Quantum ramp secret sharing from Haar scrambling

Abstract Quantum information scrambling is the phenomenon in which initially localized quantum information becomes delocalized under complex many-body dynamics. This has become a central concept in quantum chaos, quantum information theory, benchmarking protocols, and black-hole physics (Shenglong and Swingle 2022 Scrambling dynamics and out-of-time ordered correlators in quantum many-body systems: a tutorial; Pavan et al 2016 J. High Energy Phys. JHEP02(2016)004). A closely related but operationally distinct notion appears in quantum cryptography: in a quantum secret-sharing scheme, a dealer distributes a quantum secret among several parties so that only sufficiently large subsets can reconstruct it, while smaller subsets remain ignorant of it. In this work, we show that these two notions are connected at the level of quantum access structures. We introduce an information-theoretic criterion, called l -scrambling, in which every subsystem of size at most l is approximately decoupled from the reference system, thereby purifying the input secret. This gives scrambling an operational interpretation: small subsystems are unauthorized sets, while sufficiently large subsystems can recover the encoded quantum information. This establishes a quantitative bridge between scrambling, decoupling, quantum error correction, and ramp secret sharing. Our framework also provides a physically motivated route to approximate quantum ramp secret sharing and, conversely, interprets secret-sharing parameters as quantitative measures of information spreading. These results suggest applications to distributed quantum networks, NISQ-era cryptographic protocols, and the information-theoretic study of many-body quantum systems and black-hole-inspired models.