The Emergence of Probability: Deriving the Born Rule from Global Unitarity and Observer-Relative States in Quantum Cellular Automata

The Emergence of Probability: Deriving the Born Rule from Global Unitarity and Observer-Relative States in Quantum Cellular Automata Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract Quantum cellular automata (QCA) provide a discrete, local, unitary ontological picture of the universe: the global state evolves via deterministic unitary operators between discrete time steps, containing no intrinsic stochastic process. On the other hand, laboratory quantum mechanics centers on the Born rule, treating measurement outcomes as inherently probabilistic events. The tension between these two perspectives constitutes the core of the quantum measurement problem. In this paper, within the QCA framework, we introduce local observers with nite information capacity and their environmental entanglement structure, providing a purely combinatorial mechanism for probability emergence. Specically, in a QCA universe with strictly nite propagation velocity and global unitarity: the measurement process is modeled as local unitary coupling among the measured system, observer memory register, and environment; decoherence (einselection) selects stable pointer states; due to information horizon and memory capacity limitations, the observer can only access macroscopic branches compatible with their own record and cannot distinguish all microscopic QCA basis states underlying these branches. Applying environment-assisted invariance (envariance) symmetry arguments to these microscopic basis states, combined with an equal ontological weight hypothesis, we interpret the modulus of complex amplitudes as the square root of microscopic path degeneracy. Thus we prove: the subjective probability that an observer assigns to each measurement outcome equals the proportion of microscopic congurations compatible with that outcome among all accessible congurations, namely the standard Born weight pk=|ψk|2 . In the sense of rational consistency and axiomatization, we obtain the following result: in local Hilbert spaces of dimension at least three, if we require (i) global dynamics given by local QCA unitary evolution, (ii) measurement probabilities satisfying non-contextuality and environment insensitivity, (iii) no superluminal signal propagation, then Gleason-type theorems combined with QCA microscopic counting uniquely yield the Born rule as the only viable probability assignment. Wavefunction collapse is reinterpreted here as: Bayesian conditionalization and self-location update of the local observer on the global unitary state, rather than a nonlinear interruption of fundamental dynamics. Therefore, quantum probability is not an essential component of nature, but a statistical necessity for nite observers performing self-location in a discrete holographic entanglement network. Keywords: Quantum cellular automaton; Born rule; quantum measurement problem; environmentassisted invariance; decoherence; relative state interpretation; self-locating uncertainty; information horizon 1 1 Introduction & Historical Context 1.1 The Quantum Measurement Problem and Status of the Born Rule Standard quantum theory consists of two seemingly incompatible evolution laws: the time evolution of isolated systems is governed by the linear, unitary Schrödinger equation, while the measurement process is described by a nonlinear, stochastic collapse rule. Experimentally obtained frequencies conform to Born's formula pk=|⟨k|ψ⟩|2 , yet this formula is directly introduced as an independent axiom in most textbooks, without derivation from more primitive structures. Decoherence theory demonstrates that coupling between system and environment leads to rapid suppression of coherence terms in the eective description, thereby selecting stable pointer states and explaining the emergence of classical trajectories. Decoherence has been systematically developed mathematically and experimentally, occupying a central position in quantum-to-classical transition research. However, most authors acknowledge: decoherence itself does not generate specic numerical probabilities, but rather characterizes how interference terms are annihilated by the environment, assuming the Born rule holds. Thus, a natural question arises: can the Born rule be derived from more fundamental structures, without assuming probability axioms a priori? 