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The Non-Associative No-Cloning Theorem and the Fano-Token: Target Obstruction, Map Collapse, and Active Forgery Detection via the Exceptional Jordan Algebra $\mathfrak{J}_{3}(\mathbb{O})$

Ian Richard C Buckley

Published May 9, 2026• 0 citations
Publisher

Abstract

Wiesner's quantum money protocol was conceived around 1970, over a decade before Wootters and Zurek established the No-Cloning Theorem. Its security was argued operationally; No-Cloning later supplied the proof-theoretic foundation. This paper asks whether the Exceptional Jordan Algebra $\mathfrak{J}_3(\mathbb{O})$ (the Albert algebra) admits a non-associative No-Cloning theorem that is strictly stronger than the standard one, and what it implies for quantum money. The standard No-Cloning proof uses only linearity and the bilinear tensor structure; it does not use associativity, so a direct translation to $\mathfrak{J}_3(\mathbb{O})$ recovers only the standard result in new notation. The genuinely non-associative obstruction is of a different character: it concerns not the impossibility of a cloning map but the impossibility of a cloning target. We prove two theorems. Theorem 3.1 (NA No-Cloning --- Target Obstruction): the two-copy system of $\mathfrak{J}_3(\mathbb{O})$ does not exist as a Jordan algebra. Any Jordan algebra containing two commuting copies of $\mathfrak{J}_3(\mathbb{O})$ would be special (embeddable in an associative algebra), contradicting the exceptionality of $\mathfrak{J}_3(\mathbb{O})$ (Albert). There is no valid target into which an octonionic state can be cloned while remaining in the same algebraic category. Theorem 3.3 (NA No-Cloning --- Map Collapse): any Jordan algebra homomorphism $\Phi\colon \mathfrak{J}_3(\mathbb{O}) \to A_{sa}$ into the self-adjoint part of an associative algebra $A$ satisfies $\Phi([x, y, z]) = 0$ for all $x, y, z \in \mathfrak{J}_3(\mathbb{O})$. Since non-Fano associators span the 16-dimensional Peirce-$\tfrac{1}{2}$ subspace of $\mathfrak{J}_3(\mathbb{O})$, $\Phi$ collapses 16 of the 27 dimensions of $\mathfrak{J}_3(\mathbb{O})$. A dimension-collapsing map cannot clone. Together these theorems say: for octonionic quantum states, No-Cloning holds for a reason the standard theorem does not reach. The obstruction is structural, not just linear-algebraic. The Fano-Token protocol encodes a quantum banknote as a rank-1 projector in $\mathfrak{J}_3(\mathbb{O})$, the Furey projector $P_\ell$ stabilised by the $G_2$ automorphism group. As a Corollary 4.1 of Theorem 3.3 , any associative measurement operator $M$ applied to $P_\ell$ produces a defect equal to the associator $[P_\ell, M, P_\ell]$, with norm $2$ for non-Fano-compatible $M$. This converts Wiesner's passive defense into an active alarm: unauthorized measurement is detectable instantaneously, without token retrieval. The physical claim that this defect manifests as a measurable energy transition in the 731-QPU hardware is stated as Conjecture and kept separate from the algebraic results.

