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International Journal of Theoretical Physics

Quantum Secret Sharing Protocol Using Maximally Entangled Multi-qudit States

Mostafa Mansour, Zakaria Dahbi

Published Nov 5, 2020• 35 citations
Publisher
DOI
10.1007/s10773-020-04639-2

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Since the first quantum secret sharing protocol based on Greenberger–Horne–Zeilinger (GHZ) states was presented by Hillery et al, dozens of quantum secret sharing protocols based on different entangled states (e.g. Bell states, d-level Bell states and GHZ states, etc) and product states have been developed. In this paper, we present a quantum secret sharing protocol, in which the genuinely maximally entangled six-qubit states, which was found recently by Borras et al, are used. An ordinary entangled state such as a Bell state or a GHZ state, etc can be used to share at most only one bit. However, according to our protocol, Alice can share three bits among three parties by a six-qubit state, which shows that our protocol is more efficient than any previous protocol.
The no-cloning theorem prohibits the creation of identical copies of quantum information, imposing fundamental constraints on quantum technologies. A recently proposed protocol, encrypted cloning, introduced by Yamaguchi and Kempf, showed that perfect qubit clones can be produced if they are simultaneously encrypted with a single-use key. They also observed a connection between this scheme and quantum secret sharing (QSS). However, it remained an open question whether encrypted cloning could be generalised to arbitrary dimensions, and the broader relationship between the two schemes had not been formally established. In this work, we address both questions by framing encrypted clones as Absolutely Maximally Entangled (AME) states. In parallel with recent work by Ceará that utilises Zadoff-Chu sequences, we independently develop a complementary framework for arbitrary dimensions based on Weyl-Heisenberg displacement operators, both tracing back to the original qubit construction by Yamaguchi and Kempf. We analytically compute the encrypted state and prove that an encrypted qudit system comprising two signal-noise qudit pairs is equivalent to a five-party AME state in any dimension, provided the input state is uniform. We then formalise the connection to QSS by proving that a threshold QSS scheme can achieve the fundamental objectives of encrypted cloning, establishing QSS as the natural general framework within which encrypted cloning can be contextualised.
Quantum secret-sharing is a fundamental technique in quantum communication and plays a crucial role in high-dimensional quantum secret transmission. This paper investigates the teleportation of arbitrary d-dimensional multipartite quantum secrets and proposes both deterministic and probabilistic teleportation protocols. The schemes employ generalized Bell-state measurements, generalized Pauli operators, and auxiliary particles to achieve efficient and secure quantum secret-sharing. The deterministic protocol employs maximally entangled channels to achieve perfect sharing with unit success probability. Moreover, the probabilistic protocol, assisted by auxiliary particles and generalized unitary operations, achieves perfect fidelity quantum secret sharing via non-maximally entangled channels. The protocols are further extended to arbitrary d-dimensional systems, and their robustness under several common quantum noise models is analyzed numerically. Compared with previous schemes, the proposed protocols support arbitrary multipartite entangled states, high-dimensional systems, non-maximally entangled channels, and noise analysis within a unified framework. These results enrich the theory of high-dimensional quantum communication and provide practical approaches for future quantum secret-sharing and secure quantum networks.