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arXiv:2512.03929

Rethinking Collapse: Coupling Quantum States to Classical Bits with quasi-probabilities

Dagomir Kaszlikowski, Pawel Kurzynski

quant-phSubmitted Dec 3, 2025• Announced Dec 4, 2025• 0 citations
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Abstract

We propose a formulation of quantum measurement within a modified framework of frames, in which a quantum system - a single qubit - is directly coupled to a classical measurement bit. The qubit is represented as a positive probability distribution over two classical bits, a and a', denoted by p(aa'). The measurement apparatus is described by a classical bit $α= \pm 1$, initialized in the pure distribution $p(α) = \frac{1}{2}(1 + α)$. The measurement interaction is modeled by a quasi-bistochastic process $ S(bb'β\mid aa'α)$ - a bistochastic map that may include negative transition probabilities, while acting on an entirely positive state space. When this process acts on the joint initial state $p(aa')p(α)$, it produces a collapsed state $p(bb'\midβ)$, yielding the measurement outcome $β$ with the correct quantum-mechanical probability $p(β)$. This approach bypasses the von Neumann chain of infinite couplings by treating the measurement register classically, while capturing the nonclassical nature of measurement through the quasi-bistochastic structure of the interaction.

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12 pages

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We outline the Minimalistic Measurement Scheme (MMS) compatible with regular unitary evolution of a closed quantum system. Within this approach, a part of the system becomes informationally isolated (restricted) which leads to a natural emergence of the classical domain. This measurement scenario is a simpler alternative to environment-induced decoherence. In its basic version, MMS involves two ancilla qubits, AAA and XXX, entangled with each other and with the System SSS. Informational or thermodynamic cost of measurement is represented by XXX-qubit being isolated, i.e. becoming unavailable for future interactions with the rest of the system. Conditional upon this isolation, AAA-qubit, that plays the role of an Apparatus, becomes classical and records the outcome of the measurement. The procedure may be used to perform von Neumann-style projective measurements or generalized ones, that corresponds to Positive-Operator Value Measure (POVM). By repeating the same generalized measurement multiple times with different AAA- and XXX-qubits, one asymptotically approaches the wave function collapse in the basis determined by the premeasurement process. We present a simple result for the total information extracted after NNN such weak measurements. Building upon MMS, we propose a construction that maps a history of a quantum system onto a set of AAA-qubits. It resembles the Consistent History (CH) formulation of Quantum Mechanics (QM), but is distinct from it, and is built entirely within the conventional QM. In particular, consistency postulate of CH formalism is not automatically satisfied, but rather is an emerging property. Namely, each measurement event corresponds to the branching of mutually exclusive classical realities whose probabilities are additive. In a general case, however, the superposition between different histories is determined by the history density matrix.
We prove that the Hilbert space description of all joint von Neumann measurements on a quantum state can be reproduced in terms of a single measure space (Ω, F, μ) with a normalized real-valued measure μ, that is, in terms of a new general probability model, the quasi-classical probability model, developed in [Loubenets: J. Math. Phys. 53 (2012), 022201; J. Phys. A: Math. Theor. 45 (2012), 185306]. In a quasi-classical probability model for all von Neumann measurements, a random variable models the corresponding quantum observable in all joint measurements and depends only on this quantum observable. This mathematical result sheds a new light on some important issues of quantum randomness discussed in the literature since the seminal article (1935) of Einstein, Podolsky and Rosen.
Abstract In the context of the century-long debate on quantum measurement problem, the current work proposes a model that describes the process of collapse of state by quantum interaction, which resolves the controversies of the framework of quantum mechanics and describes the entire domain of quantum-to-classical world including the weak measurement and partial collapse. ‘Measurement’, being the process of physically interacting with a system in order to extracting information from it, is theorized in the current model by synthesizing the quantum interaction between system and measuring apparatus with the information entropy of such process. The model assumes Schrödinger equation to be the only guiding equation for all physical systems including the measuring apparatus, and does not presuppose ‘superposition principle’, rather derives it theoretically from the formulation. The superposed state is shown to be independent of the choice of measurement operator (observable) or basis states (pointers) of the measuring apparatus. Most interestingly, the current model explains the non-observance of ‘superposition principle’ by classical systems as the classical limit of such quantum description of measurement. Along with solving the quantum measurement problem, the work also explains weak measurement and partial collapse, which can be further extended to investigate such several emerging critical phenomena.
The measurement problem in quantum mechanics represents a disconnect between the Schr\"{o}dinger equation's unitary evolution and the non-unitary collapse of the wave function. This work resolves this issue by modeling measurement as a deterministic, thermodynamically driven process. We modify the Liouville-von Neumann equation by introducing a dual mechanism: a dephasing term that eliminates quantum coherence and a state-resolution dynamic that amplifies one outcome to certainty. An infinitesimal environmental perturbation breaks the initial symmetry, determining which outcome emerges. Numerical simulations of a qubit in superposition demonstrate that (1) coherence is conserved during unitary evolution, (2) a rapid phase transition occurs upon measurement, and (3) Born Rule statistics emerge from ensemble averaging over random environmental fluctuations. We conclude that quantum collapse need not be a separate postulate but can be modeled as a unified, deterministic dynamical law with testable experimental predictions.
Abstract In the context of the century-long debate on quantum measurement problem, the current work proposes a model that describes the process of collapse of state by quantum interaction, which resolves the controversies of the framework of quantum mechanics and describes the entire domain of quantum-to-classical world including the weak measurement and partial collapse. ‘Measurement’, being the process of physically interacting with a system in order to extracting information from it, is theorized in the current model by synthesizing the quantum interaction between system and measuring apparatus with the information entropy of such process. The model assumes Schrödinger equation to be the only guiding equation for all physical systems including the measuring apparatus, and does not presuppose ‘superposition principle’, rather derives it theoretically from the formulation. The superposed state is shown to be independent of the choice of measurement operator (observable) or basis states (pointers) of the measuring apparatus. Most interestingly, the current model explains the non-observance of ‘superposition principle’ by classical systems as the classical limit of such quantum description of measurement. Along with solving the quantum measurement problem, the work also explains weak measurement and partial collapse, which can be further extended to investigate such several emerging critical phenomena.
We develop a statistical model of microscopic stochastic deviation from classical mechanics based on a stochastic process with a transition probability that is assumed to be given by an exponential distribution of infinitesimal stationary action. We apply the statistical model to stochastically modify a classical mechanical model for the measurement of physical quantities reproducing the prediction of quantum mechanics. The system+apparatus always has a definite configuration at all times, as in classical mechanics, fluctuating randomly following a continuous trajectory. On the other hand, the wavefunction and quantum mechanical Hermitian operator corresponding to the physical quantity arise formally as artificial mathematical constructs. During a single measurement, the wavefunction of the whole system+apparatus evolves according to a Schrödinger equation and the configuration of the apparatus acts as the pointer of the measurement so that there is no wavefunction collapse. We will also show that while the outcome of each single measurement event does not reveal the actual value of the physical quantity prior to measurement, its average in an ensemble of identical measurements is equal to the average of the actual value of the physical quantity prior to measurement over the distribution of the configuration of the system.