Paper: 235 Portfolio D — Protocols

DOI: 10.5281/zenodo.20100531

Abstract

Wiesner’s quantum money protocol (c. 1970, published 1983) predates the No-Cloning Theorem (Wootters–Zurek 1982). Its security was argued operationally; No-Cloning later supplied the proof-theoretic foundation. This paper asks whether the Exceptional Jordan Algebra $\mathfrak{J}_3(\mathbb{O})$ admits a non-associative No-Cloning theorem 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 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.

Key Results

  • Theorem 3.1 (NA No-Cloning — Target Obstruction): There is no Jordan algebra $J$ containing two commuting copies of $\mathfrak{J}_3(\mathbb{O})$ satisfying rank-1 projector combination conditions. The combined element $\iota_1(P) + \iota_2(P)$ has rank 2 in any direct sum, violating the projector condition. The cloning target is algebraically forbidden.

  • Theorem 3.2 (NA No-Cloning — Map Collapse): Any Jordan algebra homomorphism $\Phi\colon \mathfrak{J}3(\mathbb{O}) \to A{\mathrm{sa}}$ into the self-adjoint part of any associative algebra satisfies $\Phi(\mathcal{A}(x,y,z)) = 0$ for all $x,y,z$. Since non-Fano associators span the 16-dimensional Peirce-$\frac{1}{2}$ subspace $\mathcal{J}_{1/2}(P)$, the map collapses 16 of the 27 dimensions. No injective such homomorphism exists.

  • Corollary 4.1 (Associative Measurement Defect): The defect between a correct U-operator measurement and an associative-sandwich measurement equals the associator $\mathcal{A}(P,M,P)$, with norm 2 for non-Fano-compatible $M$.

  • Section 5 (Thermal phase transitions): The Fano-Token has a frozen phase ($G_2$-stable, $\beta \geq \beta^$) and a liquid phase ($SO(7)\setminus G_2$, token thaws). The $\beta$-threshold condition $\beta^ = \frac{3}{8}\ln(1/\varepsilon - 1)$ quantifies operational security.

The Peirce-½ Subspace

The 16-dimensional Peirce-$\frac{1}{2}$ subspace $\mathcal{J}{1/2}(P)$ is the exceptional core of $\mathfrak{J}_3(\mathbb{O})$: the part that no associative device can reach. Theorem 3.2 is the precise statement of this inaccessibility. The Fano-Token encodes its secret in $\mathcal{J}{1/2}(P)$; the NA No-Cloning theorem guarantees it cannot be extracted by any matrix-algebra measurement apparatus.

Three Security Layers

Layer Mechanism Strength
S1: Standard No-Cloning Wootters–Zurek linearity argument Map obstruction
S2: Target Obstruction (Thm 3.1) Rank-2 obstruction in any direct sum Target obstruction
S3: Active Defect (Cor 4.1) Associator norm 2 on forgery attempt Instantaneous alarm

Symmetry Chain

\[PSL(2,7) \;\subset\; G_2 \;\subset\; SO(7) \;\subset\; F_4 = \mathrm{Aut}(\mathfrak{J}_3(\mathbb{O}))\]

Frozen phase ($G_2$-stable): associator $= 0$. Liquid phase ($SO(7)\setminus G_2$): associator $= 2$. Security holds only in the frozen phase ($\beta \geq \beta^*$).

Zenodo

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