Encrypted Cloning, Absolute Maximal Entanglement and Quantum Secret Sharing

Abstract: 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.

• The paper introduces a dimension-independent quantum encrypted cloning protocol using Weyl-Heisenberg operators, ensuring maximal mixing in local registers.
• It establishes that the generated states for two signal-noise qudit pairs are equivalent to AME(5,d) states linked to optimal quantum error-correcting codes.
• The work formally connects encrypted cloning with quantum secret sharing, demonstrating that only authorized parties can recover the original quantum state.

Encrypted Cloning, AME States, and Quantum Secret Sharing: A Technical Analysis

Overview and Motivation

The study addresses two fundamental open questions in quantum information theory: the extension of encrypted cloning protocols to arbitrary dimensions and the formal relationship between encrypted cloning and quantum secret sharing (QSS). The no-cloning theorem precludes perfect copying of quantum information, limiting quantum communication and computation primitives. Recently, encrypted cloning, as proposed for qubits by Yamaguchi and Kempf, allows perfect copies of quantum states to be generated if simultaneously encrypted with a single-use classical key, suggesting applications in secure distributed quantum computing and storage. However, extension to higher-dimensional systems (qudits) and the formalization of its connection to multipartite entanglement and QSS were unresolved.

This work generalizes encrypted cloning to arbitrary dimension $d$ via a construction based on Weyl-Heisenberg displacement operators and establishes a rigorous algebraic connection between encrypted cloning, absolutely maximally entangled (AME) states, and QSS schemes (2605.26866). The derived formalism enables encrypted cloning for any finite dimension, shows that the resulting states for two signal-noise qudit pairs correspond to AME$(5,d)$ states, and situates encrypted cloning as a constrained instance of QSS, highlighting their theoretical equivalence for certain parameter regimes.

Generalization of Encrypted Cloning to Arbitrary Dimensions

The authors present a dimension-agnostic formulation of encrypted cloning by replacing Pauli operators in the original qubit protocol with the Weyl-Heisenberg displacement operators $W(a,b)=\tau^{ab}X^a Z^b$, providing the necessary operator algebra for qudit systems. Two critical unitaries are defined:

• Encryption unitary $U_{\mathrm{enc}}^{(d,n)}$ distributes the input state's information across $n$ signal qudits, ensuring that any local reduced state is maximally mixed and thus information-theoretically concealed.
• Decryption unitary $U_{\mathrm{dec}}^{(d,n)}$ allows perfect recovery of the original state from any signal qudit accompanied by all noise qudits (the decryption key).

Key technical assertions include:

• The scheme reproduces the original qubit protocol ($d=2$) as a special case.
• For $n$ signal-noise qudit pairs, the reduced state of any single register among $A, S_1, \ldots, S_n$ is always $\mathbb{I}_d/d$, enforcing security.
• Only joint access to a signal qudit and all noise qudits enables perfect recovery via the decryption unitary, respecting the no-cloning theorem.

The use of Weyl-Heisenberg operators is justified both by their completeness as a basis on $(5,d)$0 and their algebraic properties (e.g., transpose, inversion, phase structure) being conducive to analyzing entanglement and decryption requirements.

AME States, Error Correction, and Encrypted Cloning

A central result of the paper is the identification of the encrypted state originating from two signal-noise qudit pairs ($(5,d)$1) and a uniform input as an absolutely maximally entangled state $(5,d)$2. Formally, this state exhibits maximal entanglement for any bipartition, guaranteeing every $(5,d)$3-party reduced density matrix is maximally mixed, i.e., $(5,d)$4 for $(5,d)$5.

This equivalence is not only of foundational interest; it provides immediate implications:

• The constructed states realize optimal quantum information sharing and robustness to loss: any reduction to fewer than $(5,d)$6 subsystems yields no information about the original input.
• There is a direct correspondence between $(5,d)$7 and optimal $(5,d)$8 quantum error-correcting codes, enabling the protocol's direct use in quantum memory and distributed quantum computing.
• For $(5,d)$9, the protocol cannot generate AME$W(a,b)=\tau^{ab}X^a Z^b$0 states because the Schmidt rank across balanced bipartitions is upper-bounded by $W(a,b)=\tau^{ab}X^a Z^b$1.

