Matroids and Quantum Secret Sharing Schemes

TL;DR

This paper explores the potential for a matroidal framework to characterize quantum secret sharing schemes, enabling the construction of efficient schemes for various access structures.

Contribution

It introduces the first steps toward a matroidal characterization of quantum secret sharing, linking matroid properties to efficient quantum scheme construction.

Findings

Identically self-dual matroids induce quantum secret sharing schemes.

Representability over finite fields enables efficient quantum schemes.

Provides a new perspective for designing quantum secret sharing protocols.

Abstract

A secret sharing scheme is a cryptographic protocol to distribute a secret state in an encoded form among a group of players such that only authorized subsets of the players can reconstruct the secret. Classically, efficient secret sharing schemes have been shown to be induced by matroids. Furthermore, access structures of such schemes can be characterized by an excluded minor relation. No such relations are known for quantum secret sharing schemes. In this paper we take the first steps toward a matroidal characterization of quantum secret sharing schemes. In addition to providing a new perspective on quantum secret sharing schemes, this characterization has important benefits. While previous work has shown how to construct quantum secret sharing schemes for general access structures, these schemes are not claimed to be efficient. In this context the present results prove to be useful; they enable us to construct efficient quantum secret sharing schemes for many general access structures. More precisely, we show that an identically self-dual matroid that is representable over a finite field induces a pure state quantum secret sharing scheme with information rate one.