量子数据处理等式
Quantum data processing equality
- University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该研究将量子数据处理不等式提升为精确等式，通过积分表示分解相对熵损失为恢复与边界项，并建立最优平方根恢复界，改进先前依赖。
中文摘要 AI 辅助
量子数据处理不等式断言量子相对熵在量子通道作用下是单调非增的。在此，我们通过复带上的边界和表面泊松积分表示，将这一基本不等式提升为精确等式。相对熵损失被分解为两个独立且非负的物理机制：一个操作性的Petz恢复分量，用于刻画状态的可重构性；以及一个边界项，用于度量相对模动力学的不匹配。我们进一步将该损失表示为原始状态与其平均Petz恢复状态之间的测量相对熵加上一个非负变分余项。对于Belavkin--Staszewski（BS）损失，类似的积分表示产生了一个分解，即负对数Uhlmann保真度与平均重构加上一个非负变分余项。此外，我们在任意指定的模参数下建立了恢复界，包括典型的未旋转Petz映射，并具有相对熵损失的最优平方根缩放。前置因子依赖于不同特征值的计数或对数谱量，改进了先前界中关于小特征值的逆幂依赖性。
英文摘要
The quantum data processing inequality asserts that quantum relative entropy is monotonically non-increasing under quantum channels. Here, we promote this fundamental inequality to an exact equality via boundary and surface Poisson integral representations across a complex strip. The relative-entropy loss is resolved into two independent, nonnegative physical mechanisms: an operational Petz recovery component governing state reconstructibility, and a boundary term measuring the mismatch of relative modular dynamics. We further express the loss as the measured relative entropy between the original state and its averaged Petz recovery state plus a nonnegative variational remainder. For the Belavkin--Staszewski (BS) loss, analogous integral representations yield a decomposition into negative log Uhlmann fidelity with an averaged reconstruction plus a nonnegative variational remainder. Furthermore, we establish recovery bounds at any prescribed modular parameter, including the canonical unrotated Petz map, with optimal square-root scaling in the relative entropy loss. The prefactors depend on distinct-eigenvalue counts or logarithmic spectral quantities, improving upon the inverse-power dependence on small eigenvalues in previous bounds.
