quantum-computing

Towards Unconditional Uncloneable Encryption

Pierre Botteron, Anne Broadbent, Eric Culf, Ion Nechita, Clément Pellegrini, and Denis Rochette

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A team led by Pierre Botteron, Anne Broadbent, and Eric Culf has proposed the first candidate for unconditional uncloneable encryption of a single bit, a long-standing open problem in quantum cryptography. Their approach uses quantum states constructed from anti-commuting Pauli operators to encrypt a classical message, ensuring that two adversaries cannot simultaneously decrypt it with high probability. The adversaries' success rate is shown to converge quadratically as 1/2 + 1/(2√K), where K is the number of keys. The team proved this bound for K between 2 and 7, verified it numerically up to K=17 using the NPA hierarchy, and established an asymptotic upper bound of 5/8, with a numerical upper bound of approximately 0.5980, the best known in the unconditional model.

Why it matters

This result advances the theoretical foundation of quantum cryptography by demonstrating a near-optimal unconditional security guarantee for uncloneable encryption, a primitive that could underpin future quantum-secure communication protocols without relying on computational assumptions.

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AbstractUncloneable encryption is a cryptographic primitive which encrypts a classical message into a quantum ciphertext, such that two quantum adversaries are limited in their capacity of being able to simultaneously decrypt, given the key and quantum side-information produced from the ciphertext. Since its initial proposal and scheme in the random oracle model by Broadbent and Lord [TQC 2020], uncloneable encryption has developed into an important primitive at the foundation of quantum uncloneability for cryptographic primitives. Despite sustained efforts, however, the question of unconditional uncloneable encryption (and in particular of the simplest case, called an uncloneable bit) has remained elusive. Here, we propose a candidate for the unconditional uncloneable bit problem, and provide strong evidence that the adversary's success probability in the related security game converges quadratically as ${1}/{2}+{1}/{(2\sqrt{K})}$, where $K$ represents the number of keys and ${1}/{2}$ is trivially achievable. We prove this bound's validity for $K$ ranging from $2$ to $7$ and demonstrate the validity up to $K = 17$ using computations based on the NPA hierarchy. We furthemore provide compelling heuristic evidence towards the general case. In addition, we prove an asymptotic upper bound of ${5}/{8}$ and give a numerical upper bound of $\sim 0.5980$, which to our knowledge is the best-known value in the unconditional model.Featured image: No-Cloning Game for a 1-Bit Message: Alice ($A$) encrypts a uniformly random message $m \in \{0,1\}$ using a classical key $k\in\{1,..,K\}$ into a quantum state $\rho_{m,k}\in\mathcal{B}(\mathcal{H}_A)$. She transmits it to a pirate ($P$) modeled by a CPTP map $\Phi: \mathcal{B}(\mathcal{H}_A) \to \mathcal{B}(\mathcal{H}_B \otimes \mathcal{H}_C)$. Bob ($B$) and Charlie ($C$) are then given their respective registers $\mathcal{H}_B$ and $\mathcal{H}_C$, as well as a copy of the key $k$. They output $m_{B}$, $m_{C} \in \{0, 1\}$, respectively, and collaboratively win if and only if $m_B\!=\!m_C\!=\!m$. Uncloneable-indistinguishable security holds if the winning probability is upper-bounded by $\frac{1}{2} + \text{negl}(\lambda)$ for some security parameter $\lambda$.Popular summaryThis paper explores a fundamental question in quantum cryptography: can we create encrypted information that cannot be copied, even by attackers with unlimited computational power? While ordinary digital information can always be duplicated perfectly, quantum mechanics offers the possibility of "uncloneable" encryption—messages encoded in quantum states that resist copying by design. The authors focus on the most elementary version of this challenge: protecting a single bit (0 or 1) by encrypting it into a quantum cyphertext, so that if an attacker tries to split the quantum ciphertext between two collaborators, both cannot successfully recover the message simultaneously (or with negligeable probabilty). They introduce a new candidate protocol based on special quantum states built from anti-commuting Pauli operators, a structure closely related to Clifford algebra. The goal is to make the best attack strategy no better than random guessing, up to a vanishing advantage. They prove this security exactly for small instances, verify it numerically for larger ones, and provide strong evidence that the adversaries' success approaches the ideal limit as the encoding size grows. Their results also establish the best known upper bounds for unconditional uncloneable encryption in the standard model. This work brings the field closer to a long-standing open problem: achieving fully unconditional uncloneable encryption without relying on setup assumptions.► BibTeX data@article{Botteron2026towards, doi = {10.22331/q-2026-07-08-2157}, url = {https://doi.org/10.22331/q-2026-07-08-2157}, title = {Towards {U}nconditional {U}ncloneable {E}ncryption}, author = {Botteron, Pierre and Broadbent, Anne and Culf, Eric and Nechita, Ion and Pellegrini, Cl{\'{e}}ment and Rochette, Denis}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2157}, month = jul, year = {2026} }► References [1] P. Ananth and A. Behera. A modular approach to unclonable cryptography. In Advances in Cryptology — CRYPTO 2024, volume 7, pages 3–37, 2024. DOI: 10.1007/978-3-031-68394-7_1. https://doi.org/10.1007/978-3-031-68394-7_1 [2] G. Alagic, A. Broadbent, B. Fefferman, T. Gagliardoni, C. Schaffner, and M. St. Jules. Computational security of quantum encryption.

