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Integrated Information Theory on CP²: Geometric, Algebraic, and Phenomenological Extensions for Qutrit Systems

Trent Slade

Published Dec 10, 2025• 0 citations
Publisher

Abstract

This research report develops a rigorous extension of Integrated Information Theory (IIT) from classical binary architectures to quantum systems whose internal structure is governed by the geometry of the complex projective plane CP² and the algebra of SU(3). Whereas standard IIT 3.0 assumes binary partitions and factorized probability distributions, this framework—“Qutrit Integrated Information Theory” (QIIT)—defines mechanisms as SU(3) quantum channels, states as points on CP² equipped with the Fubini–Study metric, and partitions as subalgebraic restrictions of the Lie algebra su(3). Integrated information Φ is reformulated using geometric information loss under minimal partitions of SU(3), measured via Fubini–Study distance and quantum Wasserstein metrics. QIIT predicts novel forms of irreducibility, triadic logic, curvature-driven binding, non-Abelian holonomy in recurrent thought, and expanded phenomenological structures not expressible in binary IIT. The theory reduces exactly to classical IIT when dynamics are restricted to the SU(2) isospin subalgebra (CP¹). The report provides explicit derivations, SU(3) partition taxonomies, geometric calculations on CP², generalized cause–effect repertoires, and proposals for both simulation (tensor networks) and physical experiments (photonic qutrits using Chrestenson/Fourier gates). This establishes a mathematically coherent and experimentally testable foundation for quantum extensions of IIT.

DOI
10.5281/zenodo.17874637

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This research report develops a rigorous extension of Integrated Information Theory (IIT) from classical binary architectures to quantum systems whose internal structure is governed by the geometry of the complex projective plane CP² and the algebra of SU(3). Whereas standard IIT 3.0 assumes binary partitions and factorized probability distributions, this framework—“Qutrit Integrated Information Theory” (QIIT)—defines mechanisms as SU(3) quantum channels, states as points on CP² equipped with the Fubini–Study metric, and partitions as subalgebraic restrictions of the Lie algebra su(3). Integrated information Φ is reformulated using geometric information loss under minimal partitions of SU(3), measured via Fubini–Study distance and quantum Wasserstein metrics. QIIT predicts novel forms of irreducibility, triadic logic, curvature-driven binding, non-Abelian holonomy in recurrent thought, and expanded phenomenological structures not expressible in binary IIT. The theory reduces exactly to classical IIT when dynamics are restricted to the SU(2) isospin subalgebra (CP¹). The report provides explicit derivations, SU(3) partition taxonomies, geometric calculations on CP², generalized cause–effect repertoires, and proposals for both simulation (tensor networks) and physical experiments (photonic qutrits using Chrestenson/Fourier gates). This establishes a mathematically coherent and experimentally testable foundation for quantum extensions of IIT.
Classical anthropic reasoning relies on a binary observer model that treats existence as a yes/no variable, generating persistent paradoxes in self-locating uncertainty, indexical probability, and observer-selection effects. This work introduces Qutrit Anthropics, a triadic framework that models the observer as an SU(3)-symmetric quantum-geometric entity embedded in the complex projective plane CP². Using the Fubini–Study metric, symplectic volume, and the algebraic structure of SU(3), the paper resolves classical paradoxes such as Sleeping Beauty, the Doomsday Argument, observer-moment duplication, and SIA inflation. The framework demonstrates that anthropic paradoxes arise from forcing binary logic onto a system whose natural geometry is triadic, curved, and compact. Qutrit Anthropics reframes observerhood as a bundle section over CP², establishing a unified quantum-geometric basis for probability, existence, and indexical reasoning.
This review investigates deep structural links between spectral geometry on curved manifolds and the architecture of qutrit-based quantum information. Focusing on Laplace–Beltrami operators, harmonic spectra, and eigenfunction structures on complex projective spaces such as ℂP², the study surveys how geometric mode families encode ternary logic, SU(3) representations, and qutrit control protocols. The analysis integrates differential geometry, representation theory, and quantum control, highlighting that SU(3) qutrit dynamics can often be interpreted as discrete manifestations of continuous geometric symmetries. Special attention is given to Gell-Mann generators, coherent-state constructions, spectral gaps, and geometric phases. Across the surveyed literature, a unifying theme emerges: the eigenstructure of curved manifolds provides both the natural harmonic basis and the dynamical substrate for triadic quantum information processing.
Originally conceived as a theory of consciousness, integrated information theory (IIT) provides a theoretical framework intended to characterize the compositional causal information that a system, in its current state, specifies about itself. However, it remains to be determined whether IIT as a theory of consciousness is compatible with quantum mechanics as a theory of microphysics. Here, we present an extension of IIT's latest formalism to evaluate the mechanism integrated information (φ) of a system subset to discrete, finite-dimensional quantum systems (e.g., quantum logic gates). To that end, we translate a recently developed, unique measure of intrinsic information into a density matrix formulation and extend the notion of conditional independence to accommodate quantum entanglement. The compositional nature of the IIT analysis might shed some light on the internal structure of composite quantum states and operators that cannot be obtained using standard information-theoretical analysis. Finally, our results should inform theoretical arguments about the link between consciousness, causation, and physics from the classical to the quantum.
Integrated Information Theory (IIT) has emerged as one of the leading research lines in computational neuroscience to provide a mechanistic and mathematically well-defined description of the neural correlates of consciousness. Integrated Information (ΦΦΦ) quantifies how much the integrated cause/effect structure of the global neural network fails to be accounted for by any partitioned version of it. The holistic IIT approach is in principle applicable to any information-processing dynamical network regardless of its interpretation in the context of consciousness. In this paper we take the first steps towards a formulation of a general and consistent version of IIT for interacting networks of quantum systems. A variety of different phases, from the dis-integrated (Φ=0Φ=0Φ=0) to the holistic one (extensive log⁡Φ\logΦlogΦ), can be identified and their cross-overs studied.
This work develops Spectral Algebraics, a unified framework connecting the continuous spectrum of the Laplace–Beltrami operator on curved manifolds to the discrete algebra of SU(3) qutrit systems. By treating CP² and its associated line bundles as the geometric substrate of three-state quantum logic, the analysis shows that qutrit basis states emerge naturally as the lowest-energy sections of the connection Laplacian, while triadic interactions correspond to SU(3)’s symmetric d-tensor via geometric mode-overlap integrals.The manuscript constructs this bridge from spectral geometry, heat-kernel asymptotics, and representation theory through to field-theoretic interactions and spectral flow. The result is a programmable dictionary between geometric deformation and qutrit dynamics, suggesting that computation may be engineered through spectral shaping of cavities and metamaterials. The work provides a complete theoretical foundation for implementing qutrit hardware using Laplace–Beltrami eigenmodes.
Integrated Information Theory (IIT) has emerged as one of the leading research lines in computational neuroscience to provide a mechanistic and mathematically well-defined description of the neural correlates of consciousness. Integrated Information (Φ\PhiΦ) quantifies how much the integrated cause/effect structure of the global neural network fails to be accounted for by any partitioned version of it. The holistic IIT approach is in principle applicable to any information-processing dynamical network regardless of its interpretation in the context of consciousness. In this paper we take the first steps towards a formulation of a general and consistent version of IIT for interacting networks of quantum systems. A variety of different phases, from the dis-integrated (Φ=0\Phi=0Φ=0) to the holistic one (extensive log⁡Φ\log\PhilogΦ), can be identified and their cross-overs studied.