Operational Foundations for Quaternionic and Octonionic Quantum Models: Exact Quaternionic Channels and Error Correction, Para-Linear Operators, and Categorical Closure Boundaries Beyond Associativity

Abstract: A scalar field alone does not determine a quantum theory: states, effects, processes, symmetries, composition, and discard are equally structural. Realification illustrates this point: an orthogonal complex structure J^2 = -I selects the physical real operators and the balanced composite. Quaternionic quantum mechanics has an analogous exact representation on a doubled complex space selected by an antiunitary symplectic structure Theta^2 = -I. Within that sector, a Choi fixed-point condition characterizes when a complex channel admits quaternionic Kraus operators, while exact correction of a finite-dimensional right-quaternionic code is characterized by compression coefficients in the real center and admits an explicit recovery. For octonions, nonassociativity precludes a unique continuation; para-linear, categorical, sectorial, Jordan, Clifford-envelope, and Moufang models are compared by the operational structures they retain and the additional data required to define composites, channels, or recovery. Across these cases, an exact ambient representation does not erase the complex or symplectic structure that selects the physical theory.

• The paper establishes a Choi fixed-point criterion that identifies quaternionic quantum channels within doubled complex spaces and enables Kraus reconstruction through antiunitary symmetry.
• The paper proves a right-quaternionic Knill–Laflamme theorem requiring real central compression coefficients and provides an explicit recovery channel, including degenerate codes with logical dimension at least two.
• The paper compares six octonionic frameworks and shows that para-linear, categorical, Jordan, sector-based, Clifford, and loop-register models differ in composition, complete positivity, discard, and closure properties.

This paper develops an operational framework for quantum theories over non-complex scalar algebras, with three main contributions: a Choi fixed-point criterion for quaternionic channels on a doubled complex space, a right-quaternionic Knill–Laflamme theorem with real central compression coefficients and an explicit recovery, and a systematic comparison of six operationally distinct octonionic models. Its unifying thesis is that a scalar field does not by itself determine a quantum theory: states, effects, processes, symmetries, composition, and discard are equally structural, and exact simulation in a larger ambient theory never erases the selecting structure.

The real control case

The paper first establishes a finite-dimensional control example using realification. A complex Hilbert space $\mathcal H_C = \mathbb C^d$ is embedded in $\mathbb R^{2d}$ via a state map $\mathcal S_d$ and operator map $\mathcal T_d$, with a distinguished orthogonal complex structure $\mathsf J_d^2 = -I$. Three facts carry the operational weight: the complex inner product is reconstructed from the pair $(g, \mathsf J_d)$; the physical algebra is not all of $M_{2d}(\mathbb R)$ but the commutant of $\mathsf J_d$; and the Born rule holds exactly under normalized representatives ($\rho_R := \tfrac12\mathcal T_d(\rho)$), because $\Tr_R \mathcal T_d(A) = 2\,\mathrm{Re}\,\Tr_C A$.

Composition is where the structural point becomes sharp. The ordinary real tensor product has dimension $\mathbb R^{2d}$0, while the $\mathbb R^{2d}$1-balanced quotient $\mathbb R^{2d}$2 has dimension $\mathbb R^{2d}$3 and reproduces the complex composite exactly. This reconciles two apparently conflicting lines of work: Renou et al.'s network argument against unrestricted real-amplitude quantum theory (Renou et al., 2021), and the consistent real formulations of Barrios Hita et al., Bang–Cho–Baek, and Maioli–Curado–Gazeau, which retain the hidden complex structure and modify the composite rule. Reproducing complex correlations changes the source-factorization assumption rather than invalidating the foil theory; a locality condition on chosen embeddings does not select one composite over another.

Quaternionic kinematics and exact complex representation

Finite-dimensional quaternionic quantum mechanics is formulated intrinsically: amplitudes live in the right module $\mathbb R^{2d}$4, observables are quaternionic-Hermitian matrices, reversible dynamics is $\mathbb R^{2d}$5, and probabilities use the cyclic real trace $\mathbb R^{2d}$6, which is basis-independent unlike the raw diagonal sum. Pure states are rays in $\mathbb R^{2d}$7; notably, the one-dimensional state space collapses to a single point, $\mathbb R^{2d}$8, a degeneracy that resurfaces in error correction.

The exact complex representation uses the injective real-linear embedding $\mathbb R^{2d}$9 for $\mathcal S_d$0, together with the antiunitary $\mathcal S_d$1 satisfying $\mathcal S_d$2. The key characterization (Proposition 3.3) is that $\mathcal S_d$3 lies in the image of $\mathcal S_d$4 if and only if $\mathcal S_d$5, equivalently $\mathcal S_d$6. States map bijectively onto the $\mathcal S_d$7-fixed density sector via $\mathcal S_d$8, effects map without rescaling, and every probability is preserved. Two structural consequences follow: ranks double ($\mathcal S_d$9), so a pure quaternionic state becomes a rank-two complex projector; and "doubling" means exact simulation within the constrained sector, not identification with all of $\mathcal T_d$0.

