# Entropy modulo p and quantum information

### APA

Ozols, M. (2023). Entropy modulo p and quantum information. Perimeter Institute. https://pirsa.org/23010116

### MLA

Ozols, Maris. Entropy modulo p and quantum information. Perimeter Institute, Jan. 27, 2023, https://pirsa.org/23010116

### BibTex

@misc{ pirsa_PIRSA:23010116, doi = {10.48660/23010116}, url = {https://pirsa.org/23010116}, author = {Ozols, Maris}, keywords = {Quantum Information}, language = {en}, title = {Entropy modulo p and quantum information}, publisher = {Perimeter Institute}, year = {2023}, month = {jan}, note = {PIRSA:23010116 see, \url{https://pirsa.org}} }

Maris Ozols University of Amsterdam

## Abstract

Tom Leinster recently introduced a curious notion of entropy modulo p (https://arxiv.org/abs/1903.06961). While entropy has a certain meaning in information theory and physics, mathematically it is simply a function with certain properties. Stating these as axioms, the function is unique. Surprisingly, Leinster shows that a function obeying the same axioms can also be found for "probability distributions" over a finite field, and this function is unique too.

In quantum information, mutually unbiased bases is an important set of measurements and an example of a quantum design. While in odd prime power dimensions their construction is based on a finite field, in dimension 2^n it relies on an unpleasant Galois ring. I will replace this ring by length-2 Witt vectors whose arithmetic involves only finite field operations and Leinster's entropy mod 2. This expresses qubit mutually unbiased bases entirely in terms of a finite field and allows deriving an explicit unitary correspondence between them and the affine plane over this field.

Zoom link: https://pitp.zoom.us/j/94032116379?pwd=TTI1RnByQnFuVHp1MytFUlJxckM4Zz09