Membership in moment polytopes is in NP and coNP

TL;DR

This paper proves that determining membership in moment polytopes for unitary representations of compact Lie groups is in NP and coNP, providing new insights into the computational complexity of problems related to quantum marginal and Kronecker polytopes.

Contribution

It establishes the first non-trivial complexity classification for the moment polytope membership problem, showing it lies in NP and coNP, with implications for quantum and algebraic complexity.

Findings

Membership decision problem is in NP and coNP

Applies to Kronecker polytopes and positivity of stretched Kronecker coefficients

Contrasts with NP-hardness of single Kronecker coefficient positivity

Abstract

We show that the problem of deciding membership in the moment polytope associated with a finite-dimensional unitary representation of a compact, connected Lie group is in NP and coNP. This is the first non-trivial result on the computational complexity of this problem, which naively amounts to a quadratically-constrained program. Our result applies in particular to the Kronecker polytopes, and therefore to the problem of deciding positivity of the stretched Kronecker coefficients. In contrast, it has recently been shown that deciding positivity of a single Kronecker coefficient is NP-hard in general [Ikenmeyer, Mulmuley and Walter, arXiv:1507.02955]. We discuss the consequences of our work in the context of complexity theory and the quantum marginal problem.