The antiunitary Choi fixed-point criterion

The paper's first main theorem gives a Kraus-independent membership test. For a CPTP map $\mathcal T_d$1 on $\mathcal T_d$2, the following are equivalent: (i) $\mathcal T_d$3 admits a Kraus family with each $\mathcal T_d$4 commuting with $\mathcal T_d$5; (ii) its Choi matrix satisfies $\mathcal T_d$6, where $\mathcal T_d$7 is an antiunitary involution on the Choi-vector space. When these hold, a compatible family can be reconstructed from any $\mathcal T_d$8-fixed eigenbasis of $\mathcal T_d$9, and each Kraus operator descends uniquely to $\mathsf J_d^2 = -I$0; trace preservation descends to $\mathsf J_d^2 = -I$1. An equivalent formulation is equivariance $\mathsf J_d^2 = -I$2.

Two features deserve emphasis. First, the test is genuinely quaternionic, not a disguised reality condition: the paper exhibits a channel generated by $\mathsf J_d^2 = -I$3 whose output contains the noncommuting phase $\mathsf J_d^2 = -I$4 off-diagonally, yet satisfies the fixed-point equation. Second, the criterion locates the boundary of exact simulation—a generic complex CPTP map on $\mathsf J_d^2 = -I$5 leaves the quaternionic sector. The proof mechanism specializes the fixed-real-form Choi machinery known from channel imaginarity theory ($\mathsf J_d^2 = -I$6 antecedents) to the symplectic sector $\mathsf J_d^2 = -I$7.

Exact quaternionic error correction

The second main result is a Knill–Laflamme theorem for subspace codes $\mathsf J_d^2 = -I$8 of quaternionic dimension $\mathsf J_d^2 = -I$9. Exact correctability by a quaternionic channel is equivalent to the existence of a real positive-semidefinite Gram matrix $(g, \mathsf J_d)$0 with $(g, \mathsf J_d)$1—and this reality constraint is substantive, not definitional.

The subtlety is that over $(g, \mathsf J_d)$2 there are two inequivalent readings of "acts as a scalar." Requiring $(g, \mathsf J_d)$3 forces $(g, \mathsf J_d)$4 by right-linearity. But the left-compression equation $(g, \mathsf J_d)$5 admits imaginary solutions: the paper constructs an admissible noise family with $(g, \mathsf J_d)$6 that maps two distinct code states to the same output and is provably uncorrectable. Reality of the coefficients is therefore derived within the theorem's necessity proof, via the antiunitary symmetry applied to the doubled complex condition $(g, \mathsf J_d)$7: conjugation fixes the left side but conjugates $(g, \mathsf J_d)$8, forcing $(g, \mathsf J_d)$9 since $M_{2d}(\mathbb R)$0.

Sufficiency is constructive: diagonalizing $M_{2d}(\mathbb R)$1 by a real orthogonal matrix yields mutually orthogonal syndrome projections, and the recovery $M_{2d}(\mathbb R)$2 is completed to a trace-preserving channel by a term acting only off the syndrome space. Degenerate codes require no separate hypothesis—singular $M_{2d}(\mathbb R)$3 simply records redundant error combinations. A worked example with a non-diagonal Gram matrix shows that Kraus labels need not coincide with orthogonal syndrome labels. The necessity direction proceeds through the multiplicative-domain theorem after showing that quaternionic Hermitian observables generate the full corner algebra $M_{2d}(\mathbb R)$4—a generation step that fails precisely at $M_{2d}(\mathbb R)$5, matching the fact that the one-dimensional logical system has a single normalized state and can be "corrected" trivially without satisfying the compression equations. This $M_{2d}(\mathbb R)$6 boundary is shared with the complex case, but the paper identifies exactly where the proof breaks.

A dilation proposition supplies Stinespring-type structure on a Kraus-index direct sum without invoking a bipartite quaternionic tensor product—an important distinction, since noncommutativity prevents two right-$M_{2d}(\mathbb R)$7 modules from determining a canonical composite at all.

Composition, discard, and recoverability

The paper is explicit that the preceding results are single-system statements. Balancing $M_{2d}(\mathbb R)$8 requires choosing a compatible left action on one factor, and the naive simple-tensor inner product is not well defined on $M_{2d}(\mathbb R)$9: pairing $\mathsf J_d$0 against $\mathsf J_d$1 yields $\mathsf J_d$2 versus $\mathsf J_d$3. Several consistent frameworks exist (Razon–Horwitz tensor products, ordered circuits, Euclidean Jordan composites, the complex symplectic model), and they disagree: under the universal Jordan composite, $\mathsf J_d$4, whereas the standard embedded product gives $\mathsf J_d$5.

Once a composite and discard are fixed, however, the operational theory reduces to standard complex statements. Exact erasure is characterized by block Knill–Laflamme conditions $\mathsf J_d$6 for the erased marginal being constant—and the erased state need not be maximally mixed. A no-go result shows the maximally mixed requirement is too strong: it forces Choi rank $\mathsf J_d$7, hence $\mathsf J_d$8 when $\mathsf J_d$9. Approximate correction is controlled by the information–disturbance inequality $\rho_R := \tfrac12\mathcal T_d(\rho)$0, so Proposition-level stability holds at channel level, not merely at the level of individual reduced states or codeword concurrences. None of this creates a canonical quaternionic partial trace; the model dependence resides entirely in the prior choice of subsystems and discard.