DOI
10.5281/zenodo.20100531

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The Non-Associative No-Cloning Theorem and the Fano-Token: Target Obstruction, Map Collapse, and Active Forgery Detection via the Exceptional Jordan Algebra J3(O)\mathfrak{J}_{3}(\mathbb{O})J3​(O)
Wiesner's quantum money protocol was conceived around 1970, over a decade before Wootters and Zurek established the No-Cloning Theorem. Its security was argued operationally; No-Cloning later supplied the proof-theoretic foundation. This paper asks whether the Exceptional Jordan Algebra J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) (the Albert algebra) admits a non-associative No-Cloning theorem that is strictly stronger than the standard one, and what it implies for quantum money. The standard No-Cloning proof uses only linearity and the bilinear tensor structure; it does not use associativity, so a direct translation to J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) recovers only the standard result in new notation. The genuinely non-associative obstruction is of a different character: it concerns not the impossibility of a cloning map but the impossibility of a cloning target. We prove two theorems. Theorem 3.1 (NA No-Cloning --- Target Obstruction): the two-copy system of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) does not exist as a Jordan algebra. Any Jordan algebra containing two commuting copies of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) would be special (embeddable in an associative algebra), contradicting the exceptionality of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O) (Albert). There is no valid target into which an octonionic state can be cloned while remaining in the same algebraic category. Theorem 3.3 (NA No-Cloning --- Map Collapse): any Jordan algebra homomorphism Φ ⁣:J3(O)→Asa\Phi\colon \mathfrak{J}_3(\mathbb{O}) \to A_{sa}Φ:J3​(O)→Asa​ into the self-adjoint part of an associative algebra AAA satisfies Φ([x,y,z])=0\Phi([x, y, z]) = 0Φ([x,y,z])=0 for all x,y,z∈J3(O)x, y, z \in \mathfrak{J}_3(\mathbb{O})x,y,z∈J3​(O). Since non-Fano associators span the 16-dimensional Peirce-12\tfrac{1}{2}21​ subspace of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O), Φ\PhiΦ collapses 16 of the 27 dimensions of J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O). A dimension-collapsing map cannot clone. Together these theorems say: for octonionic quantum states, No-Cloning holds for a reason the standard theorem does not reach. The obstruction is structural, not just linear-algebraic. The Fano-Token protocol encodes a quantum banknote as a rank-1 projector in J3(O)\mathfrak{J}_3(\mathbb{O})J3​(O), the Furey projector PℓP_\ellPℓ​ stabilised by the G2G_2G2​ automorphism group. As a Corollary 4.1 of Theorem 3.3 , any associative measurement operator MMM applied to PℓP_\ellPℓ​ produces a defect equal to the associator [Pℓ,M,Pℓ][P_\ell, M, P_\ell][Pℓ​,M,Pℓ​], with norm 222 for non-Fano-compatible MMM. This converts Wiesner's passive defense into an active alarm: unauthorized measurement is detectable instantaneously, without token retrieval. The physical claim that this defect manifests as a measurable energy transition in the 731-QPU hardware is stated as Conjecture and kept separate from the algebraic results.
Forty years ago, Wiesner proposed using quantum states to create money that is physically impossible to counterfeit, something that cannot be done in the classical world. However, Wiesner's scheme required a central bank to verify the money, and the question of whether there can be unclonable quantum money that anyone can verify has remained open since. One can also ask a related question, which seems to be new: can quantum states be used as copy-protected programs, which let the user evaluate some function f, but not create more programs for f? This paper tackles both questions using the arsenal of modern computational complexity. Our main result is that there exist quantum oracles relative to which publicly-verifiable quantum money is possible, and any family of functions that cannot be efficiently learned from its input-output behavior can be quantumly copy-protected. This provides the first formal evidence that these tasks are achievable. The technical core of our result is a "Complexity-Theoretic No-Cloning Theorem," which generalizes both the standard No-Cloning Theorem and the optimality of Grover search, and might be of independent interest. Our security argument also requires explicit constructions of quantum t-designs. Moving beyond the oracle world, we also present an explicit candidate scheme for publicly-verifiable quantum money, based on random stabilizer states; as well as two explicit schemes for copy-protecting the family of point functions. We do not know how to base the security of these schemes on any existing cryptographic assumption. (Note that without an oracle, we can only hope for security under some computational assumption).
Abstract Unknown quantum information cannot be perfectly copied (cloned). This statement is the bedrock of quantum technologies and quantum cryptography, including the seminal scheme of Wiesner’s quantum money, which was the first quantum-cryptographic proposal. Surprisingly, to our knowledge, quantum money has not been tested experimentally yet. Here, we experimentally revisit the Wiesner idea, assuming a banknote to be an image encoded in the polarization states of single photons. We demonstrate that it is possible to use quantum states to prepare a banknote that cannot be ideally copied without making the owner aware of only unauthorized actions. We provide the security conditions for quantum money by investigating the physically-achievable limits on the fidelity of 1-to-2 copying of arbitrary sequences of qubits. These results can be applied as a security measure in quantum digital right management.
