We provide a comprehensive study on uniformly left [Formula: see text]-bounded (respectively, [Formula: see text]-bounded) orthonormal bases in infinite-dimensional cyclic bimodules associated with c.p. maps between two von Neumann algebras [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are faithful normal states on [Formula: see text] and [Formula: see text], respectively. Separate investigations on cyclic bimodules associated with Markov maps and arbitrary c.p. maps are also provided since the results differ. This generalizes the results of the authors’ previous work on Kadison’s unitary basis problem.