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Quantum Information Protocols Through the ISA Lens: Teleportation, Superdense Coding, and Bell Tests from H¹ to H³

Ian Richard C Buckley

Published Jul 3, 2026• 0 citations
Publisher

Abstract

PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. NOTICE (2026-08-04). This record uses a magic measure that has been shown to be incorrectly signed, and it should not be cited for quantitative magic claims. The "total variation" quantity TV = sum |W(u)| over the discrete Wigner function was used with the condition TV < 1 characterising non-stabiliser states. The inequality runs the other way. For any quasi-probability distribution summing to 1, the l1 norm satisfies sum |W| >= 1 always, with equality exactly when W is non-negative. Direct computation: the stabiliser state |+> gives TV = 1.0000 and the magic state T|+> gives TV = 1.2071. The condition "TV < 1" therefore describes the empty set, and any tier defined by it has no members. Separately, TV as defined is exp(mana) -- a monotone reparameterisation of the established Wigner-negativity monotone of Veitch, Mari, Ferrie and Emerson -- and was never compared against it. Readers should consult Veitch et al. on mana and Wigner negativity, Howard and Campbell on robustness of magic, Bravyi et al. on stabiliser extent, and Leone, Oliviero and Hamma on stabiliser Renyi entropy. NOTICE (2026-08-04). A verification this record's framework relies upon has been shown to be vacuous, and the identity it claimed to establish does not hold in the naive form. The naive pentagon identity FAILS for octonionic labels. Over all 7^4 = 2401 imaginary labelings, the two independent rebracketing paths from ((ab)c)d to a(b(cd)) disagree in 1176 cases (49%). Explicit counterexample: (a,b,c,d) = (1,1,2,4) gives +e6 on one path and -e6 on the other. The verification previously relied upon (the Pachner prover of 10.5281/zenodo.19713350) is vacuous: it defines w5 := w1*w2*w3*w4 and then checks w1*w2*w3*w4*w5 = +1, i.e. x*x = 1 for x in {+1,-1}. The identical test passes 4096/4096 on RANDOM SIGNS with no octonions involved. That script's own comment concedes the equation is "tautologically true". What this does NOT mean. Octonions are non-associative by definition, so rebracketing must be path-dependent and the naive pentagon cannot hold. That is the content, not a defect. A coherent non-associative calculus must supply the associator as explicit data satisfying its own coherence condition (Mac Lane), which is what Kuperberg's spiders (Comm. Math. Phys. 1996) provide for rank-2 groups. The structural framing is defensible; the verification was measuring the wrong thing. We recast the three foundational quantum information protocols — quantum teleportation, superdense coding, and the CHSH Bell test — in the language of the Origami Instruction Set Architecture (ISA), and then lift each to its TriQ (3-qubit GHZ) and 6-731 (broken-Fano-line) extensions. In the ISA reading, teleportation is SPLIT;SPLAT with a classical wire, superdense coding is FLIP;FLOP on a Bell pair encoding $\log_2|G|$ bits via the group $G = \mathbb{Z}_2^2$, and the CHSH violation is a hybrid H¹/H² computation: the Bell pair is a stabiliser (H¹) resource but the Tsirelson-optimal measurement applies a T-gate rotation (H², TWIST=1). We prove that the Tsirelson bound $\cos^2(\pi/8)$ equals the total variation of any dark-magic (Tier 1) state in the ISA magic-type classification — the Tsirelson bound is the Wigner mass of the dark-magic sector. Lifting to the TriQ register introduces the ORBIT opcode, raises superdense coding capacity to 3 bits per qubit ($\log_2|\mathbb{Z}_2^3|$), and replaces the CHSH game with the Fano Line Verification Game whose win conditions are the $\pm 1$ values of the $6j$ symbol of $\mathbb{Z}_2^3$. The 6-731 extension (one Fano line weakened to coupling $\eta$) makes teleportation directed, reduces superdense capacity to $3 - H(\eta)$ bits per qubit, and interpolates the Fano game win probability continuously between classical and quantum bounds. Remarkably, the same asymmetry parameter $\eta$ is independently measurable by three methods — thermodynamic Carnot efficiency (Paper 325), information-theoretic channel capacity, and game-theoretic win probability — whose agreement constitutes a Pentagon-identity self-verification test for the hardware. We close with the H$^k$ Bell tower: for each cohomological depth $k$ there is a Bell-type inequality whose maximum quantum violation requires an H$^k$ resource, providing a game-theoretic ladder that extends beyond CHSH through the Fano game (H²) to a conjectured G$_2$-correlation game (H³) accessible only to the 731-ISA SPIN opcode.Appendix B provides side-by-side quantum circuit and ISA spider diagrams for all three baseline protocols. Keywords quantum teleportation, superdense coding, CHSH inequality, Bell test, Tsirelson bound, Origami ISA, H^k cohomology, stabiliser states, magic states, Fano plane, GHZ state, TriQ register, 6-731 architecture, ORBIT opcode, TWIST opcode, topological heat engine, dark magic, Wigner mass, Bell hierarchy, Mermin inequality, Pentagon equation, quantum nonlocality, quantum channel capacity, G2 correlations

