Linear rank inequalities on five or more variables

TL;DR

This paper identifies a complete set of linear rank inequalities for five variables, extending the known inequalities for fewer variables, and explores inequalities for six or more variables.

Contribution

It provides a complete list of 24 inequalities for five variables and general results for inequalities on more variables, showing new inequalities emerge beyond four variables.

Findings

24 inequalities generate all linear rank inequalities on five variables

Partial list of inequalities for six variables

Existence of new inequalities for more than four variables

Abstract

Ranks of subspaces of vector spaces satisfy all linear inequalities satisfied by entropies (including the standard Shannon inequalities) and an additional inequality due to Ingleton. It is known that the Shannon and Ingleton inequalities generate all such linear rank inequalities on up to four variables, but it has been an open question whether additional inequalities hold for the case of five or more variables. Here we give a list of 24 inequalities which, together with the Shannon and Ingleton inequalities, generate all linear rank inequalities on five variables. We also give a partial list of linear rank inequalities on six variables and general results which produce such inequalities on an arbitrary number of variables; we prove that there are essentially new inequalities at each number of variables beyond four (a result also proved recently by Kinser).