Entropy modulo p and quantum information


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


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


          @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


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