  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.
