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 subalgebra-relative entropy and its perturbation by a quantum channel with corresponding fixed point subalgebra. Following, we obtain a strong quasi-factorization inequality with constant scaling logarithmically in subalgebra index. 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.
翻译:数量密度矩阵相对的温和度是亚相位值限制的相对倍增。对于在种族环境中通过通缩有条件期望而导致的亚相位值限制,强的亚相位数表明,每种相对的异种总和至少与相交亚相交的亚相位数值相对的倍增率相同。当条件性预期不通时,被称为准致因数或近似推力的不平等将取代强大的亚相向性。倍增性或强准致因数将产生相对的倍增衰变估计,称为对数-Sobolev值的相对变异性估计,称为对数-Sobolev值,与较简单的成份数组相较。在这项工作中,我们显示了对亚相近且不相近的亚相异性变异性对比,显示的准致变异性预期在恒定点子值关系中,我们得到强烈的准致变异性不平等,在亚相位数指数中不断缩缩成正位值上。对于接近通且不相近相交相交相交的正位数位数的正位数,我们所显示的准致变相变相变式期望在正位图基化之间会适用于。