# A conceptual viewpoint on information decomposition

### APA

Perrone, P. (2016). A conceptual viewpoint on information decomposition. Perimeter Institute. https://pirsa.org/16110030

### MLA

Perrone, Paolo. A conceptual viewpoint on information decomposition. Perimeter Institute, Nov. 08, 2016, https://pirsa.org/16110030

### BibTex

@misc{ pirsa_16110030, doi = {10.48660/16110030}, url = {https://pirsa.org/16110030}, author = {Perrone, Paolo}, keywords = {Quantum Foundations}, language = {en}, title = {A conceptual viewpoint on information decomposition}, publisher = {Perimeter Institute}, year = {2016}, month = {nov}, note = {PIRSA:16110030 see, \url{https://pirsa.org}} }

Paolo Perrone Max Planck Institute for Mathematics in the Sciences

## Abstract

Can we decompose the information of a composite system into terms arising from its parts and their interactions?

For a bipartite system (X,Y), the joint entropy can be written as an algebraic sum of three terms: the entropy of X alone, the entropy of Y alone, and the mutual information of X and Y, which comes with an opposite sign. This suggests a set-theoretical analogy: mutual information is a sort of "intersection", and joint entropy is a sort of "union".

The same picture cannot be generalized to three or more parts in a straightforward way, and the problem is still considered open. Is there a deep reason for why the set-theoretical analogy fails?

Category theory can give an alternative, conceptual point of view on the problem. As Shannon already noted, information appears to be related to symmetry. This suggests a natural lattice structure for information, which is compatible with a set-theoretical picture only for bipartite systems.

The categorical approach favors objects with a structure in place of just numbers to describe information quantities. We hope that this can clarify the mathematical structure underlying information theory, and leave it open to wider generalizations.