  In this paper, we extend the notion of microstate free entropy to the bi-free
setting. In particular, using the bi-free analogue of random matrices,
microstate bi-free entropy is defined. Properties essential to an entropy
theory are developed, such as the behaviour of the entropy when transformations
on the left variables or on the right variables are performed. In addition, the
microstate bi-free entropy is demonstrated to be additive over bi-free
collections provided additional regularity assumptions are included and is
computed for all bi-free central limit distributions. Moreover, an orbital
version of bi-free entropy is examined which provides a tighter upper bound for
the subadditivity of microstate bi-free entropy and provides an alternate
characterization of bi-freeness in certain settings.