Operational choices beyond associativity

For octonions the paper argues there is no unique continuation, and organizes six routes as a decision tree rather than competing notations. The obstruction is exact: $\rho_R := \tfrac12\mathcal T_d(\rho)$1, so no faithful unital homomorphism from $\rho_R := \tfrac12\mathcal T_d(\rho)$2 into an associative algebra preserves the full product; concretely, $\rho_R := \tfrac12\mathcal T_d(\rho)$3 under the standard Fano convention.

Para-linear Hilbert theory retains genuine octonionic amplitudes via para-linear maps and regular composition, yielding spectral theory and functional calculi. But regular composition is nonassociative, so Selinger's $\rho_R := \tfrac12\mathcal T_d(\rho)$4 construction does not apply directly, and an intrinsic positive cone, complete positivity, Choi object, and discard remain to be supplied.

Cochain-twisted categories relocate nonassociativity into a categorical associator via Albuquerque–Majid cochains on $\rho_R := \tfrac12\mathcal T_d(\rho)$5. The paper proves a dagger monoidal equivalence untwisting this category to ordinary graded Hilbert spaces, so $\rho_R := \tfrac12\mathcal T_d(\rho)$6 and the twisted algebra object becomes $\rho_R := \tfrac12\mathcal T_d(\rho)$7. The model has complete compositional semantics but is operationally equivalent to complex graded quantum theory—not a new scalar theory.

Fixed quaternionic sectors $\rho_R := \tfrac12\mathcal T_d(\rho)$8 inside $\rho_R := \tfrac12\mathcal T_d(\rho)$9 inherit the entire quaternionic apparatus through Corollary 6.2, with covariance under $\Tr_R \mathcal T_d(A) = 2\,\mathrm{Re}\,\Tr_C A$0. The limitation is precise: noise moving amplitudes across incompatible sectors depends on associators and is not covered.

Jordan models keep octonionic state geometry: $\Tr_R \mathcal T_d(A) = 2\,\mathrm{Re}\,\Tr_C A$1 is the special spin factor $\Tr_R \mathcal T_d(A) = 2\,\mathrm{Re}\,\Tr_C A$2, while only $\Tr_R \mathcal T_d(A) = 2\,\mathrm{Re}\,\Tr_C A$3 (the Albert algebra) is exceptional, with pure states forming the Cayley plane. Imported boundary results show exceptional factors cannot occur as nonclassical composite factors except against classical systems, and higher spin factors cannot be added to the dagger-compact framework while preserving compact closure.

Clifford envelopes represent left multiplications $\Tr_R \mathcal T_d(A) = 2\,\mathrm{Re}\,\Tr_C A$4 generating $\Tr_R \mathcal T_d(A) = 2\,\mathrm{Re}\,\Tr_C A$5 inside $\Tr_R \mathcal T_d(A) = 2\,\mathrm{Re}\,\Tr_C A$6; channels here are channels of an encoding, measuring the associator defect $\Tr_R \mathcal T_d(A) = 2\,\mathrm{Re}\,\Tr_C A$7 rather than intrinsic octonionic dynamics. Finite Moufang registers encode the loop $\Tr_R \mathcal T_d(A) = 2\,\mathrm{Re}\,\Tr_C A$8 as labels on $\Tr_R \mathcal T_d(A) = 2\,\mathrm{Re}\,\Tr_C A$9, supporting multiplication oracles and bracketing comparisons with complex amplitudes throughout.

Limitations and open problems

The paper is candid about scope. All quaternionic results are finite-dimensional and single-system; no canonical multipartite quaternionic tensor product exists, and erasure statements apply only after a composite is externally chosen. The para-linear route lacks an intrinsic CPTP/QEC package; the cochain-twist route is shown to be operationally equivalent to complex theory; sector restriction excludes cross-sector noise; and Jordan composite restrictions are imported rather than derived. Three problems are stated explicitly: defining complete positivity, discard, Choi objects, and Stinespring dilation intrinsically for para-linear morphisms; constructing controlled dynamics moving coherently among quaternionic sectors; and formulating recovery criteria for ternary bracketing defects or multi-time nonassociative processes.

Conclusion

The paper establishes that exact ambient representation never dissolves the selecting structure: $\mathbb R^{2d}$00 inside real coordinates, $\mathbb R^{2d}$01 inside complex coordinates, and model-specific closure beyond associativity form a hierarchy in which each level answers which ambient degrees of freedom are redundant, which symmetry selects the physical sector, and which composition preserves it. Within the associative noncommutative case, the Choi fixed-point criterion and the real-coefficient Knill–Laflamme theorem with explicit right-$\mathbb R^{2d}$02-linear recovery make the quaternionic operational layer exact. Beyond associativity, the comparison prevents conflation of six genuinely different models and specifies the data missing before a global octonionic channel or recovery theory can be stated.