Information Conservation and the Emergence of Complex Quantum Amplitudes

Information Conservation and the Emergence of Complex Quantum Amplitudes Zenodo DOI: 10.5281/zenodo.17741392 Paul Cooney1, ∗ 1Independent Researcher, Innisfil, Ontario, Canada (Dated: November 27, 2025) We derive the complex Hilbert-space structure of quantum mechanics from information-theoretic principles applied to coarse-grained physical theories. Starting from an ontology of microhistories—fully specified microscopic trajectories ordered by some parameter—we define branches as equivalence classes of operationally indistinguishable microhistories. Five physically motivated axioms governing the structure of branch states—finite information capacity, convex mixtures, reversible information-preserving dynamics, local tomography, and continuous reversibility—uniquely determine complex projective space as the pure-state manifold. The Born rule emerges from symmetry constraints on branch multiplicities combined with Gleason’s theorem. No prior assumptions about complex numbers, superposition, or quantum probability are made. The reconstruction unifies operational, information-geometric, and path-integral perspectives, demonstrating that quantum theory is the unique effective description of any physical system admitting a coarsegrained microhistory structure with the specified information-theoretic properties. I. INTRODUCTION Quantum mechanics employs complex vector spaces, unitary evolution, and Born-rule probabilities. While extraordinarily successful empirically, the physical origin of these mathematical structures remains conceptually opaque. Why complex rather than real or quaternionic amplitudes? Why unitary dynamics? Why squared-amplitude probabilities? ∗paul.co[email protected]to.ca 2 Several reconstruction programs address these questions from different starting points. Hardy’s operational axioms [1], the Chiribella–D’Ariano–Perinotti (CDP) purification postulate [2], and the Masanes–M¨uller causality-based approach [3] successfully recover complex Hilbert space from physically motivated constraints on preparations, transformations, and measurements. This work takes a complementary perspective. We begin with an ontological assumption: the fundamental description consists of microhistories—fully specified microscopic configurations ordered by some parameter λ. Physical observations access only coarse-grained information, partitioning microhistories into branches: equivalence classes yielding identical predictions for all accessible measurements. We show that five information-theoretic axioms governing branch structure uniquely force complex Hilbert space as the effective state space. The reconstruction proceeds in four steps: 1. Sections II–III: Define microhistories, operational observables, and branch equivalence classes. 2. Section IV: State five information-theoretic axioms constraining branch dynamics. 3. Section V: Prove that real and quaternionic Hilbert spaces violate local tomography, and show that information geometry forces a unique K¨ahler structure isomorphic to complex projective space. 4. Section VI: Derive the Born rule from branch-multiplicity symmetry and Gleason’s theorem. Section VII discusses conceptual implications and connections to other reconstruction programs. The appendices provide technical details on information geometry and branch symmetries. a. Notation. We use natural units ℏ=c= 1 except where explicitly restored for clarity. Microhistories are denoted H, branches [H]orB, the ordering parameter λ, and the effective state space S. 3 II. MICROHISTORIES AND OPERATIONAL OBSERVABLES A. The Ontological Layer We posit that the fundamental description of a physical system consists of microhistories: fully specified microscopic configurations evolving along an ordering parameter λ. For concreteness, λmay represent: •physical time tin non-relativistic mechanics, •proper time τalong worldlines in relativity, •a discrete update index in cellular automata, •any monotonic parameter defining a total ordering on microscopic states. [Microhistory] A microhistory is a trajectory H={X(λ)}λf λi,(1) where X(λ) specifies the complete microscopic state at each value of the ordering parameter λ∈[λi, λf]. The microscopic ontology X(λ) is intentionally left general: it may represent field configurations, particle positions and momenta, spin configurations, or any other complete specification. We require only that the set H([λi, λf]) of dynamically admissible microhistories is