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The Born Perceptron: Mimicking Quantum Mechanics and Distributed Measurements in Time

Aman Chawla

Published Mar 15, 2026• 0 citations
Publisher

Abstract

This paper develops a neural architecture capable of reproducing several structural aspects of quantum mechanics. The proposed model, termed the Born Perceptron, extends the classical perceptron by structuring outputs as value–index pairs and interpreting normalized activations as probability distributions over indexed outcomes. By allowing indices to inhabit geometric manifolds rather than discrete labels, the perceptron naturally represents probabilistic sampling over state spaces. When stacked sequentially, such perceptrons generate stochastic trajectories reminiscent of quantum state evolution under repeated measurement. Introducing complex-valued activations yields probabilities determined by squared magnitudes, reproducing the Born rule. The framework is further extended to treat measurements distributed in time, particularly position samplings. In this regime, the neural system produces sequences of sampled positions whose joint probabilities resemble those derived in path-integral formulations of quantum mechanics. The resulting architecture offers a classical computational model that captures several conceptual features of quantum theory including measurement, collapse, trajectory evolution, and amplitude-based probability rules.

DOI
10.5281/zenodo.19032101

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Abstract We propose an exercise in which one attempts to deduce the formalism of quantum mechanics solely from phenomenological observations. The only assumed inputs are obtained through sequential probing of quantum systems; no presuppositions about the underlying mathematical structures are permitted. We demonstrate that it is indeed possible to derive, on this basis, a complete and fully functional formalism rooted in the structures of Hilbert spaces. However, the resulting formalism—the bi -trajectory formalism—differs significantly from the standard state-focused formulation. In Part I of the paper, we analyze the outcomes of various experiments involving sequential measurements of quantum observables. These outcomes are quantitatively described by phenomenological multi-time probability distributions, estimated from experimental data. Our first conclusion is that the theory describing these experiments must be non-classical: the measured sequences cannot be interpreted as sampling of a uni -trajectory representing the system’s observable. The non-classical nature of the investigated systems manifests in a range of observed phenomena, including quantum interference, the quantum Zeno effect, and uncertainty relations between the measured observables.

