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arXiv:2004.08749 · Quantum

Emergence of the Born rule in quantum optics

Brian R. La Cour, Morgan C. Williamson

Published Oct 26, 2020• 10 citations
arXivHTMLPDF

Abstract

The Born rule provides a fundamental connection between theory and observation in quantum mechanics, yet its origin remains a mystery. We consider this problem within the context of quantum optics using only classical physics and the assumption of a quantum electrodynamic vacuum that is real rather than virtual. The connection to observation is made via classical intensity threshold detectors that are used as a simple, deterministic model of photon detection. By following standard experimental conventions of data analysis on discrete detection events, we show that this model is capable of reproducing several observed phenomena thought to be uniquely quantum in nature, thus providing greater elucidation of the quantum-classical boundary.

DOI
10.22331/q-2020-10-26-350

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The Born rule provides a fundamental connection between theory and observation in quantum mechanics, yet its origin remains a mystery. We consider this problem within the context of quantum optics using only classical physics and the assumption of a quantum electrodynamic vacuum that is real rather than virtual. The connection to observation is made via classical intensity threshold detectors that are used as a simple, deterministic model of photon detection. By following standard experimental conventions of data analysis on discrete detection events, we show that this model is capable of reproducing several observed phenomena thought to be uniquely quantum in nature, thus providing greater elucidation of the quantum-classical boundary.
This paper describes a simple, causally deterministic model of quantum measurement based on an amplitude threshold detection scheme. Surprisingly, it is found to reproduce many phenomena normally thought to be uniquely quantum in nature. To model an NNN -dimensional pure state, the model uses NNN complex random variables given by a scaled version of the wave vector with additive complex noise. Measurements are defined by threshold crossings of the individual components, conditioned on single-component threshold crossings. The resulting detection probabilities match or approximate those predicted by quantum mechanics according to the Born rule. Nevertheless, quantum phenomena such as entanglement, contextuality, and violations of Bell’s inequality under local measurements are all shown to be exhibited by the model, thereby demonstrating that such phenomena are not without classical analogs.
The very old problem of the statistical content of quantum mechanics (QM) is studied in a novel framework. The Born's rule (one of the basic postulates of QM) is derived from theory of classical random signals. We present a measurement scheme which transforms continuous signals into discrete clicks and reproduces the Born's rule. This is the scheme of threshold type detection. Calibration of detectors plays a crucial role. Creation of such a detection model provides a possibility to represent the basic quantum phenomena as classical field phenomena completed by measurements with threshold type and properly calibrated detectors. Finally, the wave-particle duality has been resolved in favor of the purely wave description. It is well known that coincidence probability plays an important role in the debate on the possibility to reduce quantum theory (especially quantum optics) to classical field theory. Our model matches well the predictions of quantum theory and experiment. It violates the prediction of the classical field model which does not take into account the presence of threshold type detectors with proper calibration.
The predictions of quantum mechanics are probabilistic. Quantum probabilities are extracted using a postulate of the theory called the Born rule, the status of which is central to the problem of quantum mechanics. Efforts to justify the Born rule from other physical principles, and thus elucidate the measurement process, have involved lengthy statistical or information-theoretic arguments. Here we show that Bohm's deterministic formulation of quantum mechanics allows the Born rule for measurements on a single system to be derived, without any statistical assumptions. We solve a simple example where the creation of an ensemble of identical quantum states, together with position measurements on those states, are described by Bohm's quantum dynamics. The calculated measurement outcomes agree with the Born-rule probabilities, which are thus a consequence of deterministic evolution. Our results demonstrate that quantum probabilities can emerge from simple dynamical laws alone, and they support the view that there is no underlying indeterminism in quantum phenomena.
The Born rule, P(x)=∣ψ(x)∣2P(\mathbf{x})=|\psi(\mathbf{x})|^2P(x)=∣ψ(x)∣2 is one of the most successful relations in modern physics. Yet its physical meaning remains conceptually distinctive. Schrödinger's equation describes the continuous evolution of a complex wave amplitude, whereas an actual detector records discrete and localized events. The Born rule connects these two descriptions with remarkable precision, but the physical mechanism responsible for the quadratic relation between wave amplitude and event frequency is not contained explicitly in the Schrödinger equation itself. This paper revisits the historical and physical relationship between Schrödinger's wave mechanics and Born's statistical interpretation of scattering. The purpose is not to deny the empirical validity of the Born rule, but to identify the physical gap between continuous wave evolution and discrete detection statistics. We distinguish three levels of description: the preparation of localized quantum excitations, their continuous propagation and scattering, and the final microscopic detector event. This leads to a central question: \boxedtext{Why should the rate of discrete detection events be proportional to ∣ψ∣2|\psi|^2∣ψ∣2?} The paper argues that the answer cannot be obtained merely by defining ∣ψ∣2|\psi|^2∣ψ∣2 to be a probability density. Instead, a physical theory must explain how a continuously distributed complex amplitude produces a localized event rate. We investigate the possibility that the quadratic dependence emerges from a local interaction functional, energy-transfer mechanism, or bilinear response of the detector to the incident quantum amplitude. A particularly important distinction is made between the wave amplitude itself and the experimentally recorded event frequency. A single detection event is localized, whereas the Born distribution emerges only through repeated preparations and statistical accumulation. The paper therefore proposes that the Born rule may be viewed as the macroscopic statistical image of a microscopic event-generation process. No completed derivation of Born's rule is claimed here. Rather, the paper formulates the precise dynamical problem that such a derivation must solve and identifies several mathematical and experimental constraints that any successful mechanism must satisfy.
The Born rule stands as the fundamental probability postulate of quantum mechanics, yet its first-principle derivation remains a longstanding open problem. This paper presents a rigorous, collapse-free, and observer-independent derivation of the Born rule. Starting from four physically natural and minimal axioms—unitary dynamics, uniqueness of classical records, unitary invariance, and consistency with the classical limit and ergodicity—we uniquely deduce the Born rule from first principles. We prove that the Born rule is not an independent postulate, but a necessary mathematical consequence of the core axiomatic structure. Our approach is self-contained, assumption-free, and purely unitary, establishing that the Born rule is canonically determined by the fundamental mathematical and physical principles of quantum mechanics.
The use of Bohmian mechanics as a practical tool for modeling non-relativistic quantum phenomena of matter provides clear evidence of its success, not only as a way to interpret the foundations of quantum mechanics, but also as a computational framework. In the literature, it is frequently argued that such a realistic view-based on deterministic trajectories cannot account for phenomena involving the "creation" and "annihilation" of photons. In this paper, by revisiting and rehabilitating earlier proposals, we show how quantum optics can be modeled using Bohmian trajectories for electrons in physical space, together with well-defined electromagnetic fields evolving in time. By paying special attention to an experiment demonstrating partition noise for photons, and to how the Born rule emerges in this context, the paper pursues two main goals. First, it vindicates the pedagogical use of this simple Bohmian framework to compute, understand, and visualize quantum electrodynamics phenomena. Second, given that measurements are ultimately indicated on matter pointers, it clarifies what it means to measure photon or electromagnetic-field properties, even when they are considered non-ontic elements.