1.2 Existing Born Rule Derivation Schemes A broad literature has formed around Born rule derivation. Gleason's theorem proves in Hilbert spaces of dimension at least three: if probabilities assigned to projection measures satisfy noncontextuality and countable additivity, then these probabilities must be realized by some density operator through the trace formula, essentially equivalent to the Born rule. Deutsch, Wallace, and others, under the Everett interpretation, use classical decision theory to argue that utility maximization for rational agents across many-world branches requires Born weights. Zurek's environment-assisted invariance (envariance) approach takes entanglement symmetry between system and environment as the starting point: in perfect entangled states, certain unitary transformations applied to the system can be compensated by applying another unitary transformation to the environment, leaving the overall state invariant. Using this symmetry and path subdivision for equal-amplitude superposition states, one can formally derive pk∝ |ψk|2 . Other works attempt to derive the Born rule from non-contextual probability assumptions, no-signaling principles, etc. Meanwhile, criticisms of these derivations point out: all known schemes explicitly or implicitly introduce assumptions approximately equivalent to the Born rule, such as continuity, additivity, or independence from measurement context. Therefore, a strictly no additional axiom derivation remains an open problem. 1.3 QCA and Cellular Automaton Interpretation In parallel, discussions about whether quantum theory can be reduced to discrete, local, deterministic cellular automaton dynamics continue to develop. 't Hooft's cellular automaton interpretation treats quantum states as statistical envelopes over a more fundamental classical automaton, seeking an essentially deterministic microscopic theory. On the other hand, quantum cellular automaton (QCA) models constructed by D'Ariano, Bisio, Perinotti, and others demonstrate that on discrete lattices satisfying locality, homogeneity, and isotropy, Weyl, Dirac, and Maxwell equations can emerge in the continuum limit, making QCA a concrete mathematical realization of the universe as quantum computation vision. 2 In the QCA universe picture, the global state |Ψt⟩ evolves according to a local unitary operator U across discrete time steps: |Ψt+1⟩=U|Ψt⟩ . Dynamics are strictly deterministic, with no intrinsic randomness. The origin of probabilistic features manifested in measurements is the most pressing foundational question in this picture. 1.4 Goals and Basic Strategy of This Paper This paper's goal is to provide an emergent derivation of the Born rule under the following premises: 1. The universe is described by a local, translation-invariant, unitary QCA; 2. Observers are local subsystems on the QCA with nite information capacity, whose records are realized as pointer states selected by environmental decoherence; 3. Probability is understood as the observer's self-locating uncertainty on the global unitary state, not an ontological stochastic process. Within this framework, measurement is modeled as a local entanglement process among measured system S , observer memory register O , and environment E . Subsequent environment-induced decoherence makes interference between dierent measurement outcome branches invisible to O 's accessible operators. We introduce an equal ontological weight hypothesis: in the QCA's ontological basis, each orthogonal microscopic conguration (or path) has equal ontological weight. Exploiting QCA's discreteness, we write complex amplitudes as square roots of degeneracies of equal-weight microscopic state superpositions, interpreting probability as the normalized ratio of the number of microscopic congurations compatible with a given macroscopic record. To make this construction universal and unique, this paper proves under support from Gleason's theorem, no-signaling constraints, and envariance symmetry: in a QCA universe, as long as local measurement probabilities satisfy non-contextuality and environment insensitivity, the Born rule is the only probability assignment compatible with global unitarity, locality, and observer niteness. 2 Model & Assumptions 2.1 QCA Universe and Ontological Basis Let the universe be modeled as a quantum cellular automaton dened on a discrete lattice set Λ⊂Zd . Each site x∈Λ carries a nite-dimensional Hilbert space Hx≃Cdcell , and the global Hilbert space is the tensor product H=Nx∈ΛHx . QCA dynamics are given by a family of discrete-time unitary operators U satisfying: 1. Locality: U can be decomposed into nite-depth circuits of local unitary gates acting on nite neighborhoods; 2. Homogeneity and translation invariance: local rules are identical between sites; 3. Finite propagation velocity: there exists a LiebRobinson velocity vLR such that any local perturbation's support is conned within an eective light cone of radius approximately vLRt at time t . Selecting an eigenbasis {|γx⟩} for each cell, the global ontological basis consists of tensor products |γ⟩=Nx|γx⟩ . In this paper, microscopic conguration and ontological basis vector are synonymous. Assumption 1 (Equal Ontological Weight (A1)) . In QCA ontology, each orthogonal ontological basis state |γ⟩ has equal basic weight; any probabilistic statement arises from counting or weighting these basis states under certain conditions. 