DOI
10.5281/zenodo.21158961

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PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. NOTICE (2026-08-04). This record uses a magic measure that has been shown to be incorrectly signed, and it should not be cited for quantitative magic claims. The "total variation" quantity TV = sum |W(u)| over the discrete Wigner function was used with the condition TV < 1 characterising non-stabiliser states. The inequality runs the other way. For any quasi-probability distribution summing to 1, the l1 norm satisfies sum |W| >= 1 always, with equality exactly when W is non-negative. Direct computation: the stabiliser state |+> gives TV = 1.0000 and the magic state T|+> gives TV = 1.2071. The condition "TV < 1" therefore describes the empty set, and any tier defined by it has no members. Separately, TV as defined is exp(mana) -- a monotone reparameterisation of the established Wigner-negativity monotone of Veitch, Mari, Ferrie and Emerson -- and was never compared against it. Readers should consult Veitch et al. on mana and Wigner negativity, Howard and Campbell on robustness of magic, Bravyi et al. on stabiliser extent, and Leone, Oliviero and Hamma on stabiliser Renyi entropy. NOTICE (2026-08-04). A verification this record's framework relies upon has been shown to be vacuous, and the identity it claimed to establish does not hold in the naive form. The naive pentagon identity FAILS for octonionic labels. Over all 7^4 = 2401 imaginary labelings, the two independent rebracketing paths from ((ab)c)d to a(b(cd)) disagree in 1176 cases (49%). Explicit counterexample: (a,b,c,d) = (1,1,2,4) gives +e6 on one path and -e6 on the other. The verification previously relied upon (the Pachner prover of 10.5281/zenodo.19713350) is vacuous: it defines w5 := w1w2w3w4 and then checks w1w2w3w4w5 = +1, i.e. xx = 1 for x in {+1,-1}. The identical test passes 4096/4096 on RANDOM SIGNS with no octonions involved. That script's own comment concedes the equation is "tautologically true". What this does NOT mean. Octonions are non-associative by definition, so rebracketing must be path-dependent and the naive pentagon cannot hold. That is the content, not a defect. A coherent non-associative calculus must supply the associator as explicit data satisfying its own coherence condition (Mac Lane), which is what Kuperberg's spiders (Comm. Math. Phys. 1996) provide for rank-2 groups. The structural framing is defensible; the verification was measuring the wrong thing. We recast the three foundational quantum information protocols — quantum teleportation, superdense coding, and the CHSH Bell test — in the language of the Origami Instruction Set Architecture (ISA), and then lift each to its TriQ (3-qubit GHZ) and 6-731 (broken-Fano-line) extensions. In the ISA reading, teleportation is SPLIT;SPLAT with a classical wire, superdense coding is FLIP;FLOP on a Bell pair encoding log⁡2∣G∣\log_2|G|log2​∣G∣ bits via the group G=Z22G = \mathbb{Z}_2^2G=Z22​, and the CHSH violation is a hybrid H¹/H² computation: the Bell pair is a stabiliser (H¹) resource but the Tsirelson-optimal measurement applies a T-gate rotation (H², TWIST=1). We prove that the Tsirelson bound cos⁡2(π/8)\cos^2(\pi/8)cos2(π/8) equals the total variation of any dark-magic (Tier 1) state in the ISA magic-type classification — the Tsirelson bound is the Wigner mass of the dark-magic sector. Lifting to the TriQ register introduces the ORBIT opcode, raises superdense coding capacity to 3 bits per qubit (log⁡2∣Z23∣\log_2|\mathbb{Z}_2^3|log2​∣Z23​∣), and replaces the CHSH game with the Fano Line Verification Game whose win conditions are the ±1\pm 1±1 values of the 6j6j6j symbol of Z23\mathbb{Z}_2^3Z23​. The 6-731 extension (one Fano line weakened to coupling η\etaη) makes teleportation directed, reduces superdense capacity to 3−H(η)3 - H(\eta)3−H(η) bits per qubit, and interpolates the Fano game win probability continuously between classical and quantum bounds. Remarkably, the same asymmetry parameter η\etaη is independently measurable by three methods — thermodynamic Carnot efficiency (Paper 325), information-theoretic channel capacity, and game-theoretic win probability — whose agreement constitutes a Pentagon-identity self-verification test for the hardware. We close with the Hk^kk Bell tower: for each