The Birkhoff theorem for unitary matrices of prime dimension

TL;DR

This paper extends Birkhoff's theorem to unitary matrices of prime dimension, showing that such matrices with equal line sums form a convex set analogous to the doubly stochastic matrices.

Contribution

It proves a Birkhoff-like theorem for unitary matrices of prime dimension, establishing a new geometric characterization in quantum information theory.

Findings

Unitary matrices of prime dimension with equal line sums form a convex set.

The convex polytope's vertices correspond to permutation matrices.

The theorem generalizes classical Birkhoff's theorem to quantum matrices.

Abstract

The Birkhoff's theorem states that any doubly stochastic matrix lies inside a convex polytope with the permutation matrices at the corners. It can be proven that a similar theorem holds for unitary matrices with equal line sums for prime dimensions.