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An Information Theoretical Model for Quantum Secret Sharing

Hideki Imai, Jörn Müller‐Quade, Anderson C. A. Nascimento, Pim Tuyls, Andreas Winter

Published Jan 1, 2005• 13 citations
OpenAlex

Abstract

Similarly to earlier models for quantum error correcting codes, we introduce a quantuminformation theoretical model for quantum secret sharing schemes. This model providesnew insights into the theory of quantum secret sharing. By using our model, among otherresults, we give a shorter proof of Gottesman's theorem that the size of the shares in aquantum secret sharing scheme must be as large as the secret itself. Also, we introducedapproximate quantum secret sharing schemes and showed robustness of quantum secretsharing schemes by extending Gottesman's theorem to the approximate case.

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Similarly to earlier models for quantum error correcting codes, we introduce a quantuminformation theoretical model for quantum secret sharing schemes. This model providesnew insights into the theory of quantum secret sharing. By using our model, among otherresults, we give a shorter proof of Gottesman's theorem that the size of the shares in aquantum secret sharing scheme must be as large as the secret itself. Also, we introducedapproximate quantum secret sharing schemes and showed robustness of quantum secretsharing schemes by extending Gottesman's theorem to the approximate case.
Similarly to earlier models for quantum error correcting codes, we introduce a quantum information theoretical model for quantum secret sharing schemes. This model provides new insights into the theory of quantum secret sharing. By using our model, among other results, we give a shorter proof of Gottesman's theorem that the size of the shares in a quantum secret sharing scheme must be as large as the secret itself. Also, we introduced approximate quantum secret sharing schemes and showed robustness of quantum secret sharing schemes by extending Gottesman's theorem to the approximate case.
In this paper we introduce a quantum information theoretical model for quantum secret sharing schemes. We show that quantum information theory provides a unifying framework for the study of these schemes. We prove that the information theoretical requirements for a class of quantum secret sharing schemes reduce to only one requirement (the recoverability condition) as a consequence of the no-cloning principle. We give also a shorter proof of the fact that the size of the shares in a quantum secret sharing scheme must be at least as large as the secret itself.
Abstract Quantum secret sharing has wide application prospects in quantum communication, quantum network and other fields. It can provide a secure and reliable information sharing method for multi-party communication. In this paper, we mainly study the connection mechanism between the quantum secret-sharing scheme and quantum error-correcting codes. Firstly, we give the necessary and sufficient conditions for the existence of quantum error-correcting codes based on group algebra theory. Secondly, we give the corresponding quantum secret sharing scheme model with local quantum operation and classical communication capability can be constructed by any given quantum error-correcting codes. The advantage of this model is that the corresponding quantum secret-sharing scheme can be found for any quantum error-correcting code.
We present a general scheme for sharing quantum secrets, and an extension to sharing classical secrets, which contain all known quantum-secret-sharing schemes. In this framework, we show the equivalence of existence of both schemes, that is, the existence of a scheme sharing a quantum secret implies that the extended classical-secret-sharing scheme works, and vice versa. As a consequence of this, we find schemes sharing classical secrets for arbitrary access structures. We then clarify the relationship to quantum error correction and observe several restrictions thereby imposed, which for example indicates that for pure-state threshold schemes the share size qqq must scale with the number of players nnn as $q\ensuremath{\ge}\sqrt{n}$. These results also provide a way of searching for quantum-error-correcting codes.
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. We introduce an information-theoretic model for the rate analysis of QSS and its relation to compound quantum channels, following a similar approach as of Zou et al. (2015) on classical secret sharing. We establish a regularized characterization for the QSS capacity, and determine the capacity for QSS with dephasing noise.