Analogues of Entropy in Bi-Free Probability Theory: Microstates

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