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In this paper, a notion of non-microstate bi-free entropy with respect tocompletely positive maps is constructed thereby extending the notions ofnon-microstate bi-free entropy and free entropy with respect to a completelypositive map. By extending the operator-valued bi-free structures to allow formore analytical arguments, a notion of conjugate variables is constructed usingboth moment and cumulant expressions. The notions of free Fisher informationand entropy are then extended to this setting and used to show minima of theFisher information and maxima of the non-microstate bi-free entropy atbi-R-diagonal elements.

Bi-Free Entropy with Respect to Completely Positive Maps