Quantum Secret Sharing Rates
Gabrielle Lalou, Husein Natur, Uzi Pereg
TL;DR
The paper develops an information-theoretic framework for quantum secret sharing over noisy broadcast channels by mapping QSS access structures to a compound quantum channel with an informed decoder. It proves that the QSS capacity equals the quantum capacity of the induced compound channel, characterized by a regularized coherent-information quantity $I_c(\\mathscr{A})$, and demonstrates a closed-form capacity for dephasing channels in a $(3,4)$ QSS. A key technical result is the equality $E(\\mathcal{J}) = Q(\\mathcal{J})$, shown via a teleportation-based argument that removes the need for classical communication. The work reveals that secrecy in QSS arises directly from the no-cloning theorem once reliable reconstruction is ensured, and it lays groundwork for explicit capacity-approaching QSS schemes and extensions to other cryptographic primitives.
Abstract
This paper studies the capacity limits for quantum secret sharing (QSS). The goal of a QSS scheme is to distribute a quantum secret among multiple participants, such that only authorized parties can recover it through collaboration, while no information can be obtained without such collaboration. Following the approach of Zou et al. (2015) on classical secret sharing, we introduce an information-theoretic model for the rate analysis of QSS and its relation to compound quantum channels. We establish a regularized characterization for the QSS capacity, and determine the capacity for QSS with dephasing noise.