cohomological depth kkk there is a Bell-type inequality whose maximum quantum violation requires an Hk^kk resource, providing a game-theoretic ladder that extends beyond CHSH through the Fano game (H²) to a conjectured G2_22​-correlation game (H³) accessible only to the 731-ISA SPIN opcode.Appendix B provides side-by-side quantum circuit and ISA spider diagrams for all three baseline protocols. Keywords quantum teleportation, superdense coding, CHSH inequality, Bell test, Tsirelson bound, Origami ISA, H^k cohomology, stabiliser states, magic states, Fano plane, GHZ state, TriQ register, 6-731 architecture, ORBIT opcode, TWIST opcode, topological heat engine, dark magic, Wigner mass, Bell hierarchy, Mermin inequality, Pentagon equation, quantum nonlocality, quantum channel capacity, G2 correlations
Le formalisme de la mécanique quantique sur l'espace des phases est utilisé dans une étude restreinte de la mesure quantique et de l'intrication, et d'une manière plus étendue du protocole de la téléportation à variables continues. Le but est d'explorer partiellement la relation mutuelle entre ce formalisme et l'information quantique. Au lieu des quasi-distributions de Wigner utilisées actuellement dans le cas des variables continues, nous utilisons de vraies densités de probabilité qui sont définies positives et qui rendent compte des mesures quantiques non précises au moyen d'une fonction d'onde propre représentant un appareil de mesure quantique non idéal. Ceci peut être réalisé sur un espace des phases non relativiste (ou relativiste). La formule obtenue est semblable à une formule connue en optique quantique mais contient l'effet de l'instrument de mesure. Elle a été appliquée à trois cas. Dans le premier cas, les deux appareils de mesure, correspondant aux deux parties intriquées et partagées par Alice et Bob, ne sont pas intriqués et sont décrits par deux fonctions d'ondes propres gaussiennes identiques, relatives au groupe de Weyl-Heisenberg. Elles conduisent à une densité de probabilité identique à la fonction de Husimi qui est analysée et comparée à la fonction de Wigner. Une nouvelle expression de la fidélité de la téléportation d'un état cohérent est obtenue en fonction des variances des quadratures. Dans le deuxième cas, les deux appareils de mesure sont intriqués dans un état comprimé du vide à deux modes. Dans le troisième cas, les appareils de mesure sont dans un état intriqué comprimé, obtenu par la combinaison de deux états gaussiens. L'observation générale est que les états des appareils de mesure partagés par Alice et Bob influencent la fidélité de la téléportation à travers leur imprécision et leur intrication.
Fibrational Tensor Codes (FTCs): Holonomic Quantum Computation, G2G_2G2​ Fiber Bundles, and the Cayley-Dickson Hardware Hierarchy
NOTICE (2026-08-04). A verification this record's framework relies upon has been shown to be vacuous, and the identity it claimed to establish does not hold in the naive form. The naive pentagon identity FAILS for octonionic labels. Over all 7^4 = 2401 imaginary labelings, the two independent rebracketing paths from ((ab)c)d to a(b(cd)) disagree in 1176 cases (49%). Explicit counterexample: (a,b,c,d) = (1,1,2,4) gives +e6 on one path and -e6 on the other. The verification previously relied upon (the Pachner prover of 10.5281/zenodo.19713350) is vacuous: it defines w5 := w1*w2*w3*w4 and then checks w1*w2*w3*w4*w5 = +1, i.e. x*x = 1 for x in {+1,-1}. The identical test passes 4096/4096 on RANDOM SIGNS with no octonions involved. That script's own comment concedes the equation is "tautologically true". What this does NOT mean. Octonions are non-associative by definition, so rebracketing must be path-dependent and the naive pentagon cannot hold. That is the content, not a defect. A coherent non-associative calculus must supply the associator as explicit data satisfying its own coherence condition (Mac Lane), which is what Kuperberg's spiders (Comm. Math. Phys. 1996) provide for rank-2 groups. The structural framing is defensible; the verification was measuring the wrong thing. Additional note. This record predates a change of vocabulary and, in places, of substance. Opcode names used here were subsequently revised (ORBIT to RESOLVE, LABEL to PROJECT, BIND to FUSE, MERGE to JOIN), so the terminology does not match the current reference. Readers wanting the current definitions should consult the maintained opcode reference rather than this record. This working paper introduces Fibrational Tensor Codes (FTCs), an exploratory quantum error correction paradigm that relocates the burden of error correction from macroscopic spatial topology — as in surface codes — to microscopic gauge topology. Rather than arranging qubits on 2D associative grids, the FTC architecture maps the