Sharper bounds improve understanding of complex Grothendieck constant
Sharper Bounds for the Complex Grothendieck Constant
Data Structures and Algorithms
Summary
There is a special number called the complex Grothendieck constant that helps measure how certain mathematical inequalities behave when complex numbers are involved. This paper improves the known range where this number lies by giving a tighter upper and lower limit. The authors used clever ideas from related mathematical cases to make the upper limit smaller than the best known before. This helps make mathematical comparisons involving complex numbers more precise.
What this means in practice
- •For quantum algorithm developers: Use tighter bounds on the complex Grothendieck constant to improve performance guarantees in quantum information protocols involving complex vector spaces.
- •For optimization engineers: Refine semidefinite programming relaxations in complex-valued optimization problems based on improved constant bounds.
A theory result. No direct application yet.
Authors
Steven Heilman, Chris Jones, Giulio Malavolta
Abstract
We show that $1.4<K_G^{\mathbb{C}}<1.404898554746$, where $K_G^{\mathbb{C}}$ is the complex Grothendieck constant. The upper bound improves on Haagerup's bound of $1.40490913\ldots$ from 1987, and the lower bound improves on Davie's bound of $1.33807$ from 1984. The upper bound combines ideas from the real and noncommutative complex cases of Grothendieck's inequality.