Analogues of entropy in bi-free probability theory: Non-microstate

Charlesworth, Ian; Skoufranis, Paul

doi:10.1016/j.aim.2020.107367

“…It is worthy to note that the microstate bi-free entropy for bi-free central limit distributions has the same form as Gaussian distributions with respect to the Shannon entropy and the free central limit distributions with respect to free entropy. Furthermore, the same value is obtained for non-microstate bi-free entropy in our sister paper [9].…”

Section: Introductionsupporting

confidence: 81%

“…In particular, such theory may be of interest in relation to the recent work [13] which develops a random matrix approach to the Peterson-Thom conjecture via tensors of random matrices, a topic which bi-free probability theory provides substantial information on. In our sister paper [9] a notion of non-microstate bi-free entropy is developed.…”

Section: Introductionmentioning

confidence: 99%

Analogues of Entropy in Bi-Free Probability Theory: Microstates

Charlesworth

¹

,

Skoufranis

²

2021

International Mathematics Research Notices

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…It is worthy to note that the microstate bi-free entropy for bi-free central limit distributions has the same form as Gaussian distributions with respect to the Shannon entropy and the free central limit distributions with respect to free entropy. Furthermore, the same value is obtained for non-microstate bi-free entropy in our sister paper [7].…”

Section: Introductionsupporting

confidence: 81%

“…Question 8.7. Does the microstate bi-free entropy from [7] agree with the above non-microstate bi-free entropy for tracially bi-partite collections?…”

Section: Open Questionsmentioning

confidence: 84%

See 1 more Smart Citation

Analogues of Entropy in Bi-Free Probability Theory: Microstates

Charlesworth¹,

Skoufranis²

2019

Preprint

Self Cite

0

View full text Add to dashboard Cite

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 and is computed for all bi-free central limit distributions.

show abstract

Bi-free entropy with respect to completely positive maps

Katsimpas¹,

Skoufranis²

2023

Can. J. Math.-J. Can. Math.

Self Cite

0

View full text Add to dashboard Cite

In this paper, a notion of non-microstate bi-free entropy with respect to completely positive maps is constructed thereby extending the notions of non-microstate bi-free entropy and free entropy with respect to a completely positive map. By extending the operator-valued bi-free structures to allow for more analytical arguments, a notion of conjugate variables is constructed using both moment and cumulant expressions. The notions of free Fisher information and entropy are then extended to this setting and used to show minima of the Fisher information and maxima of the non-microstate bi-free entropy at bi-R-diagonal elements.

show abstract

Analogues of entropy in bi-free probability theory: Non-microstate

Cited by 3 publications

References 22 publications

Analogues of Entropy in Bi-Free Probability Theory: Microstates

Analogues of Entropy in Bi-Free Probability Theory: Microstates

Analogues of Entropy in Bi-Free Probability Theory: Microstates

Bi-free entropy with respect to completely positive maps

Contact Info

Product

Resources

About