We obtain formulas for Petz–Rényi and Umegaki relative entropy from the idea of distribution of a positive self-adjoint operator. Classical results on Rényi and Kullback–Leibler divergences are applied to obtain new results and new proofs for some known results about Petz–Rényi and Umegaki relative entropy. Most important among these, is a necessary and sufficient condition for the finiteness of the Petz–Rényi $\alpha$-relative entropy. All of the results presented here are valid in both finite and infinite dimensions. In particular, these results are valid for states in Fock spaces and thus are applicable to continuous variable quantum information theory.

我们从正自伴随算子的分布思想中得到了 Petz–Rényi 和 Umegaki 相对熵的公式。应用 Rényi 和 Kullback-Leibler 散度的经典结果来获得有关 Petz-Rényi 和 Umegaki 相对熵的一些已知结果的新结果和新证明。其中最重要的是 Petz-Rényi 有限性的充分必要条件$\alpha$-相对熵。这里给出的所有结果在有限和无限维度上都是有效的。特别是，这些结果对于福克空间中的状态有效，因此适用于连续变量量子信息论。