Non-Threshold Quantum Secret Sharing Schemes in the Graph State Formalism

TL;DR

This paper extends the graph state formalism for quantum secret sharing to general access structures, establishing equivalences with quantum codes and enabling the construction of diverse QSS schemes.

Contribution

It introduces methods to realize arbitrary access structures in graph state QSS and links quantum codes to secret sharing schemes.

Findings

Equivalence between $[[n,1]]$ binary quantum codes and graph state QSS.

Restricted equivalence between CSS codes and graph state QSS.

Construction of quantum secret sharing schemes with arbitrary access structures.

Abstract

In a recent work, Markham and Sanders have proposed a framework to study quantum secret sharing (QSS) schemes using graph states. This framework unified three classes of QSS protocols, namely, sharing classical secrets over private and public channels, and sharing quantum secrets. However, most work on secret sharing based on graph states focused on threshold schemes. In this paper, we focus on general access structures. We show how to realize a large class of arbitrary access structures using the graph state formalism. We show an equivalence between $[[n,1]]$ binary quantum codes and graph state secret sharing schemes sharing one bit. We also establish a similar (but restricted) equivalence between a class of $[[n,1]]$ Calderbank-Shor-Steane (CSS) codes and graph state QSS schemes sharing one qubit. With these results we are able to construct a large class of quantum secret sharing schemes with arbitrary access structures.