well-defined for every interval. Crucially, no quantum or linear structure is assumed at this stage. Microhistories may evolve deterministically, stochastically, or via discrete update rules. The dynamics need not preserve any norm, admit superposition, or satisfy locality. B. Operational Observables Physical measurements access only coarse-grained, macroscopic observables, not full microhistories. [Observable] An observable O∈ O is a functional O:H([λi, λf]) ×[λi, λf]→R,(2) 4 extracting a real number O[H, λ] from a microhistory Hat ordering value λ. The set Ois determined by experimental capabilities. Two microhistories differing only in degrees of freedom inaccessible to available detectors yield identical values for all O∈ O. In a quantum field theory, X(λ) might specify field values ϕ(x, λ) at all spacetime points. Observables O∈ O include only spatially averaged quantities, correlation functions at detector resolution, and integrated energies—not point values ϕ(x0, λ0). [Double-Slit Experiment] Consider a particle traversing a double-slit apparatus. Microhistories Hspecify the complete trajectory through space, including which slit (if any) the particle passes through. If no which-path detector is present, observables O∈ O include only the final screen position xscreen—not the trajectory details. All microhistories ending at the same screen location belong to the same branch, regardless of which slit they passed through. This naturally explains interference: the branch structure coarse-grains over path details, and the effective dynamics (quantum mechanics) must account for contributions from all trajectories in the equivalence class. This coarse-graining is not a limitation but a fundamental feature of real experiments: measurement devices have finite resolution, finite temporal response, and finite sensitivity. III. OPERATIONAL EQUIVALENCE AND BRANCHES A. Operational Indistinguishability Two microhistories are operationally equivalent if no accessible measurement can distinguish them, either presently or in the future. [Operational Equivalence] Two microhistories H, H′∈ H([λi, λf]) are operationally equivalent at λ∗, written H∼λ∗H′, if: 1. For all observables O∈ O, O[H, λ∗]=O[H′, λ∗],(3) 2. For all λ>λ∗, the probability distributions over future observable values—conditioned on the state at λ∗and any allowed interventions—are identical for Hand H′. Condition (1) ensures present observables agree. Condition (2) ensures future predictions agree. Together, they capture the operational content of a physical state: once Hand H′ 5 agree on all presently measurable information and generate identical conditional probabilities for all future measurements, they represent the same physical situation for all practical purposes. [Connection to Decoherence] In systems with environmental interactions, condition (2) is typically satisfied when microscopic histories have been entangled with large environmental degrees of freedom. Decoherence selects preferred “pointer states” [4], and operationally equivalent microhistories correspond to those lying within the same pointer basis. However, our framework does not require decoherence—operational equivalence is defined purely by predictive indistinguishability. B. Branches as Equivalence Classes [Branch] A branch B(λ∗) is an equivalence class of microhistories under ∼λ∗: B(λ∗)=[H] = {H′∈ H([λi, λf]) : H′∼λ∗H}.(4) The set of all branches at λ∗is B(λ∗)=H([λi, λf])/∼λ∗.(5) Branches are the effective states of the theory. They retain exactly the operationally relevant information while discarding inaccessible microscopic details. An observer who knows which branch the system occupies at λ∗can predict all future measurement outcomes with the same accuracy as an observer with complete microhistory information. [Classical Phase Space] In classical mechanics, microhistories are phase-space trajectories (q(t), p(t)). If measurement precision is finite, branches are small phase-space cells. For perfect precision, each branch contains a single trajectory. [Quantum Path Integral] In the path-integral formulation, microhistories are field configurations ϕ(x, t). Branches correspond to sets of paths contributing to the same effective amplitude after environmental tracing. The branch structure depends on which observables are accessible. [Spin Measurement] Consider a spin-1/2 particle prepared in state |ψ⟩=α|↑⟩ +β|↓⟩ along the z-axis, where |α|2+|β|2= 1. Before measurement, there exists a single branch containing all microscopic spin