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arXiv:2607.28602

Pauli Encodings & Unclonable Encryption

Pierre Botteron, Sébastien Designolle, Omar Fawzi

quant-phSubmitted Jul 30, 2026• Announced Jul 31, 2026• 0 citations• Revised Jul 31, 2026 (v2)
v1 · Jul 30, 2026v2 · Jul 31, 2026
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Abstract

The unclonable bit question asks whether quantum encryption can prevent an adversary from producing two systems that both reveal the plaintext once the key is disclosed. We introduce and study Pauli Encodings, a simple class of one-bit encryption schemes whose ciphertexts are normalized eigenspace projectors of Pauli strings. For every Pauli Encoding with K Pauli strings, we prove a universal lower bound $1/2+1/(2\sqrt{K})$ on the optimal monogamy-of-entanglement winning probability, together with sharper bounds for several structured families. We then establish two limitations of natural approaches to unclonable security. First, if the Pauli strings are restricted to strings of X and Z of length n, the encoding is not secure. Second, we identify a universal 3/4 obstruction showing that arguments based only on pairwise guessing marginals cannot establish unclonable-indistinguishable security. When the Pauli strings all pairwise anticommute, the protocol becomes the one studied in [Quantum 10, 2157 (2026)]. We exploit the symmetry of this protocol to solve the third level of the natural semidefinite programming relaxation obtaining an asymptotic upper bound of approximately 0.5556 on the winning probability. Finally, we prove strong unclonable-indistinguishable security against bounded-local-dimension adversaries and strong indistinguishability security for several efficient Pauli families. First-level NPA computations provide additional numerical evidence towards the strong unclonable-indistinguishable security.