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University of Chicago Press, 1990.Cited by[1] Pierre Botteron, "Nonlocal Games Through Communication Complexity and Quantum Cryptography", arXiv:2510.09457, (2025). [2] Anne Broadbent, Eric Culf, and Denis Rochette, "Optimal Untelegraphable Encryption and Implications for Uncloneable Encryption", arXiv:2510.00903, (2025). [3] Archishna Bhattacharyya and Eric Culf, "Uncloneable encryption from decoupling", Nature Physics 22 2, 315 (2026). [4] Andrea Coladangelo, Qipeng Liu, and Ziyi Xie, "The curious case of "XOR repetition" of monogamy-of-entanglement games", arXiv:2509.01831, (2025). [5] Archishna Bhattacharyya, Anne Broadbent, and Eric Culf, "The uncloneable bit exists", arXiv:2603.08916, (2026). [6] Prabhanjan Ananth, "Classically impossible cryptography: Quantum encryption", Nature Physics 22 2, 184 (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-07-08 19:18:19). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-07-08 19:18:17: Could not fetch cited-by data for 10.22331/q-2026-07-08-2157 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractUncloneable encryption is a cryptographic primitive which encrypts a classical message into a quantum ciphertext, such that two quantum adversaries are limited in their capacity of being able to simultaneously decrypt, given the key and quantum side-information produced from the ciphertext. Since its initial proposal and scheme in the random oracle model by Broadbent and Lord [TQC 2020], uncloneable encryption has developed into an important primitive at the foundation of quantum uncloneability for cryptographic primitives. Despite sustained efforts, however, the question of unconditional uncloneable encryption (and in particular of the simplest case, called an uncloneable bit) has remained elusive. Here, we propose a candidate for the unconditional uncloneable bit problem, and provide strong evidence that the adversary's success probability in the related security game converges quadratically as ${1}/{2}+{1}/{(2\sqrt{K})}$, where $K$ represents the number of keys and ${1}/{2}$ is trivially achievable. We prove this bound's validity for $K$ ranging from $2$ to $7$ and demonstrate the validity up to $K = 17$ using computations based on the NPA hierarchy. We furthemore provide compelling heuristic evidence towards the general case. In addition, we prove an asymptotic upper bound of ${5}/{8}$ and give a numerical upper bound of $\sim 0.5980$, which to our knowledge is the best-known value in the unconditional model.Featured image: No-Cloning Game for a 1-Bit Message: Alice ($A$) encrypts a uniformly random message $m \in \{0,1\}$ using a classical key $k\in\{1,..,K\}$ into a quantum state $\rho_{m,k}\in\mathcal{B}(\mathcal{H}_A)$. She transmits it to a pirate ($P$) modeled by a CPTP map $\Phi: \mathcal{B}(\mathcal{H}_A) \to \mathcal{B}(\mathcal{H}_B \otimes \mathcal{H}_C)$. Bob ($B$) and Charlie ($C$) are then given their respective registers $\mathcal{H}_B$ and $\mathcal{H}_C$, as well as a copy of the key $k$. They output $m_{B}$, $m_{C} \in \{0, 1\}$, respectively, and collaboratively win if and only if $m_B\!=\!m_C\!=\!m$. Uncloneable-indistinguishable security holds if the winning probability is upper-bounded by $\frac{1}{2} + \text{negl}(\lambda)$ for some security parameter $\lambda$.Popular summaryThis paper explores a fundamental question in quantum cryptography: can we create encrypted information that cannot be copied, even by attackers with unlimited computational power? While ordinary digital information can always be duplicated perfectly, quantum mechanics offers the possibility of "uncloneable" encryption—messages encoded in quantum states that resist copying by design. The authors focus on the most elementary version of this challenge: protecting a single bit (0 or 1) by encrypting it into a quantum cyphertext, so that if an attacker tries to split the quantum ciphertext between two collaborators, both cannot successfully recover the message simultaneously (or with negligeable probabilty). They introduce a new candidate protocol based on special quantum states built from anti-commuting Pauli operators, a structure closely related to Clifford algebra. The goal is to make the best attack strategy no better than random guessing, up to a vanishing advantage. They prove this security exactly for small instances, verify it numerically for larger ones, and provide strong evidence that the adversaries' success approaches the ideal limit as the encoding size grows. Their results also establish the best known upper bounds for unconditional uncloneable encryption in the standard model. This work brings the field closer to a long-standing open problem: achieving fully unconditional uncloneable encryption without relying on setup assumptions.► BibTeX data@article{Botteron2026towards, doi = {10.22331/q-2026-07-08-2157}, url = {https://doi.org/10.22331/q-2026-07-08-2157}, title = {Towards {U}nconditional {U}ncloneable {E}ncryption}, author = {Botteron, Pierre and Broadbent, Anne and Culf, Eric and Nechita, Ion and Pellegrini, Cl{\'{e}}ment and Rochette, Denis}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2157}, month = jul, year = {2026} }► References [1] P. Ananth and A. Behera. A modular approach to unclonable cryptography. In Advances in Cryptology — CRYPTO 2024, volume 7, pages 3–37, 2024. DOI: 10.1007/978-3-031-68394-7_1. https://doi.org/10.1007/978-3-031-68394-7_1 [2] G. Alagic, A. Broadbent, B. Fefferman, T. Gagliardoni, C. Schaffner, and M. St. Jules. Computational security of quantum encryption.

Hadronic Journal Supplement, 8: 329–345, 1993. [51] J. Watrous. The Theory of Quantum Information.

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