This direct mapping to error-correcting codewords, supported by results in the literature [e.g., see Phys. Rev. A 69, 052330], highlights the protocol's utility for robust quantum information storage and processing.

Formal Connection to Quantum Secret Sharing

The manuscript proves, using both operator algebra and the Choi-Jamilkowski isomorphism, that the encrypted cloning protocol for $W(a,b)=\tau^{ab}X^a Z^b$2 signal-noise qudit pairs is equivalent to a pure state QSS $W(a,b)=\tau^{ab}X^a Z^b$3 threshold scheme for $W(a,b)=\tau^{ab}X^a Z^b$4-dimensional secrets and shares. The construction proceeds as follows:

• By partially encrypting a $W(a,b)=\tau^{ab}X^a Z^b$5-dimensional Bell state (encryption on one register only), one generates an $W(a,b)=\tau^{ab}X^a Z^b$6 state, which via known results [Phys. Rev. A 86, 052335] establishes a bijective mapping to QSS threshold schemes.
• Any authorized set (size $W(a,b)=\tau^{ab}X^a Z^b$7) can recover the secret, and any unauthorized set (size $W(a,b)=\tau^{ab}X^a Z^b$8) gains no information, consistent with QSS access/adversary structure requirements.

Crucially, the analysis demonstrates that encrypted cloning is a strict subset of QSS: the latter allows more general access structures, whereas encrypted cloning restricts decryption to specified noise qudits. Thus, QSS provides a unifying and more general operational framework within which encrypted cloning is naturally embedded.

Strong Technical Results and Key Claims

• Dimension independence: The encrypted cloning protocol and its mapping to AME states and QSS are rigorously shown for arbitrary $W(a,b)=\tau^{ab}X^a Z^b$9, not only the qubit case.
• Maximal hiding: For $U_{\mathrm{enc}}^{(d,n)}$0, the protocol guarantees all two-party marginals are maximally mixed, substantiated by explicit computation of all $U_{\mathrm{enc}}^{(d,n)}$1-qudit marginals and by a Schmidt rank argument.
• Inability to generalize AME structure for $U_{\mathrm{enc}}^{(d,n)}$2: Using rank counting arguments, the authors prove that the protocol cannot yield higher-party AME states for $U_{\mathrm{enc}}^{(d,n)}$3, i.e., the encrypted $U_{\mathrm{enc}}^{(d,n)}$4-party state lacks maximal mixing on $U_{\mathrm{enc}}^{(d,n)}$5-qudit marginals for any $U_{\mathrm{enc}}^{(d,n)}$6.
• Loss-tolerance: The protocol allows (partial) recovery when some noise qudits are lost, by replacing the missing decryption key with an available signal qudit, as established for both $U_{\mathrm{enc}}^{(d,n)}$7 and general $U_{\mathrm{enc}}^{(d,n)}$8.

Implications and Future Directions

Pragmatically, the established formalism provides new approaches for high-dimensional quantum secret sharing, distributed quantum memory, and secure, loss-tolerant multiparty quantum protocols. The explicit operator constructions facilitate analysis and simulation on quantum platforms supporting qudit logic (e.g., superconducting circuits, trapped ions with higher-level structure).

The mapping to AME states opens the door to utilizing encrypted cloning protocols for constructing optimal quantum error-correcting codes in arbitrary dimensions and for engineering entanglement resources for quantum communication.

Potential future research avenues include:

• Detailed analysis of the stabilizer structure of the encrypted states and their utility in fault-tolerant quantum computing.
• Exploration of encrypted qudit cloning protocols as primitives for blind quantum computation, leveraging their key-hiding properties for secure delegated computation.
• Investigation of scheme performance and error thresholds on next-generation superconducting or photonic hardware for practical realization.
• Generalization to access structures beyond threshold schemes, potentially broadening protocol flexibility for secure multiparty computation.

Conclusion

This paper achieves an overview between encrypted cloning, multipartite entanglement theory, and QSS by supplying a rigorous qudit protocol grounded in Weyl-Heisenberg algebra and demonstrating its equivalence to $U_{\mathrm{enc}}^{(d,n)}$9 states and threshold QSS for uniform inputs (2605.26866). These results unify several previously disconnected strands of quantum information theory and lay the groundwork for both theoretical advancement and application to secure, distributed quantum technologies.