configurations consistent with this preparation. Upon measuring Sz, the system splits into two branches: 6 •Branch B↑: all microhistories yielding outcome +ℏ/2, •Branch B↓: all microhistories yielding outcome −ℏ/2. Within each post-measurement branch, microscopic details may vary (environmental degrees of freedom, exact detector dynamics), but all microhistories in B↑agree on future predictions: subsequent measurements of Szwill yield +ℏ/2 with certainty. The branches are operationally distinguishable because they predict different outcomes for future Szmeasurements. Crucially, if we instead measure Sx, the system splits into different branches B→and B←. The branch structure is observable-dependent—there is no unique partition of microhistories into branches independent of measurement context. This contextuality is a feature, not a bug: it reflects the operational definition of equivalence. C. Why This Framework? The microhistory-branch construction provides a physical interpretation for effective quantum states: they are equivalence classes of microscopic configurations indistinguishable to observers with finite resolution. This answers the question “what is a quantum state?” without invoking wavefunction collapse, hidden variables, or many worlds. Moreover, the framework is general: it applies to any theory admitting a microscopic ontology and a coarse-graining procedure. The reconstruction that follows will show that if branches satisfy five physically natural information-theoretic constraints, then the effective state space must be a complex Hilbert space with unitary dynamics and Born-rule probabilities. Quantum mechanics emerges not from fundamental quantum postulates but from universal features of information compression. IV. INFORMATION-THEORETIC AXIOMS We now formalize constraints on the effective state space Sof branches. These axioms describe how information stored in branches can be mixed, transformed, and composed. Importantly, we do not assume linearity, complex structure, inner products, or the Born rule. These will emerge as consequences. 7 A. Axiom 1: Finite Information Capacity [Finite Information] For any finite spatial region and any finite set of accessible observables, the number of operationally distinguishable branches is finite or countably infinite. The effective state space Sadmits a finite-dimensional real vector-space embedding. a. Physical Motivation. No finite apparatus can distinguish infinitely many mutually exclusive macroscopic states. Measurement precision and detector sensitivity impose fundamental limits on information capacity. Axiom IV A encodes the Bekenstein–Holevo bound [5] at a pre-quantum level: finite regions store finite information. b. Mathematical Content. Sis a compact convex subset of RNfor some finite N. This excludes pathological infinite-information theories while permitting classical probability theory (S= probability simplex) and quantum mechanics (S= density operators). B. Axiom 2: Convex Mixtures [Convexity] If ω1, ω2∈ S are preparable states, then for all p∈[0,1], the convex combination ω=p ω1+(1−p)ω2(6) represents a state preparable by selecting ω1with probability pand ω2with probability 1−p. a. Physical Motivation. Classical uncertainty over preparation procedures combines linearly. If an experimenter prepares ω1on Mondays and ω2on Tuesdays, the ensemble state on a randomly chosen day is the mixture (6). This is not quantum superposition—it is the operational fact that ignorance about which preparation occurred is represented by a weighted average. b. Mathematical Content. Sis a convex set. Pure states (extremal preparations) lie on the boundary ∂S; mixed states lie in the interior. C. Axiom 3: Reversible Information-Preserving Dynamics Underlying microscopic dynamics typically conserve distinguishability: deterministic evolution preserves phase-space volume (Liouville’s theorem), and stochastic dynamics preserve 8 ensemble information. At the coarse-grained level, this manifests as norm preservation. [Reversible Dynamics] There exists a one-parameter family of transformations {T∆λ: S → S} satisfying: 1. T0= id, 2. T∆λ1+∆λ2=T∆λ2◦T∆λ1(group property), 3. Each T∆λis bijective (reversibility), 4. There exists a norm ∥·∥ on Ssuch that ∥T∆λω∥=∥ω∥ ∀ω∈ S,∆λ. (7) a. Physical Motivation. Reversible microscopic evolution conserves global distinguishability. Observers cannot lose the ability to differentiate initially distinguishable states through reversible processes. Axiom IV C ensures that the effective theory respects this conservation at the coarse-grained level. b. Mathematical Content. The norm ∥·∥ is not specified a priori. In classical probability, ∥ρ∥=Pi|ρi|(total variation). In quantum mechanics, ∥ρ∥= Tr(ρ) (trace norm). The specific form will be determined by Axioms IV D and IV E. D. Axiom 4: Local Tomography To describe composite systems, we require that joint information is recoverable from local correlations. [Local Tomography] For subsystems Aand Bwith state spaces SAand SB, the composite system AB has state space SAB satisfying: 1. Tensor product structure. For pure states ωA∈ SA,ωB∈ SB, the product state ωA⊗ωB∈ SAB exists. 