Cayley-Dickson division algebra hierarchy directly onto hardware substrates: a Quaternionic routing layer (4 physical qubits, [[4,2,2]][[4,2,2]][[4,2,2]], PG(1,2)PG(1,2)PG(1,2)) and an Octonionic computational layer (7 physical qubits, PG(2,2)PG(2,2)PG(2,2), the Fano plane). The central conjecture proposes that the 7-qubit Hilbert space, isomorphic as a vector space to the Clifford algebra Cℓ(0,7)\mathbb{C}\ell(0,7)Cℓ(0,7), can be physically confined to the 8-dimensional minimal left ideal identified by C. Furey in her construction of the complex octonions. This confinement — achieved by appropriate physical coupling rather than discrete stabilizer constraints — would yield a topologically protected 8-dimensional logical qudit (the Logical Quoct) in which local errors are converted to heralded erasures via an Associator Penalty, enabling active feed-forward correction at approximately twice the rate of blind Pauli-flip correction. The key architectural principle is that non-associativity is confined strictly to the continuous Berry phase of the encoding map within a G2G_2G2​ fiber bundle; the logical gate group remains fully associative. The paper formally falsifies 7-qubit triorthogonality (Theorem 5.1), establishing that transversal CCZ requires scaling to the 15-qubit PG(3,2)PG(3,2)PG(3,2) Reed-Muller geometry, and identifies six concrete open problems required to advance this framework from architectural proposal to engineering specification.
PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. A practitioner's primer for quantum computing researchers. No knowledge of octonions, sheaf theory, or category theory is assumed. Knowledge of stabiliser states, the Clifford group, and T-gates is assumed. The central message: the structure of the 3-qubit Pauli group is secretly governed by the Fano plane PG(2,2), and the Fano plane is the geometry of the 731 instruction set architecture (731 ISA) — a unified framework for quantum computation that organises every opcode as a Pachner move on a simplicial complex. The framework reveals three levels of quantum computational resource:• Level 0 (Clifford): stabiliser states; Gottesman-Knill simulable; the Fano lines are the free sector.• Level 1 (Standard magic): T-gates; Wigner function negativity; non-Clifford but accessible to the Origami ISA (regime 2).• Level 2 (Associamancy): the SPIN opcode; the Schur boundary; genuinely complex irreps of PSL(2,7); not accessible to any associative hardware. The Origami ISA (five opcodes: SPLIT, SPLAT, TWIST, FLIP, FLOP) subsumes Bell sampling, qudit stabiliser learning, and the abelian hidden subgroup problem as special cases. The 731 ISA adds SPIN and BIND, unlocking Level 2. This primer is a companion to "The Topology of Risk" (doi:10.5281/zenodo.20642983), which covers the same H⁰/H¹/H² framework applied to financial risk. Keywords: quantum computing, Fano plane, Origami ISA, 731 ISA, stabiliser states, magic states, associamancy, Schur boundary, primer, tutorial, hidden subgroup problem, MBQC, photonics
Protocoles de chiffrement quantiques de plusieurs parties en environnements réalistes Cette thèse se situe à l’interface entre la théorie quantique et les expériences, en mettant l’accent sur les protocoles pratiques qui peuvent être mises en place en utilisant la technologie présente. Notre objectif est de contribuer à la construction d’un cadre général pour la conception et la mise en oeuvre de schémas de chiffrement quantiques qui permettront d’améliorer la sécurité des futurs réseaux de télécommunication. En outre, cette thèse veut avancer la recherche pertinente sur la physique quantique et l’informatique, en améliorant notre compréhension du phénomène d’intrication. Les corrélations des états intriqués ne peuvent pas être reproduites par des moyens classiques, ce qui permet d’effectuer des tâches (par exemple la téléportation et le codage super-dense) qui sont autrement impossibles. Il est donc d’une importance majeure d’être capables de vérifier si un état quantique est intriqué. Dans cette thèse, nous montrons comment vérifier efficacement si une source physique peut créer des états intriqués multipartites et les partager avec de nombreuses parties, dont certains sont malhonnêtes et collaborent avec la source. Ce protocole pourrait s’avérer essentiel pour tout type de calcul quantique entre les parties méfiantes et pourrait aussi faciliter la délégation sécurisée des tâches de calcul aux serveurs quantiques puissants qui ne sont pas fiables. Finalement, nous étudions le lien entre la théorie des jeux et la non-localité quantique, dans le cadre de jeux bayésiens. Nous examinons comment l’intrication partagé aide les joueurs gagner un jeu avec une probabilité plus élevée que les ressources classiques pourraient atteindre.
