# The asymptotic induced matching number of hypergraphs: balanced binary strings

## TL;DR

This paper calculates the asymptotic induced matching number of specific hypergraphs related to balanced binary strings, using advanced algebraic methods, with implications for tensor theory, quantum information, and computational complexity.

## Contribution

It introduces a new lower bound for the asymptotic induced matching number of certain hypergraphs using the higher-order Coppersmith-Winograd method.

## Key findings

- Determines the asymptotic induced matching number for hypergraphs of balanced binary strings.
- Establishes the asymptotic subrank of tensors supported by these hypergraphs.
- Provides an optimal protocol for entanglement distillation in quantum information theory.

## Abstract

We compute the asymptotic induced matching number of the $k$-partite $k$-uniform hypergraphs whose edges are the $k$-bit strings of Hamming weight $k/2$, for any large enough even number $k$. Our lower bound relies on the higher-order extension of the well-known Coppersmith-Winograd method from algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam. Our result is motivated by the study of the power of this method as well as of the power of the Strassen support functionals (which provide upper bounds on the asymptotic induced matching number), and the connections to questions in tensor theory, quantum information theory and theoretical computer science. Phrased in the language of tensors, as a direct consequence of our result, we determine the asymptotic subrank of any tensor with support given by the aforementioned hypergraphs. In the context of quantum information theory, our result amounts to an asymptotically optimal $k$-party stochastic local operations and classical communication (slocc) protocol for the problem of distilling GHZ-type entanglement from a subfamily of Dicke-type entanglement.