Abstract
Building on work of Kontsevich, we introduce a definition of the entropy of a finite probability distribution in which the "probabilities" are integers modulo a prime p. The entropy, too, is an integer mod p. Entropy mod p is shown to be uniquely characterized by a functional equation identical to the one that characterizes ordinary Shannon entropy. We also establish a sense in which certain real entropies have residues mod p, connecting the concepts of entropy over R and over Z/pZ. Finally, entropy mod p is expressed as a polynomial which is shown to satisfy several identities, linking into work of Cathelineau, Elbaz-Vincent and Gangl on polylogarithms.
Original language | English |
---|---|
Pages (from-to) | 279 – 314 |
Number of pages | 28 |
Journal | Communications in Number Theory and Physics |
Volume | 15 |
Issue number | 2 |
DOIs | |
Publication status | Published - 18 Jun 2021 |