PENDING QUALITY AUDIT (2026-08). The file is restricted while this record is reviewed as part of a systematic audit of the author's corpus. It has not yet been assessed. Metadata and DOI remain public, and access can be requested. We articulate a structural parallel between the MIP*=RE theorem and the Origami Instruction Set Architecture (ISA). The algebraic rigidity of entangled-prover verification — exact-or-fail Pauli basis tests, low-degree tests, Mermin Magic Square constraints — corresponds precisely to the Frobenius axiom SPLIT∘SPLAT=1, the Pentagon identity, and the seven-point Fano plane PG(2,2) in the ISA framework. Six numerical experiments confirm the parallels: Pentagon unitarity holds exactly for all j ∈ {1/2,...,15/2}; Frobenius fidelity degrades as (1-p)^{2(2j+1)} under depolarising noise; the GHZ stabiliser group forms a perfect quantum Fano strategy satisfying all seven constraints simultaneously while no classical strategy can; G₂ non-associativity produces a sharp excluded-volume step function above j=7/2. This version adds a synthesis remark showing that the numbers 3 and 7 — the two dimensions where the cross product exists, gyroscopes are stable, and choreographic orbits can close — are not two separate coincidences but the same theorem: the Fano plane PG(2,F₂) has 7 points and 3 points per line because these are the imaginary unit counts of the octonions O and quaternions H respectively, forced by the unique projective plane of order q=2 over the field F₂. The broken Fano (6/7 architecture) and the magic/non-associativity conjecture are restated in this unified language. The practical implication is a conjecture that fault-tolerant quantum computing overhead is governed by the regime of the code's F-move amplitudes, with MSD-free computation possible at rung 1 and higher. Keywords MIP* Equals RE, Interactive Proof Systems, Quantum Entanglement, Pentagon Identity, Frobenius Axiom, Fano Plane, PG(2,2), GHZ Stabiliser, Origami ISA, Wigner 6j Symbol, Fault-Tolerant Quantum Computing, Magic State Distillation, Regime Ladder, Complexity Theory, Quantum Prover, Local Pauli Test, Low-Degree Test, G₂ Non-Associativity, Excluded Volume, Depolarising Noise, Rung-1, Quantum Verification, Adelic Simplicial Architecture
Cette thèse s inscrit dans le cadre de l information quantique avec des variables continues, en utilisant des états quantiques du champ électromagnétique. En combinant les outils propres aux variables discrètes, ou la lumière est décrite en termes de photons, avec les outils des variables continues, où la lumière est décrite en termes de quadratures, nous pouvons étudier théoriquement et produire expérimentalement des états non-classiques, ainsi que des protocoles élémentaires d information quantique. Ainsi, nous avons produit expérimentalement un état chat de Schrödinger , superposition quantique de deux états lumineux quasi-classiques, sur lequel nous avons appliqué une porte quantique introduisant une phase dans la superposition. Nous avons ensuite analysé la qualité de cette porte en utilisant un modèle simple de notre expérience. Nous nous sommes ensuite intéressés aux corrélations quantiques, mesurées par la discorde quantique, pour une classe d états particulièrement importants en information quantique. Nous avons quantifié la précision de nos mesures en les comparant aux bornes de Cramér-Rao classique et quantique. Enfin, nous avons étudié théoriquement l utilisation d un amplificateur quantique non-déterministe en cryptographie quantique. Cet amplificateur possède la propriété de pouvoir amplifier des états quantiques sans en amplifier le bruit quantique associé. Ainsi, nous avons montré qu il permet une amélioration de la distance maximale de transmission d une clé secrète, ainsi qu une amélioration de la résistance au bruit introduit par le canal quantique.