Quantum tokens are underlying primitives for quantum money and network proposals, which leverage the no-cloning theorem to realize unforgeable authentication. A relevant but overlooked type of attack to such architectures is one in which a hacker steals the classical side information about the token states from the issuing agent (e.g., a bank), enabling the forgery of fake tokens without violating the no-cloning theorem. Our proposal addresses this threat by removing the classical side information about the token states, where instead a copy of the token is stored at the bank, i.e. a quantum vault. This copy can be accessed by anyone to perform authentication, consuming the token pair in the process. We benchmark the protocol and identify quality parameters within a hardware agnostic framework employing IBMQ processors, such that the protocol is applicable to arbitrary quantum platforms. By comparing the efficiency with which genuine tokens are produced and authenticated against a possible estimate-and-forge attack, we quantify the security of the protocol. We achieve probabilities lower than 10−410^{-4}10−4 for false-negative errors (Type 1) and 10−1010^{-10}10−10 for successful attacks (Type 2) when considering quantum bills composed of 200 tokens, even in the worst performing processor and most conservative uncertainty estimate. The quantum vault not only protects both user and bank symmetrically with the same quantum principles, but provides a step towards public key authentication, since any untrusted party can have access granted by the bank to the tokens' authentication interface without being able to clone them. Besides this bank-mediated form of publicly accessible verifiability, our proposal achieves traceability and revocability, since the bank retains the vault copy of the tokens.
Unlike classical money, which is hard to forge for practical reasons (e.g. producing paper with a certain property), quantum money is attractive because its security might be based on the no-cloning theorem. The first quantum money scheme was introduced by Wiesner circa 1970. Although more sophisticated quantum money schemes were proposed, Wiesner’s scheme remained appealing because it is both conceptually clean as well as relatively easy to implement. We show efficient adaptive attacks on Wiesner’s quantum money scheme [1] (and its variant by Bennett et al. [2]), when valid money is accepted and passed on, while invalid money is destroyed. We propose two attacks, the first is inspired by the Elitzur-Vaidman bomb testing problem [3, 4], while the second is based on the idea of protective measurements [5]. It allows us to break Wiesner’s scheme with 4 possible states per qubit, and generalizations which use more than 4 states per qubit. The attack shows that Wiesner’s scheme can only be safe if the bank replaces valid notes after validation.
Secret sharing is ubiquitous throughout cryptography. All possible access structures are classically feasible, and in the case of threshold access structures, the protocols are even very efficient. However, when moving to the quantum setting, the no-cloning theorem shows that many access structures are impossible. In fact, no-cloning exactly characterizes feasibility: for thresholds, quantum secret sharing is possible if and only if the threshold ttt is strictly more than n/2n/2n/2. In this work, we propose a variant of quantum secret sharing (QSS) where two or more identical copies of the input state are provided, but the output is only required to recover one copy. This notion circumvents the simple one-copy no-cloning obstruction, though the natural kkk-copy generalization still gives much milder obstructions. For thresholds using kkk copies, no-cloning implies that QSS is impossible whenever t≤n/(k+1)t\leq n/(k+1)t≤n/(k+1). It is tempting to hypothesize that no-cloning continues to exactly characterize the many-copy case. However, we show that this is not the case. We give positive results showing that multiple copies allow for going slightly beyond the single-copy obstruction: for thresholds, we construct QSS whenever t>(n−k+1)/2t>(n-k+1)/2t>(n−k+1)/2. On the other hand, we give a novel obstruction we call the Clique Path obstruction, which applies to arbitrary access structures. For thresholds, it shows that QSS is impossible whenever t≤(n−1)/kt\leq (n-1)/kt≤(n−1)/k. Our upper and lower bounds exactly match for k=2k=2k=2. Both our results leverage connections to the kkk-colorability of certain graphs derived from the access structure. We leave closing the gap for k≥3k\geq 3k≥3 copies as a fascinating direction for future work. Secret sharing is closely related to error correction, which can also be considered in the many-copy setting. Our results imply similar obstructions for quantum error correction for erasure channels.
Unlike classical money, which is hard to forge for practical reasons (e.g. producing paper with a certain property), quantum money is attractive because its security might be based on the no-cloning theorem. The first quantum money scheme was introduced by Wiesner circa 1970. Although more sophisticated quantum money schemes were proposed, Wiesner's scheme remained appealing because it is both conceptually clean and relatively easy to implement. We show efficient adaptive attacks on Wiesner's quantum money scheme [Wie83] (and its variant by Bennett et al. [BBBW83]), when valid money is accepted and passed on, while invalid money is destroyed. We propose two attacks, the first is inspired by the Elitzur-Vaidman bomb testing problem [EV93, KWH+95], while the second is based on the idea of protective measurements [AAV93]. It allows us to break Wiesner's scheme with 4 possible states per qubit, and generalizations which use more than 4 states per qubit.