This paper aims to establish a general theoretical framework that, starting fromtwo basic axioms about the realizability of the physical world, deduces the necessary form of algebraic structures required to describe quantum systems. These twoaxioms are: Generalized Information Flow Constraint (G1) and Finite-Step Computability (G2). We prove that for open quantum systems (non-Hermitian systems)incorporating dissipation and gain, their observable algebra must be organized intoa generalized factor with a specific “weight” structure. Under the requirement offinite generation (G2), the “relative size” index of subalgebra embeddings exhibitsdiscreteness, which constitutes the mathematical foundation for the emergent generalized braiding algebras and dissipative quantum error-correcting codes in a classof non-Hermitian topological orders. This work not only provides a unified algebraicperspective for understanding non-Hermitian topological matter but also proposesexperimental verification ideas based on dissipative topological superconductingsystems and noisy quantum processors.
This paper aims to establish a general theoretical framework that, starting fromtwo basic axioms about the realizability of the physical world, deduces the necessary form of algebraic structures required to describe quantum systems. These twoaxioms are: Generalized Information Flow Constraint (G1) and Finite-Step Computability (G2). We prove that for open quantum systems (non-Hermitian systems)incorporating dissipation and gain, their observable algebra must be organized intoa generalized factor with a specific “weight” structure. Under the requirement offinite generation (G2), the “relative size” index of subalgebra embeddings exhibitsdiscreteness, which constitutes the mathematical foundation for the emergent generalized braiding algebras and dissipative quantum error-correcting codes in a classof non-Hermitian topological orders. This work not only provides a unified algebraicperspective for understanding non-Hermitian topological matter but also proposesexperimental verification ideas based on dissipative topological superconductingsystems and noisy quantum processors.
Based on the two fundamental principles—Information Conservation (A1) andFinite-Step Computability (A2)—this paper derives the core structures of operatoralgebras, namely subfactors, the Jones index, and planar algebras. The researchdemonstrates that the factor structure of the observable algebra in quantum systems, the finite index of subfactor inclusions, and its discrete set of possible values(the Jones index) are not pure mathematical abstractions, but rather the inevitablealgebraic forms of organization for physical systems under information constraintsand realizability requirements. Information Conservation necessitates the existenceof a unique finite faithful trace on the algebra, forcing the observable algebra tomanifest as a type II factor. Computability requires the algebraic description tobe finitely generated, which locks the index of a subfactor inclusion to be finite andquantizes its spectrum into the classical Jones index set. This discrete spectrum directly corresponds to planar algebras (e.g., Temperley–Lieb algebras) which encodethe braiding statistics of anyons in topological quantum systems and the algebraicproperties of fault-tolerant quantum error-correcting codes. Furthermore, this paper proposes two operable experimental verification schemes: validating Jones polynomial evaluations by measuring the statistical phases of anyon braiding in quantumHall systems, and indirectly determining the effective dimension parameter relatedto the Jones index by analyzing the scaling behavior of logical error rates in surface codes implemented on superconducting quantum processors. These schemesestablish a falsifiable, unified bridge between abstract algebra, topological order,and quantum information.
