AI 中文总结
本文引入测地量子f-散度，证明其满足数据处理不等式，确定其等号情况及可逆测地量子马尔可夫链的坍缩，还分析了参数单调性等性质，为量子信息领域提供了新的散度工具。
AI 中文摘要
我们引入测地量子f-散度D_f^t，其中0≤t≤1，该散度由标准相对模算子与极大相对模算子之间的仿射不变测地线得到。对于对易态，它可约化为经典f-散度。对数生成元产生连接Umegaki熵与Belavkin-Staszewski（BS）熵的测地相对熵，而幂生成元产生连接Petz族与几何族的(t,α)-Rényi散度。我们的第一个主要结果是，对于每个有限算子凸生成元f和每个t∈[0,1]，D_f^t满足数据处理不等式（DPI）。特别地，当0<α<1或1<α≤2时，这为测地相对熵和(t,α)-Rényi散度提供了DPI。我们的第二个主要结果确定了DPI中的等号情况：对于可逆态，每个决定等号的算子凸生成元，在每个非极大参数0≤t<1时，恰好对应Petz充分类；在t=1时，该类变为通常更大的极大类或BS类，该类也与二次生成元对应的散度在所有t下的等号类重合。我们的第三个主要结果是可逆测地量子马尔可夫链的相应坍缩：对于三种有序条件互信息构造中的每一种，t-量子马尔可夫链在0≤t<1时恰好是量子马尔可夫链，而在t=1时，它们是可逆BS量子马尔可夫链，这是一个可能严格更大的类。我们还确定了参数单调性区间以及与(α,z)族的交集，证明了强化的数据处理和重构估计，比较了三种条件方向，在正低本征值假设下建立了连续性界并给出了互补的不连续性结果，还给出了单位成本容量的解释。
英文摘要
We introduce the geodesic quantum $f$-divergences $D_f^t$, $0\leq t\leq1$, obtained from the affine-invariant geodesic between the standard and maximal relative modular operators. They reduce to the classical $f$-divergence for commuting states. The logarithmic generator yields geodesic relative entropies joining the Umegaki and Belavkin-Staszewski entropies, while the power generators yield $(t,α)$-Rényi divergences joining the Petz and geometric families. Our first main result is data processing of $D_f^t$ for every finite operator-convex generator $f$ and every $t\in[0,1]$. In particular, this gives DPI for the geodesic relative entropies and for the $(t,α)$-Rényi divergences when $0<α<1$ or $1<α\leq2$. Our second main result identifies equality in the DPI: for invertible states, every equality-determining operator-convex generator has, at each nonmaximal parameter $0\leq t<1$, exactly the Petz sufficiency class; at $t=1$, this changes to the generally larger maximal, or BS, class, which also coincides with the equality class of the divergences associated with the quadratic generator for every $t$. Our third main result is the corresponding collapse of invertible geodesic quantum Markov chains: for each of the three ordered conditional-mutual-information constructions, the $t$-quantum Markov chains are precisely the quantum Markov chains for $0\leq t<1$, whereas at $t=1$ they are the invertible BS quantum Markov chains, a class that can be strictly larger. We also determine parameter-monotonicity regimes and the intersections with the $(α,z)$ family. We prove strengthened data-processing and reconstruction estimates; we compare the three conditional orientations; we establish continuity bounds under positive lower-eigenvalue assumptions together with complementary discontinuity results; and we give a capacity-per-unit-cost interpretation.
Comments116 pages, 8 figures