2. Local determination. Joint statistics of all locally accessible measurements on A and Buniquely determine the global state in SAB. 9 a. Physical Motivation. If Alice measures Aand Bob measures B, their joint statistics ⟨OA⊗OB⟩contain all operationally accessible information about the composite system. Local tomography asserts that no “hidden correlations” exist beyond what is detectable through local and jointly local measurements. b. Discriminatory Power. Local tomography is satisfied by classical and complexquantum theories but fails for real and quaternionic quantum mechanics [6, 7]. This axiom therefore plays a crucial role in selecting complex Hilbert space uniquely. E. Axiom 5: Continuous Reversibility To exclude discrete classical theories (finite state spaces with permutation dynamics), we impose continuity of reversible transformations. [Continuous Reversibility] The reversible transformations {T∆λ}act transitively and continuously on the pure-state manifold P=∂S. That is: 1. For any pure states ω1, ω2∈ P, there exists ∆λsuch that T∆λω1=ω2. 2. The map (∆λ, ω)7→ T∆λωis continuous in both arguments. a. Physical Motivation. Physical systems exhibit continuous symmetries. Rotations of spin states, time translations, and phase transformations all act smoothly on state space. Axiom IV E ensures that Pis a connected, continuously symmetric manifold, enabling information-geometric analysis. F. Summary The five axioms collectively express: •Finite information resolution (Axiom IV A), •Linear representation of classical uncertainty (Axiom IV B), •Conservation of distinguishability (Axiom IV C), •Absence of hidden correlations (Axiom IV D), •Continuous symmetry (Axiom IV E). 16 By Axiom IV B (convexity), probability assignments are linear over mixtures. By Axiom IV E (continuous reversibility), probabilities vary continuously with the state. Combined with the rational-amplitude result (19), Gleason’s theorem uniquely fixes p(Pk) = ⟨ψ|Pk|ψ⟩=|⟨k|ψ⟩|2.(22) [Dimension Restriction] Gleason’s theorem requires dim(H)≥3. For dim(H)=2 (qubits), additional symmetry arguments or appeal to composite systems suffice to derive the Born rule [11]. E. Summary The Born rule emerges from: 1. Branch-multiplicity symmetry: Equal amplitudes ⇒equal probabilities, 2. Rational extension: Ancilla construction for rational amplitude ratios, 3. Gleason’s theorem: Noncontextuality, additivity, and continuity uniquely determine p(k) = |⟨k|ψ⟩|2. No separate postulate is required—the Born rule is a consequence of the informationtheoretic structure imposed by Axioms IV A–IV E. VII. DISCUSSION AND CONCEPTUAL IMPLICATIONS A. What Has Been Derived We have shown that if a physical theory admits: •an ontology of microhistories, •a coarse-graining into operationally indistinguishable branches, •finite information capacity, convex mixtures, reversible norm-preserving dynamics, local tomography, and continuous symmetry, 17 then the effective state space is necessarily a complex Hilbert space with unitary evolution and Born-rule probabilities. Quantum mechanics is not fundamental—it is the unique effective description of any system satisfying these constraints. B. Interpretation of Quantum States Effective quantum states |ψ⟩are informational summaries of branch equivalence classes. The wavefunction does not represent a physical wave but encodes: •which macroscopic configurations are compatible with observations, •the relative weights (multiplicities) of microhistories within each configuration, •conditional probabilities for future measurements. This resolves the “what is ψ?” question without invoking collapse, hidden variables, or many worlds: ψis a compressed representation of operationally relevant microhistory information. C. Relation to Decoherence The branch framework naturally accommodates decoherence-based interpretations. When a system interacts with a large