arXiv:2410.14410•Journal of Physics A Mathematical and Theoretical•Aug 10, 2026•1 citation
This paper develops a neural architecture capable of reproducing several structural aspects of quantum mechanics. The proposed model, termed the Born Perceptron, extends the classical perceptron by structuring outputs as value–index pairs and interpreting normalized activations as probability distributions over indexed outcomes. By allowing indices to inhabit geometric manifolds rather than discrete labels, the perceptron naturally represents probabilistic sampling over state spaces. When stacked sequentially, such perceptrons generate stochastic trajectories reminiscent of quantum state evolution under repeated measurement. Introducing complex-valued activations yields probabilities determined by squared magnitudes, reproducing the Born rule. The framework is further extended to treat measurements distributed in time, particularly position samplings. In this regime, the neural system produces sequences of sampled positions whose joint probabilities resemble those derived in path-integral formulations of quantum mechanics. The resulting architecture offers a classical computational model that captures several conceptual features of quantum theory including measurement, collapse, trajectory evolution, and amplitude-based probability rules.
The Schrödinger equation and the Born rule constitute two of the most successful structures in quantum mechanics. The former governs the continuous evolution of the wave function, while the latter connects its squared amplitude to the statistical distribution of experimental outcomes. Yet these two structures operate at different descriptive levels: the Schrödinger state is continuous, spatially extended, and phase-sensitive, whereas individual experiments terminate in localized physical events. This paper develops a theoretical framework for reconstructing the physical relation between these two levels. Rather than proceeding only in the conventional direction from Schrödinger dynamics to Born probabilities, we introduce an inverse explanatory strategy: from Born back to Schrödinger. Born statistics are treated as an empirical constraint from which the physical role of the wave function is reconsidered. We propose that, for the single-particle regime examined here, the wave function may be interpreted functionally as a dynamical propagation structure rather than as the localized particle itself. Its amplitude, phase, conserved current, gauge-covariant propagation momentum, and associated velocity field jointly define a structured family of possible motions. This interpretation leads to three separations: propagation from localization, propagation from physical population, and arrival from detector registration. A physical preparation generates a source measure (\mu_0^{\rm phys}); Schrödinger dynamics generates a propagation map (\Phi_\Psi); and microscopic detector dynamics is represented by a response map (\mathcal R_D). The Born problem can then be formulated as the closure condition RD ⁣[(ΦΨ)∗μ0phys]=?μBorn,D. \boxed{\mathcal R_D\!\left[(\Phi_\Psi)_*\mu_0^{\rm phys}\right] \overset{?}{=} \mu_{\rm Born,D}.} RD​[(ΦΨ​)∗​μ0phys​]=?μBorn,D​.​ This relation is not claimed here as a derived law, but as the central dynamical problem that a physical reconstruction of quantum probability must solve without inserting the Born measure at either boundary. The paper synthesizes six preceding studies on wave-packet localization, Born's scattering interpretation, the emergence of probability, the physical meaning of the wave function, source-to-detector transport, and unified propagation dynamics. It places the resulting framework in dialogue with Madelung and Hamilton–Jacobi formulations, Bohmian mechanics, decoherence, Quantum Darwinism, structural and symmetry-based derivations of the Born rule, Everettian approaches, objective-collapse theories, and detector models. The resulting proposal is best understood as a propagation-first theoretical reconstruction program: Schrödinger dynamics organizes possible propagation, while Born statistics constrain how that propagation becomes physically realized. Keywords: Schrödinger equation; Born rule; wave function; quantum foundations; propagation structure; quantum probability; quantum trajectories; source-to-detector transport; detector response; wave–particle duality; Bohmian mechanics; quantum measurement
We formulate a discrete two-state stochastic process with elementary rules that give rise to Born statistics and reproduce the probabilities from the Schrödinger equation under an associated Hamiltonian matrix, which we identify. We define the probability to observe a state, classical or quantum, in proportion to the number of \textit{events} at that state--number of ways a walker may materialize at a point of observation at time t through a sequence of transitions starting from known initial state at t=0. The quantum stochastic process differs from its classical counterpart in that the quantum walker is a pair of qubits, each transmitted independently through all possible paths to a point of observation, and whose recombination may produce a positive or negative event (classical events are never negative). We represent the state of the walker via a square matrix of recombination events, show that the indeterminacy of the qubit state amounts to rotations of this matrix, and show that the Born rule counts the number qubits on this matrix that remain invariant over a full rotation.
The Born Rule is central to quantum mechanics, yet it remains a postulate: a probabilistic rule grafted onto the deterministic evolution of the Schrödinger equation. This work derives the Born Rule as an emergent statistical law grounded in thermodynamic principles. We propose that quantum probability reflects the survival likelihood of a branch under decoherence, determined by its informational reversibility, structural fragility, and exposure to entropy.In this formulation, reversibility quantifies the mutual information retained during measurement, entropy reflects environmental disruption, and buffering represents the system’s capacity to absorb it. In low-entropy, high-buffering conditions typical of laboratory systems, the survival probability of a branch closely approximates the squared amplitude of its wavefunction — recovering the Born Rule as a limiting case.This approach reframes quantum amplitude as a measure of structural persistence and interprets wavefunction collapse as a form of thermodynamic selection. The result unifies concepts from measurement theory, thermodynamics, and information theory into a coherent physical framework, while also predicting conditions under which deviations from the Born Rule could arise — offering new directions for theoretical and experimental inquiry.
We develop a dynamical framework for quantum measurement based on stochastic but unitary evolution in projective state space. Random Hamiltonians drawn from the Gaussian Unitary Ensemble generate stochastic unitary dynamics of the quantum state, while equivalence classes reflecting finite detector resolution define classical observables as well as classical configuration-space and phase-space submanifolds. When the evolution is constrained to the phase-space submanifold, free Schrödinger dynamics reduces to Newtonian motion, while stochastic motion constrained to the classical configuration-space submanifold yields ordinary Brownian motion in classical space. Transition probabilities under the stochastic dynamics satisfy the Born rule, whereas the constrained classical evolution produces the normal probability distributions characteristic of classical measurements. We show that, in this setting, measurement, state reduction, and the quantum-classical transition emerge from unitary dynamics alone, without invoking nonunitary collapse or additional postulates. Entanglement and EPR correlations arise geometrically from the evolution of joint states in composite state space, preserving locality in spacetime. The framework provides a unified dynamical account of measurement and classicality compatible with the structure of quantum mechanics.
An alternative formulation of quantum mechanics is proposed based on a principle that can be considered a generalization of the Born rule. The principle is composed of a mathematical expression and an associated interpretation, and establishes a correlation between the positions of a particle at two different times. Under reasonable conditions for the wave function, this correlation implies that the particles follow quasiclassical trajectories. It is also shown that the Born rule is equivalent to a particular case of the evolved principle. These features allow the principle to provide a unified explanation of the results of the statistical experiments and of the quasiclassical macroscopic evolution. There is a strong analogy between the quantum principle and a probabilistic principle which is necessary to derive empirical predictions from the mathematical formalism of probability theory. This principle is referred to by some authors as Cournot's principle, while other authors use the equivalent notion of typicality. In this paper probability theory and quantum mechanics are formulated in such a way as to explicitly include the two principles and to emphasize the very similar conceptual structure of the two theories.
Abstract We propose an exercise in which one attempts to deduce the formalism of quantum mechanics solely from phenomenological observations. The only assumed inputs are obtained through sequential probing of quantum systems; no presuppositions about the underlying mathematical structures are permitted. We demonstrate that it is indeed possible to derive, on this basis, a complete and fully functional formalism rooted in the structures of Hilbert spaces. However, the resulting formalism—the bi -trajectory formalism—differs significantly from the standard state-focused formulation. In Part I of the paper, we analyze the outcomes of various experiments involving sequential measurements of quantum observables. These outcomes are quantitatively described by phenomenological multi-time probability distributions, estimated from experimental data. Our first conclusion is that the theory describing these experiments must be non-classical: the measured sequences cannot be interpreted as sampling of a uni -trajectory representing the system’s observable. The non-classical nature of the investigated systems manifests in a range of observed phenomena, including quantum interference, the quantum Zeno effect, and uncertainty relations between the measured observables.