Analogues of Entropy in Bi-Free Probability Theory: Microstates

Abstract

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.

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

“…Note this agrees with the microstate bi-free entropy of ({S k } n k=1 , {S k } n+m k=n+1 ) obtained in [7] and that n+m 2 log(2πe) is n + m times the free entropy of a single semicircular operator with variance one.…”

“…In Section 6 we define the non-microstate bi-free entropy (see Definition 6.1) as an integral of the Fisher information of perturbations by the independent bi-free Brownian motion. The non-microstate bi-free entropy of every self-adjoint bi-free central limit distribution is computed and agrees with the microstate bi-free entropy as seen by [7]. Furthermore, natural properties desired for an entropy theory are demonstrated for the non-microstate bi-free entropy and a lower bound based on the non-microstate free entropy of the system obtained by modifying all right variables to be left variables is obtained.…”

“…Of course, the most natural question is Question 9.3. Does the microstate bi-free entropy from [7] agree with the above non-microstate bi-free entropy for tracially bi-partite collections?…”

“…We note that we have used a specific bi-free Brownian motion in Definition 6.1, namely the one defined by completely independent bi-free central limit distributions. This appears to be the optimal choice as this choice of bi-free central limit distribution has the maximal microstate bi-free entropy among all bi-free central limit distributions (see [7]) and minimizes the inequality in the bi-free Cramer-Rao inequality (Proposition 5.10). We note other non-microstate bi-free entropies are possible by selecting different bi-free Brownian motions.…”

“…As a diagrammatical view of the free conjugate variables is possible using non-crossing partitions, in this paper we extend this diagrammatical view using [4,5] to develop a notion of non-microstate bi-free entropy. In our sister paper [7] a notion of microstate bi-free entropy is developed.…”