  In this paper, we extend the notion of non-microstate free entropy to the
bi-free setting. Using a diagrammatic approach involving bi-non-crossing
diagrams, bi-free difference quotients are constructed as analogues of the free
partial derivations. Adjoints of bi-free difference quotients are discussed and
used to define bi-free conjugate variables. Notions of bi-free Fisher
information and non-microstate entropy are defined and properties of free
entropy are extended to the bi-free setting.