3 This assumption corresponds to the ontological sequence equivalence idea in 't Hooft's cellular automaton schemes, while harmonizing with translation invariance of the ontological basis and uniformity of local rules in QCA. 2.2 SystemObserverEnvironment Partition In the global QCA universe, select a nite region ΛS as the measured system S , another nite region ΛO as the observer and measurement apparatus O , and the remaining sites as environment E . The corresponding Hilbert space decomposes as H ≃ HS⊗HO⊗HE. The observer's Hilbert space HO is divided into record subspace and remaining degrees of freedom: HO≃ HM⊗HO,rest, where an orthogonal basis {|Mj⟩} of HM realizes classical measurement outcome memory. Decoherence theory and environment-induced superselection (einselection) show that for macroscopic apparatuses, there exists a set of pointer states that remain robust under environmental monitoring, playing the role of classical records in actual measurements. Assumption 2 (Finite Information Capacity (A2)) . The observer record subspace HM has nite dimension; their knowledge and memory of the universe must be encoded on a nite number of distinguishable records |Mj⟩ . We take log2dim HM as the upper bound on classical information the observer can store. 2.3 Measurement Interaction and Decoherence Consider the measured system's initial state |ψS⟩=X k αk|sk⟩S, where {|sk⟩} is the eigenbasis of the observable to be measured (or corresponding pointer state basis). Observer and environment initially are in some reference state |Ψ0⟩=|ψS⟩⊗|Mready⟩O⊗|E0⟩E. The measurement process is realized by a segment of local unitary evolution on the QCA, abstractly representable as a unitary operator Umeas supported on ΛS∪ΛO∪ΛE,near within nite time. Its ideal form satises Umeas|sk⟩S⊗|Mready⟩O⊗|E0⟩E=|sk⟩S⊗|Mk⟩O⊗|˜ Ek⟩E. Linear extension to superposition states yields preand post-measurement states |Ψpre⟩=X k αk|sk⟩S⊗|Mready⟩O⊗|E0⟩E, |Ψpost⟩=Umeas|Ψpre⟩=X k αk|sk⟩S⊗|Mk⟩O⊗|˜ Ek⟩E. 4 Subsequently, under QCA global evolution, the environment continues to couple with the S  O composite, producing further decoherence, causing environmental states of dierent k branches to become nearly orthogonal: ⟨˜ Ek(t)|˜ Eℓ(t)⟩ ≈ δkℓ, while maintaining robustness of pointer states |Mk⟩ . Thus, for observer-local observables, the global pure state of systemapparatusenvironment is statistically indistinguishable from a classical mixed state. 2.4 Probability Axioms and Symmetry Assumptions To structurally derive specic probability weights, we need several general requirements on how observers distribute subjective probabilities among branches. Assumption 3 (Non-contextuality (A3)) . Let {Pk} be a mutually exclusive complete set of projection operators in the subspace HS⊗HM with PkPk=I . The probability pk that an observer assigns to outcome k depends only on the projection of the current state onto Pk , not on how this measurement is embedded in a larger Hilbert space. This assumption is identical to the noncontextuality requirement for probability measures in Gleason's theorem. Assumption 4 (Environment Insensitivity (A4)) . For any measurement scheme, as long as the systemobserver reduced state ρSO is unchanged, the observer's probability distribution over outcomes does not change due to changes in the environment state. This assumption is consistent with the envariance derivation and the Everett framework principle that pure environment transformations should not aect local probabilities. Assumption 5 (Finite Rationality and Continuity (A5)) . For a given measurement basis {|sk⟩} , if state |ψ⟩ is replaced by a small-norm perturbation |ψ′⟩ , reasonable probability assignments pk(ψ) and pk(ψ′) should not undergo discontinuous jumps. Formally, pk is a continuous function of the state. Assumption 6 (No Superluminal Signaling Constraint (A6)) . The combination of QCA dynamics and measurement rules must not allow classical information transmission faster than light using entanglement and local operations. This principle has been used in extensive literature to constrain nonlinear quantum evolution and unconventional probability rules. Through these structural assumptions, this paper will derive the Born rule within the QCA universe and discuss its uniqueness. 