Quantum Secret Sharing and Error Correction vs No-Cloning

2026-09-30

AI Summary

This study addresses the fundamental limitation imposed by the quantum no-cloning theorem on the access structures of quantum secret sharing (QSS), which constrains them to conventional threshold schemes. To overcome this barrier, we propose a novel paradigm that reconstructs a single quantum state from multiple input copies. By introducing the graph-theoretic obstacle of “clique paths” and integrating k-colorability analysis with quantum error-correcting code theory, we demonstrate that multi-copy mechanisms can partially circumvent no-cloning restrictions. Specifically, we establish tight bounds for multi-copy QSS access structures and construct new protocols that transcend traditional threshold constraints. Furthermore, we reveal that quantum error-correcting codes operating over erasure channels are subject to analogous physical limitations, thereby deepening the theoretical understanding of the fundamental security boundaries in quantum information.

Abstract

Secret sharing is ubiquitous throughout cryptography. All possible access structures are classically feasible, and in the case of threshold access structures, the protocols are even very efficient. However, when moving to the quantum setting, the no-cloning theorem shows that many access structures are impossible. In fact, no-cloning exactly characterizes feasibility: for thresholds, quantum secret sharing is possible if and only if the threshold $t$ is strictly more than $n/2$. In this work, we propose a variant of quantum secret sharing (QSS) where two or more identical copies of the input state are provided, but the output is only required to recover one copy. This notion circumvents the simple one-copy no-cloning obstruction, though the natural $k$-copy generalization still gives much milder obstructions. For thresholds using $k$ copies, no-cloning implies that QSS is impossible whenever $t\leq n/(k+1)$. It is tempting to hypothesize that no-cloning continues to exactly characterize the many-copy case. However, we show that this is not the case. We give positive results showing that multiple copies allow for going slightly beyond the single-copy obstruction: for thresholds, we construct QSS whenever $t>(n-k+1)/2$. On the other hand, we give a novel obstruction we call the Clique Path obstruction, which applies to arbitrary access structures. For thresholds, it shows that QSS is impossible whenever $t\leq (n-1)/k$. Our upper and lower bounds exactly match for $k=2$. Both our results leverage connections to the $k$-colorability of certain graphs derived from the access structure. We leave closing the gap for $k\geq 3$ copies as a fascinating direction for future work. Secret sharing is closely related to error correction, which can also be considered in the many-copy setting. Our results imply similar obstructions for quantum error correction for erasure channels.

Problem

• Quantum Secret Sharing
• No-Cloning Theorem
• Threshold Access Structures
• Quantum Error Correction
• Multi-copy Setting

Innovation

• Quantum Secret Sharing
• No-Cloning Theorem
• Clique Path Obstruction
• Threshold Access Structures
• Quantum Error Correction