Approximate reconstructability of quantum states and noisy quantum secret sharing schemes

TL;DR

This paper formalizes approximate quantum secret sharing, showing that authorized sets can nearly reconstruct the secret while unauthorized sets gain minimal information, based on quantum channel capacities.

Contribution

It introduces a formal cryptographic framework for approximate quantum secret sharing and characterizes reconstructability via entanglement-assisted capacities of quantum channels.

Findings

Approximate reconstructability linked to low information leakage.

Reconstruction condition expressed through quantum channel capacities.

Framework applicable to secure quantum communication protocols.

Abstract

We introduce and analyse approximate quantum secret sharing in a formal cryptographic setting, wherein a dealer encodes and distributes a quantum secret to players such that authorized structures (sets of subsets of players) can approximately reconstruct the quantum secret and omnipotent adversarial agents controlling non-authorized subsets of players are approximately denied the quantum secret. In particular, viewing the map encoding the quantum secret to shares for players in an authorized structure as a quantum channel, we show that approximate reconstructability of the quantum secret by these players is possible if and only if the information leakage, given in terms of a certain entanglement-assisted capacity of the complementary quantum channel to the players outside the structure and the environment, is small.