebraic restriction appears throughout quantum information. For subalgebra restrictions given by commuting conditional expectations in tracial settings, strong subadditivity shows that the sum of relative entropies to each is at least as large as the relative entropy to the intersection subalgebra. When conditional expectations do not commute, an inequality known as quasi-factorization or approximate tensorization replaces strong subadditivity. Multiplicative or strong quasi-factorization yields relative entropy decay estimates known as modified logarithmic-Sobolev inequalities for complicated quantum Markov semigroups from those of simpler constituents. In this work, we show multiplicative comparisons between particular forms of relative entropy, including subalgebra-relative entropy and certain perturbations to it. Following, we obtain a strong quasi-factorization inequality with constant scaling logarithmically in subalgebra index and sometimes avoiding explicit index dependence. For conditional expectations that nearly commute and are not too close to a set with larger intersection algebra, the shown quasi-factorization is asymptotically tight in that the constant approaches one. We apply quasi-factorization to uncertainty relations between incompatible bases and to conditional expectations arising from graphs. Quasi-factorization yields bounds of optimal asymptotic order on mixing processes described by finite graphs.