Quantum ramp secret sharing from Haar scrambling

TL;DR

This paper establishes an equivalence between Haar scrambling and quantum secret sharing, revealing a ramp secret-sharing scheme that can be efficiently implemented and has applications in quantum cryptography, distributed networks, and fundamental physics.

Contribution

It demonstrates the equivalence between Haar scrambling and quantum secret sharing, leading to a new ramp secret-sharing scheme with broad applications.

Findings

Haar scrambling is equivalent to a ramp secret-sharing scheme.

Varying initial state purity yields different secret-sharing types.

The protocol can be efficiently implemented in polynomial time.

Abstract

Quantum information scrambling has emerged as a powerful tool for studying the dynamics of chaotic quantum many-body systems, assessing benchmarking protocols, and even investigating exotic black hole models. During quantum information scrambling, localized quantum information disperses across the entire system, hiding from the observers who can only access part of it. On the other side, we have a fundamental cryptographic primitive called secret-sharing schemes, where a dealer shares a quantum secret with a group of parties, such that any subset of parties above a specific threshold size can reconstruct it. In this paper, we demonstrate that two protocols, Haar scrambling, which serves as a baseline for other scrambling techniques, and quantum secret sharing, are indeed equivalent. The scheme one gets out of this is of a ramp secret-sharing nature rather than a threshold scheme. We expect this because of the inherent randomness present in Haar unitaries. Moreover, by varying the purity of the initial states, we demonstrate that it is possible to obtain all possible types of ramp secret-sharing schemes. Furthermore, utilizing complexity theoretic arguments, we argue that the protocol can be implemented efficiently, i.e., in polynomial time. Finally, we explore potential applications, ranging from security implications for distributed quantum networks to cryptography in the Noisy Intermediate-Scale Quantum (NISQ) era, and draw insights for fundamental physics, such as many-body physics and quantum black holes.