This paper aims to provide a unified physical origin explanation for topological order that transcends the Landau paradigm, starting from principled axioms.Based on two fundamental axioms—Information Conservation (A1) and Finite-StepComputability (A2)—we argue that topological order is not an accidental productof specific models, but rather a macroscopic quantum structure that inevitablyemerges from quantum many-body systems under the joint action of A1 (globalinformation constraints) and A2 (local interaction rules) in order to achieve robustnon-local information storage and processing. This paper systematically demonstrates that core features such as topological degeneracy, anyonic fractional statistics, and topological entanglement entropy can be logically derived from these twoaxioms, and integrates topological order with many-body dynamics from a unifiedspatiotemporal perspective, further proposing numerical and experimental testingschemes. This framework provides a new principled foundation for understandingthe nature, classification, and role of topological order in quantum computing andquantum gravity
This paper proposes and develops a research framework for deriving fundamental mathematical structures from underlying physical axioms. We prove that,starting from two basic principles that a self-referential computational system mustsatisfy Information Conservation (A1) and Computability (A2) the mathematicaldescription of its evolutionary process is necessarily organized into a specific category. The objects of this category correspond to discrete semantic levels of thesystem state, and the morphisms are completely positive trace-preserving (CPTP)maps, which directly embody the constraint of information conservation. The tensor product structure and dagger structure possessed by the category originate,respectively, from the parallel operations permitted by computability and the formal time-reversal symmetry implied by information conservation. Ultimately, wesystematically derive a complete dagger-compact category. Distinct from purelymathematical constructions, the credibility of this framework is rooted in an empirically testable physical prediction: in a quantum processor, the complexity growthdynamics of the system state should exhibit a periodic oscillation pattern related tothe logarithm of the system size (ln N). This explicit prediction provides a crucialfalsification condition for the theory.
The indispensability of the complex structure in quantum mechanics has beenexperimentally confirmed, but its algebraic origin—the fact that the complex number field is the only commutative, algebraically closed division algebra over the realnumbers—has not yet been rigorously linked to an interpretive framework of quantum reality. This paper proposes an axiomatic system for a latent world algebra,decomposing the descriptive structure of quantum states into a direct sum of anobservable subalgebra and a latent world subalgebra, and defining the multiplication rules and commutation relations between them. It is proved that this algebraicstructure is a concrete realization of a C*-algebra, where self-adjoint elements correspond to the algebraic representation of observables, and non-self-adjoint elementsencode the algebraic information of modal dimensions. Furthermore, the square-lawclassification of the latent world subalgebra and its correspondence with Cliffordalgebras are explored, interpreting the difference between J2 = −I (complex structure) and J2 = +I (real structure) as two distinct physical modalities. On thisbasis, the algebraic framework is extended to quaternionic and octonionic cases,arguing that the non-associativity of octonions may correspond to the non-localityof quantum entanglement. Finally, topological invariants such as Chern numbersand characteristic classes are represented as numerical values obtained from thepairing of central cohomology classes of the latent world algebra with K-theory,establishing an equivalence relation between topological transitions and algebraicprojection mappings. This study provides a rigorous mathematical foundation forunderstanding the algebraic necessity of quantum structures and opens a formalizedpath for exploring hypercomplex quantum theories.
This paper proposes an axiomatic framework based on three criteria—structural stability (S), information efficiency (E), and dynamical accessibility (D)—to rigorously verify the emergence of Z topological order in the Kitaev Toric Code model. These criteria provide operational, computable, and falsifiable definitions of “emergence” from three dimensions: physical robustness, information-theoretic compression efficiency, and experimental realizability. In the Toric Code model, topological degeneracy leads to an enhancement of the information efficiency criterion with increasing system size, exhibiting the characteristic of “scale-enhanced order.” Numerical verification shows that for systems with size L ≥ 2, all three criteria exceed the empirical threshold (> 0.5), and the comprehensive emergence strength parameter β′ also grows with system size, confirming the robustness of Z topological order as a stable emergent phase. This work provides a quantifiable theoretical tool for the study of emergence in topological quantum matter and emphasizes the central role of mathematical rigor in interdisciplinary research.
We present a mathematically rigorous framework for non-Hermitian topological phases in open quantum-biological systems, enabling room-temperature, topologically protected information transfer between quantum states and biological molecular dynamics. The theory is built on generalized non-Hermitian Chern numbers, Berry phase matching, and unitary conformal mapping, with a strict fidelity constraint of F≥99.95% and a non-Hermitian backscattering-free condition for edge-state transport. This work unifies non-Hermitian quantum mechanics, topological condensed matter physics, and biological quantum dynamics, offering testable predictions for coherence enhancement in photosynthetic complexes, topological protection in ion channels, and DNA topological information encoding. All results are derived from open axiomatic quantum theory with no confidential content, providing a new theoretical foundation for next-generation quantum-biological interfaces.