3 Main Results (Theorems and Statements) For clarity of presentation, this section rst states the main theorems and propositions, with proof details in later sections and appendices. Theorem 7 (Relative State Structure of QCA Measurement) . In a QCA universe satisfying A1 A2, consider any nite-dimensional measured system S and observer memory register M , along with a local unitary operator Umeas implementing ideal measurement. Then for any initial state |ψS⟩=X k αk|sk⟩S,|Ψ0⟩=|ψS⟩⊗|Mready⟩O⊗|E0⟩E, 5 the global state produced by measurement and subsequent decoherence can be written in Schmidt decomposition form |Ψpost⟩=X k αk|sk⟩S⊗|Mk⟩O⊗|Ek⟩E, where {|Ek⟩} are nearly orthogonal in the environment Hilbert space. The observer's subjective which branch they are in can be modeled as self-locating uncertainty over this set of labeled terms. Theorem 8 (Equal-Amplitude Branches and Equal Probability in Discrete QCA) . Under Theorem 1's setup, if |αk|2=Nk/N are rational numbers with N=PkNk a positive integer, then there exists a ne-grained decomposition of the environment Hilbert space such that |Ψpost⟩=1 √NX k Nk X j=1|sk⟩S⊗|Mk⟩O⊗|εk,j⟩E, where all |εk,j⟩ are mutually orthogonal, and each term has the same complex amplitude modulus 1/√N . If we further assume: 1. Each microscopic conguration |sk, Mk, εk,j⟩ has equal ontological weight (A1); 2. Any permutation of the environment subspace is physically envariant, with corresponding branches equivalent from the observer's perspective; then the observer's subjective probability of self-locating in a branch carrying record Mk is pk=Nk N=|αk|2. Theorem 9 (Continuum Limit and General Born Rule) . Under assumptions A1A5, generalizing Theorem 2's rational ratios Nk/N to general complex amplitudes: for any normalized state |ψS⟩=X k αk|sk⟩, there exists a sequence of rational numbers {N(n) k/N(n)} such that N(n) k/N(n)→ |αk|2 . For each n , construct corresponding equal-amplitude QCA microscopic states according to Theorem 2, and set probability assignment p(n) k=N(n) k/N(n) . Continuity assumption A5 guarantees the limit pk= limn→∞ p(n) k exists and satises pk=|αk|2. Therefore, the Born rule holds for all pure states. Theorem 10 (Gleason-Type Uniqueness and Non-contextuality) . On local Hilbert spaces of dimension at least three, assuming probabilities p(P) assigned to each projection operator P satisfy: 1. Non-negativity and normalization; 2. If {Pi} are mutually orthogonal with PiPi=I , then Pip(Pi)=1 ; 3. Non-contextuality (A3); then there exists a unique density operator ρ such that p(P) = tr(ρP) . This is Gleason's theorem statement. In a QCA universe, combined with A4A6, we can take ρ as the systemobserver reduced state, uniquely selecting Born-type probabilities pk=|αk|2 , and ruling out all unconventional probability rules incompatible with the no-signaling principle. 6 Theorem 11 (Eective Collapse and Information Horizon) . In the above framework, the global QCA state always evolves by unitary operators; there is no physical nonlinear collapse. However, for any local observer and their accessible operator algebra, replacing the global state |Ψpost⟩=X k αk|sk⟩⊗|Mk⟩⊗|Ek⟩ with the conditionalized state |Ψ(k) eff ⟩=|sk⟩⊗|Mk⟩⊗|Ek⟩ and using pk=|αk|2 as weight, is completely equivalent for any future experimental observable statistical consequence. This eective collapse can be interpreted as Bayesian updating performed by the observer within their information horizon, not a change in global dynamics. 4 Proofs This section provides proof outlines for Theorems 15, with more detailed constructions and technical details in the appendices. 4.1 Theorem 1: Relative State Structure in QCA Proof outline: 1. Realization of measurement unitarity. Under the QCA framework, any nite-dimensional systemapparatus composite can realize the required unitary Umeas by splicing local gate arrays within a nite spacetime block, consistent with universality results for implementing arbitrary nitedimensional unitaries in nite quantum circuits. 2. Schmidt decomposition and decoherence. For a bipartite system HSO ⊗HE , any pure state has a Schmidt decomposition. Appropriately choosing pointer basis {|Mk⟩} after measurement, we can write |Ψpost⟩ as |Ψpost⟩=X k αk|sk⟩⊗|Mk⟩⊗|Ek⟩, where |Ek⟩ become nearly orthogonal through environmentsystem interaction in short time. Decoherence theory and model calculations show that at macroscopic apparatus scales, pointer state interference terms rapidly decay on observable timescales, with negligible contribution to subsequent dynamics. 