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The unclonable bit question asks whether quantum encryption can prevent an adversary from producing two systems that both reveal the plaintext once the key is disclosed. We introduce and study Pauli Encodings, a simple class of one-bit encryption schemes whose ciphertexts are normalized eigenspace projectors of Pauli strings. For every Pauli Encoding with K Pauli strings, we prove a universal lower bound 1/2+1/(2K)1/2+1/(2\sqrt{K})1/2+1/(2K​) on the optimal monogamy-of-entanglement winning probability, together with sharper bounds for several structured families. We then establish two limitations of natural approaches to unclonable security. First, if the Pauli strings are restricted to strings of X and Z of length n, the encoding is not secure. Second, we identify a universal 3/4 obstruction showing that arguments based only on pairwise guessing marginals cannot establish unclonable-indistinguishable security. When the Pauli strings all pairwise anticommute, the protocol becomes the one studied in [Quantum 10, 2157 (2026)]. We exploit the symmetry of this protocol to solve the third level of the natural semidefinite programming relaxation obtaining an asymptotic upper bound of approximately 0.5556 on the winning probability. Finally, we prove strong unclonable-indistinguishable security against bounded-local-dimension adversaries and strong indistinguishability security for several efficient Pauli families. First-level NPA computations provide additional numerical evidence towards the strong unclonable-indistinguishable security.
We give, to our knowledge, the first plain-model, one-time information-theoretically secure, efficient unclonable encryption scheme for one classical bit. Previous work by Bhattacharyya and Culf (Nature Physics, 2026) and Bhattacharyya, Broadbent, and Culf either only showed 1/poly(λ)1/\mathsf{poly}(λ)1/poly(λ) security loss or required inefficient encryption/decryption operations. We avoid both of these caveats; in doing so, we obtain (to our knowledge) the first plain-model construction of many-time secure 1→21 \to 21→2 unclonable encryption for arbitrary polynomial-length messages, assuming the existence of pseudorandom function-like states (Bartusek and Goldin). The key is a uniformly random non-identity phase-free Pauli on nnn qubits, and bit aaa is encrypted as a random (−1)a(-1)^a(−1)a eigenstate of that Pauli. The scheme is exponentially secure; we prove that the probability that both receivers recover the bit is at most 12+122n/(4n−1)=12+O(2−n/2).\frac{1}{2}+\frac{1}{2}\sqrt{{2^n}/({4^n-1})} = \frac{1}{2} + O\left(2^{-n/2}\right).21​+21​2n/(4n−1)​=21​+O(2−n/2). By a lower bound due to Broadbent, Culf, and Rochette, this is the best probability bound achievable with nnn-qubit ciphertexts (up to the constant hidden in the O(⋅)O(\cdot)O(⋅)). The main conceptual idea is to leverage, in a precise spectral sense, the balanced commutation-anticommutation structure of the Pauli group. The proof is intricate but completely elementary and makes use of standard spectral bound techniques. The main technical workhorse is a standalone linear-algebraic lemma that informally relates the positivity of two different operators, each capturing the intuition that if the two receivers can individually decrypt unusually often then they must also disagree often. GPT-5.6 Sol Ultra found this proof in an extended conversation with the author and drafted a preliminary version of this paper. The author is fully accountable for the correctness of this paper.
Uncloneable 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 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 ∼ 0.5980 , which to our knowledge is the best-known value in the unconditional model.
We give an unconditional construction of information-theoretically secure one-time private-key unclonable encryption for one-bit messages, with efficient encryption and decryption and exponentially small unclonable-indistinguishability advantage. Previous work by Bhattacharyya and Culf (Nature Physics, 2026) and Bhattacharyya, Broadbent, and Culf (arXiv:2603.08916) either only showed inverse-polynomial security loss or required inefficient encryption/decryption operations; we avoid both of these caveats. Our scheme's key is a uniformly random non-identity phase-free Pauli on nnn qubits, and bit aaa is encrypted as a random (−1)a(-1)^a(−1)a eigenstate of that Pauli. We prove that the probability that both receivers recover the bit is at most 1/2+O(2−n/2).1/2 + O\left(2^{-n/2}\right).1/2+O(2−n/2). By a lower bound due to Broadbent, Culf, and Rochette (arXiv:2510.00903), this is the best probability bound achievable with nnn-qubit ciphertexts while ensuring perfect correctness (up to the constant hidden in the O(⋅)O(\cdot)O(⋅)).
In this note, we consider the setting of uncloneable encryption satisfying uncloneable indistinguishability, a form of symmetric key encryption that prevents the cloning of ciphertexts in a very strong sense. Our goal is to minimize the assumptions under which (many-time secure) uncloneable encryption is known to exist, assuming the existence of an information-theoretic "uncloneable bit", i.e. a one-time secure uncloneable encryption scheme for one-bit messages. We observe that if a t -> t' uncloneable bit exists, then the following implications hold. 1. If many-time secure symmetric key encryption exists, then many-time secure t -> t' uncloneable encryption for arbitrary-length messages exists. Since many-time secure uncloneable encryption implies many-time secure symmetric key encryption, this result is tight. 2. If pseudorandom unitaries exist, then many-time secure t -> t' uncloneable encryption for arbitrary-length messages with identical copy security exists. These results together show that many-time secure uncloneable encryption may follow from concrete assumptions in "microcrypt", the world of unstructured quantum cryptography that plausibly exists even if P = NP.
We establish quantum uncloneable encryption with unconditional security, preventing two non-communicating adversaries from simultaneously decrypting a single ciphertext −-− even when both are given the key. Our construction achieves security that approaches the ideal limit at a rate that is exponentially small in the security parameter, without employing any assumptions. Our proof invokes unitary invariance of the shared entangled state and simplifies the adversarial strategies by enforcing this symmetry. Crucially, it then rules out the sender being highly correlated with two non-communicating adversaries at once by an approximation property that we develop, for such unitarily invariant states, which yields a near-optimal bound on the probability of cloning. Consequently, no coordinated strategy beats random guessing of the encrypted bit, establishing unconditional uncloneability. This reveals the existence of an uncloneable bit in Nature and delineates a fundamental, physically enforced cryptographic primitive unavailable in classical settings.
We investigate the notion of untelegraphable encryption (UTE), a quantum encryption primitive that is a special case of uncloneable encryption (UE), where the adversary’s capabilities are restricted to producing purely classical information rather than arbitrary quantum states. We present an unconditionally secure construction of UTE that achieves untelegraphable-indistinguishability security, together with natural multi-ciphertext and bounded collusion-resistant extensions, without requiring any additional assumptions. We also extend this to the unbounded case, assuming pseudo-random unitaries, yielding everlasting security. Furthermore, we derive results on UE using approaches from UTE in the following ways: first, we provide new lower bounds on UTE, which give new lower bounds on UE; second, we prove an asymptotic equivalence between UTE and UE in the regime where the number of adversaries in UE grows. These results suggest that UTE may provide a new path toward achieving a central open problem in the area: indistinguishability security for UE in the plain model.