environment, interference between macroscopically distinct configurations is dynamically suppressed. The resulting pointer states correspond precisely to operationally distinguishable branches. However, our reconstruction does not require decoherence. Operational equivalence is defined by predictive indistinguishability, which may arise through decoherence, fundamental coarse-graining, or any other mechanism that partitions microhistories into classes with identical future statistics. D. Relation to Other Reconstructions •Hardy [1]: Begins with operational theories of preparations, transformations, and effects. Our microhistory-branch framework provides a physical ontology underlying 18 Hardy’s abstract operations. •CDP [2]: Uses purification as a central postulate. Our Axiom IV D implies purification: any mixed state on Acan be realized as a marginal of a pure state on A⊗B. •Masanes–M¨uller [3]: Emphasizes causality and information-theoretic consistency. Our approach complements this by grounding operational axioms in a microhistory ontology. The present work unifies these perspectives: operational axioms, information geometry, and microhistory-based path integrals all lead to the same conclusion—complex Hilbert space is inevitable. E. What is Not Derived The reconstruction establishes the mathematical structure of quantum theory but does not address: •Specific Hamiltonians: The generator Hof time evolution must come from additional physical input (e.g., symmetries, field content). •Planck’s constant ℏ:The normalization of the action is not determined by informationtheoretic axioms alone. •Measurement problem: We clarify the information structure of outcomes but do not select a specific interpretation (Copenhagen, Everett, GRW, etc.). •Microscopic dynamics: The evolution of microhistories X(λ) remains modeldependent. Thus, the reconstruction explains why quantum mechanics has its form but not what the world is made of or which Hamiltonian governs it. F. Broader Implications a. Layered View of Physics. The derivation suggests a two-tier structure: 19 1. Microscopic ontology: Microhistories X(λ) evolving according to some (potentially non-quantum) dynamics. 2. Effective theory: Branches [H] obeying complex Hilbert-space rules due to information compression. Quantum mechanics resides at the effective level. Its mathematical form is universal—any microhistory theory with the right information-theoretic properties yields the same effective description. b. Testability. The reconstruction itself is not directly testable (it is a structural result), but it makes quantum mechanics more falsifiable: if future experiments reveal violations of local tomography or reversibility, quantum theory would need revision—not as a fundamental postulate, but as an emergent effective description whose underlying assumptions have been violated. c. Extensions. Natural directions for future work include: •Generalization to indefinite causal structure [12], •Application to quantum field theory and holographic duality, •Exploration of alternative information measures (non-monotone metrics, non-convex state spaces), •Connection to specific microhistory models (e.g., lattice theories, discrete causal sets, cellular automata). VIII. CONCLUSION We have derived the complex Hilbert-space structure of quantum mechanics—including unitary dynamics and the Born rule—from five physically motivated information-theoretic axioms applied to coarse-grained microhistory ensembles. The reconstruction shows that quantum theory is not a fundamental postulate but the unique effective framework for any physical system admitting a branch structure satisfying finite information capacity, convexity, reversible information preservation, local tomography, and continuous symmetry. 20 This provides a conceptually transparent answer to foundational questions: quantum states are informational summaries of operationally indistinguishable microhistories; complex amplitudes emerge from K¨ahler geometry induced by information conservation; and Born-rule probabilities follow from branch-multiplicity symmetry. The result unifies operational, information-geometric, and path-integral perspectives, offering a coherent picture of why nature employs the mathematical structures of quantum theory. ACKNOWLEDGMENTS The author thanks the quantum foundations community for decades of work clarifying the operational and information-theoretic underpinnings of quantum mechanics. [1] L. 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