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In this paper, we extend the notion of non-microstate free entropy to thebi-free setting. Using a diagrammatic approach involving bi-non-crossingdiagrams, bi-free difference quotients are constructed as analogues of the freepartial derivations. Adjoints of bi-free difference quotients are discussed andused to define bi-free conjugate variables. Notions of bi-free Fisherinformation and non-microstate entropy are defined and properties of freeentropy are extended to the bi-free setting.

Analogues of Entropy in Bi-Free Probability Theory: Non-Microstate