3. Relative states and self-locating uncertainty. Everett's relative state interpretation holds that after measurement, the global state is a set of labeled branches, each label corresponding to an observer record. Although the observer is intrinsic to one branch, they can view their uncertainty as self-locating uncertainty of which branch am I in during the time window between branch formation and record reading. This structure requires no additional assumptions, arising directly from QCA unitary evolution and pointer states selected by decoherence. Thus Theorem 1 is proved. □ 4.2 Theorem 2: Equal-Amplitude Branches and Microscopic Counting Theorem 2 is the combinatorial core of the Born rule. Step 1: Rational amplitude squared and environment subdivision. 7 Assume |αk|2=Nk/N where Nk and N are integers with N=PkNk . In Hilbert space, we can introduce an auxiliary environment subspace, splitting each term αk|sk, Mk, Ek⟩ into Nk equal-amplitude orthogonal states: αk|sk, Mk, Ek⟩=rNk Neiθk|sk, Mk, Ek⟩=1 √N Nk X j=1 eiθk|sk, Mk, εk,j⟩, where {|εk,j⟩}Nk j=1 are orthogonal in the environment Hilbert space, and for all k and j form part of an orthogonal basis. Such decomposition can always be realized by extending environment dimension, corresponding in the QCA picture to encoding microscopic labels on additional cells or internal degrees of freedom. Summing over all k : |Ψpost⟩=1 √NX k Nk X j=1 eiθk|sk, Mk, εk,j⟩. Overall phases eiθk are irrelevant for probability and can be ignored. Step 2: Envariance and equal probability. Consider a given k . Within the environment Hilbert space, any permutation of {|εk,j⟩}Nk j=1 can be realized by an environment unitary Vk . Dene the joint unitary transformation V=O k Vk acting on the environment while keeping systemobserver identity, then the global state is formally invariant: V|Ψpost⟩=|Ψpost⟩. This is precisely Zurek's environment-assisted invariance (envariance): when systemobserver and environment are perfectly entangled, certain transformations on the environment can compensate transformations on the system, leaving the overall state invariant, forcing the observer to assign equal probabilities to related branches. In this case, for any xed k , the only dierence among various j labels lies in the environment state |εk,j⟩ , which is inaccessible to the observer. If the observer assigns dierent subjective probabilities to these microscopic dierences, this would cause probability changes under operations that only alter the environment without changing systemobserver observables, directly contradicting A4 (environment insensitivity). Therefore, the only choice satisfying envariance and A4 is: assign equal probability across all j under the same k . Let qk,j be the probability of observer self-location in microscopic branch (k, j) . Then for any xed k : qk,1=qk,2=··· =qk,Nk≡qk. The overall normalization condition is X k Nk X j=1 qk,j = 1 ⇒X k Nkqk= 1. By symmetry and A1's equal ontological weight, we can directly take qk,j = 1/N . Combining these: qk=1 N, pk≡ Nk X j=1 qk,j =Nk N. 8 Thus pk=|αk|2. Theorem 2 is proved. □ 4.3 Theorem 3: Continuous Extension to General Amplitudes For any pure state |ψS⟩=Pkαk|sk⟩ , let rk=|αk|2 satisfying Pkrk= 1 . Since rationals are dense in reals, we can construct a rational approximation sequence r(n) k=N(n) k N(n)→rk,X k r(n) k= 1 (for example, take r(n) k as rk rounded to n decimal places and renormalized). For each n , construct equal-amplitude microscopic states and probability assignment p(n) k=r(n) k according to Theorem 2. Continuity assumption A5 requires: when ∥ψ(n)−ψ∥ → 0 , we have p(n) k→pk . Since r(n) k→rk : pk= lim n→∞ p(n) k= lim n→∞ r(n) k=rk=|αk|2. Therefore, the Born rule holds in general. □ 4.4 Theorem 4: Gleason-Type Uniqueness and No-Signaling Theorems 2 and 3 provide constructive derivations of the Born rule for specic measurement scenarios. To demonstrate uniqueness, we invoke Gleason-type results and no-signaling constraints. Gleason's theorem shows: in Hilbert spaces of dimension at least three, if probability measure µ(P) assigned to projection operator P satises: 1. µ(P)≥0 and µ(I)=1 ; 2. For any countable family of mutually orthogonal projections {Pi} : µ(PiPi) = Piµ(Pi) ; then there must exist a unique density operator ρ such that µ(P) = tr(ρP) . In the QCA universe, measurement probabilities satisfying non-contextuality and additivity are uniquely given by the Born rule via Gleason's theorem. Combined with no-signaling constraints, any alternative probability rule violates either non-contextuality or causality. □ 4.5 Theorem 5: Eective Collapse The argument for Theorem 5 is essentially a rigorous formulation of the eective collapse idea from decoherence theory in the QCA context. Due to decoherence-induced orthogonality of environment states in dierent branches, local observables cannot distinguish between the full superposition and an eective mixture. This justies treating collapse as Bayesian updating within the observer's information horizon. □ 9