We introduce an quantum entropy for bimodule quantum channels on finite von Neumann algebras, generalizing the remarkable Pimsner-Popa entropy. The relative entropy for Fourier multipliers of bimodule quantum channels establishes an upper bound of the quantum entropy. Additionally, we present the Araki relative entropy for bimodule quantum channels, revealing its equivalence to the relative entropy for Fourier multipliers and demonstrating its left/right monotonicities and convexity. Notably, the quantum entropy attains its maximum if there is a downward Jones basic construction. By considering R\'{e}nyi entropy for Fourier multipliers, we find a continuous bridge between the logarithm of the Pimsner-Popa index and the Pimsner-Popa entropy. As a consequence, the R\'{e}nyi entropy at $1/2$ serves a criterion for the existence of a downward Jones basic construction.

翻译：我们引入了有限冯·诺依曼代数上双模量子通道的量子熵，推广了著名的Pimsner-Popa熵。双模量子通道的傅里叶乘子的相对熵为量子熵建立了上界。此外，我们提出了双模量子通道的Araki相对熵，揭示了其与傅里叶乘子相对熵的等价性，并证明了其左/右单调性和凸性。值得注意的是，当存在向下Jones基本构造时，量子熵达到最大值。通过考虑傅里叶乘子的Rényi熵，我们发现了Pimsner-Popa指数的对数与Pimsner-Popa熵之间的连续桥梁。作为推论，在$1/2$处的Rényi熵可作为存在向下Jones基本构造的判据。