The Finite Geometry of Breaking Quantum Secrets

TL;DR

This paper employs finite geometry to unify the study of quantum secret sharing and contextuality, revealing geometric structures that govern entanglement and enabling explicit secret-breaking protocols.

Contribution

It introduces a finite geometric framework for analyzing quantum codes, linking tensor factorizations to geometric structures that elucidate secret sharing and contextuality.

Findings

Finite geometric structures underpin quantum secret sharing schemes.

Explicit secret-breaking protocols are derived for (3,5) and (4,7) schemes.

A novel geometric perspective on contextual configurations is proposed.

Abstract

Using a finite geometric framework for studying the pentagon and heptagon codes we show that the concepts of quantum secret sharing and contextuality can be studied in a nice and unified manner. The basic idea is a careful study of the respective $2+3$ and $3+4$ tensorial factorizations of the elements of the stabilizer groups of these codes. It is demonstrated in detail how finite geometric structures entailing a specific three-qubit (resp. four-qubit) embedding of binary symplectic polar spaces of rank two (resp. three), corresponding to these factorizations, govern issues of contextuality and entanglement needed for a geometric understanding of quantum secret sharing. Using these results for the $(3,5)$ and $(4,7)$ threshold schemes explicit secret breaking protocols are derived. Our results hint at a novel geometric way of looking at contextual configurations.