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Physica Scripta

Quantum secret sharing based on quantum codes

G.H. Wang, Yucheng Chen

Published Oct 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ae126a

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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.
Quantum secret sharing(QSS) is a procedure of sharing classical information or quantum information by using quantum states. This paper presents how to use a [2 k − 1, 1, k ] quantum error-correcting code (QECC) to implement a quantum ( k , 2 k − 1) threshold scheme. It also takes advantage of classical enhancement of the [2 k − 1, 1, k ] QECC to establish a QSS scheme which can share classical information and quantum information simultaneously. Because information is encoded into QECC, these schemes can prevent intercept-resend attacks and be implemented on some noisy channels.
Quantum secret-sharing and quantum error-correction schemes rely on multipartite decoding protocols, yet the nonlocal operations involved are challenging and sometimes infeasible. Here we construct a quantum secret-sharing protocol with a reduced number of quantum communication channels between the players. We introduce a scheme based on embedding a classical linear code into a quantum error-correcting code and then mapping the latter to a quantum secret-sharing protocol. In contrast to the Calderbank-Shor-Steane construction, we do not impose any restriction on the classical code; our protocol works with any arbitrary linear code. Our work paves the way towards the more general problem of simplifying the decoding of quantum error-correcting codes.
Quantum secret sharing is a fundamental technique in quantum cryptography. However, in practical quantum networks, it still faces several bottlenecks, such as high quantum communication cost and low transmission efficiency. To reduce the communication cost, ramp quantum secret sharing schemes have been proposed in existing studies. Nevertheless, intermediate sets in such schemes may leak partial information about the secret. To address this problem, this paper presents a method for detecting intermediate sets. Based on this method, we further propose a communication-efficient perfect quantum secret sharing scheme with eavesdropping detection capability.We analyze the communication cost under different numbers of participating parties and determine the range of participants that minimizes the reconstruction communication cost. The results verify the communication efficiency and security of the proposed scheme.
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.
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.
Secret sharing of a quantum state, or quantum secret sharing, in which a dealer wants to share certain amount of quantum information with a few players, has wide applications in quantum information.The critical criterion in a threshold secret sharing scheme is confidentiality; with less than the designated number of players, no information can be recovered.Furthermore, in a quantum scenario, one additional critical criterion exists; the capability of sharing entangled and unknown quantum information.Here by employing a six-photon entangled state, we demonstrate a quantum threshold scheme, where the shared quantum secrecy can be efficiently reconstructed with a state fidelity as high as 93%.By observing that any one or two parties cannot recover the secrecy, we show that our scheme meets the confidentiality criterion.Meanwhile, we also demonstrate that entangled quantum information can be shared and recovered via our setting, which demonstrates that our implemented scheme is fully quantum.Moreover, our experimental setup can be treated as a decoding circuit of the 5-qubit quantum error-correcting code with two erasure errors.