Generalized Semi-Quantum Secret Sharing Schemes

TL;DR

This paper develops a systematic framework for quantum secret sharing using stabilizer codes, enabling the conversion of ramp schemes into perfect schemes with minimal classical communication.

Contribution

It introduces a method to determine access structures and uses the information group to transform ramp schemes into perfect secret sharing schemes efficiently.

Findings

Provides a systematic way to determine access structures.

Introduces the information group concept for quantifying intermediate information.

Enables conversion of ramp schemes into perfect schemes with minimal classical bits.

Abstract

We investigate quantum secret sharing schemes constructed from $[[n,k,\delta]]_D$ non-binary stabilizer quantum error correcting codes with carrier qudits of prime dimension $D$. We provide a systematic way of determining the access structure, which completely determines the forbidden and intermediate structures. We then show that the information available to the intermediate structure can be fully described and quantified by what we call the *information group*, a subgroup of the Pauli group of $k$ qudits, and employ this group structure to construct a method for hiding the information from the intermediate structure via twirling of the information group and sharing of classical bits between the dealer and the players. Our scheme allows the transformation of a ramp (intermediate) quantum secret sharing scheme into a semi-quantum perfect secret sharing scheme with the same access structure as the ramp one but without any intermediate subsets, and is optimal in the amount of classical bits the dealer has to distribute.