Recently, Yamaguchi and Kempf [Phys. Rev. Lett. 136:010801, arXiv:2501.02757] proved that encrypted qubits can be cloned. In this work, we generalize the encrypted cloning protocol and prove that it also applies to higher-order quantum systems. Given that a straightforward generalization of the protocol using the exponential of the shift and phase operators fails to satisfy the unitary requirement for a quantum gate, we propose a different approach. We introduce a new operator to be used in the encryption process and show that it is unitary. We adapt the decryption operator from the reference paper to fit in the framework of multi-level quantum systems. We analyze the circuit implementation of the proposed operators and show that the overhead imposed by larger dimensions scales linearly with qudit dimension.
Recently, Yamaguchi and Kempf [Phys. Rev. Lett. 136:010801, arXiv:2501.02757] proved that encrypted qubits can be cloned. In this work, we generalize the encrypted cloning protocol and prove that it also applies to higher-order quantum systems. Given that a straightforward generalization of the protocol using the exponential of the shift and phase operators fails to satisfy the unitary requirement for a quantum gate, we propose a different approach. We introduce a new operator to be used in the encryption process and show that it is unitary. We adapt the decryption operator from the reference paper to fit in the framework of multi-level quantum systems. We analyze the circuit implementation of the proposed operators and show that the overhead imposed by larger dimensions scales linearly with qudit dimension.
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.
We show that encrypted cloning of unknown quantum states is possible. Any number of encrypted clones of a qubit can be created through a unitary transformation, and each of the encrypted clones can be decrypted through a unitary transformation. The decryption of an encrypted clone consumes the decryption key; i.e., only one decryption is possible, in agreement with the no-cloning theorem. Encrypted cloning represents a new paradigm that provides a form of redundancy, parallelism, or scalability where direct duplication is forbidden by the no-cloning theorem. For example, a possible application of encrypted cloning is to enable encrypted quantum multicloud storage.
The no-cloning theorem forbids the creation of identical copies of qubits, thereby imposing strong limitations on quantum technologies. A recently-proposed protocol, encrypted cloning, showed, however, that the creation of perfect clones is theoretically possible - if the clones are simultaneously encrypted with a single-use decryption key. It has remained an open question, however, whether encrypted cloning is stable under hardware noise and thus practical as a quantum primitive. This is nontrivial because spreading quantum information widely could dilute it until barely exceeding the noise level, leading to catastrophic fidelity decay. Given the complexity of hardware noise, theory and classical simulation are insufficient to settle this. Here, we settle this question experimentally, on IBM Heron-R2 superconducting processors using up to 154 qubits. We find that encrypted cloning is stable under hardware noise, even when used as a module, namely in parallel, series or interleaved, while preserving pre-existing entanglement. This establishes it as a versatile quantum primitive for practical use, and it necessitates a refinement to our understanding of the no-cloning theorem: quantum information can be spread at will, in theory and in practice, without dilution or degradation, if encrypted or obscured. The actual constraint is that the decryption mechanism must be single-use.
Not all unitary operations upon a set of qubits can be implemented by sequential interactions between each qubit and an ancillary system. We analyze the specific case of sequential quantum cloning, 1-->M, and prove that the minimal dimension D of the ancilla grows linearly with the number of clones M. In particular, we obtain D=2M for symmetric universal quantum cloning and D=M+1 for symmetric phase-covariant cloning. Furthermore, we provide a recipe for the required ancilla-qubit interactions in each step of the sequential procedure for both cases.
Encrypted cloning offers a means of introducing redundancy into quantum storage while respecting the no-cloning theorem: an unknown state is encoded into multiple signal-noise pairs, and only authorized subsets can recover the original information. However, the leakage properties of unauthorized subsets particularly for higher-dimensional systems (qudits) have remained unexplored. In this work, we systematically classify the informative subsets of the storage register in the qudit encrypted-cloning protocol. We focus on unauthorized subsets of size n that contain exactly one qudit from each signal-noise pair. We show that the presence or absence of information leakage is determined by the solution set of a system of congruences whose coefficients depend on the dimension d and on the numbers of signal and noise qudits in the subset. The reduced state is completely uninformative if and only if the congruence system admits only the trivial solution; otherwise, it retains a residual dependence on the input state through specific generalized Pauli operators. Low-dimensional examples (n=1,2,3) are worked out explicitly, and the complete classification is expressed in terms of a greatest-common-divisor condition. Our results extend the parity-based classification known for qubits (d=2) to arbitrary finite dimensions, revealing a dimension-dependent boundary of confidentiality in encrypted cloning.
Quantum encrypted cloning, introduced by Yamaguchi and Kempf, is a Pauli-based protocol that distributes an unknown input qubit into multiple encrypted signal-noise pairs in such a way that redundancy is created without violating the no-cloning theorem, since at most one clone can later be perfectly recovered through an appropriate decoding procedure. In previous work we showed that unauthorized subsets of the storage register are not, in general, completely uninformative, and we identified a parity-dependent leakage pattern. In the present work we extend the analysis to subsets that also include the transformed source qubit A. Exploiting the purity of the global encoded state and the complementarity between storage-only subsets and subsets containing A, we derive a full classification of the informativeness of all sets of the form H={A}∪C. We show that these subsets are fully informative in the generic case. Two exceptions arise. First, if all pairs are incomplete and |C|<n, then the reduced state is completely uninformative. Second, if |C|=n, n is odd, and the number q of signal qubits in C is even, then the reduced state is partially informative. In this latter case, the residual dependence on the input state is confined to the y-component of the Bloch vector. These results provide a complete parity-based characterization of leakage for subsets containing the transformed input qubit.