  In this paper, a notion of non-microstate bi-free entropy with respect to
completely positive maps is constructed thereby extending the notions of
non-microstate bi-free entropy and free entropy with respect to a completely
positive map. By extending the operator-valued bi-free structures to allow for
more analytical arguments, a notion of conjugate variables is constructed using
both moment and cumulant expressions. The notions of free Fisher information
and entropy are then extended to this setting and used to show minima of the
Fisher information and maxima of the non-microstate bi-free entropy at
bi-R-diagonal elements.
