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arXiv:2104.12801 · Foundations of Physics

A Locally Deterministic, Detector-Based Model of Quantum Measurement

Brian R. La Cour

Published Sep 2, 2014• 13 citations
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Abstract

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 $$N$$ -dimensional pure state, the model uses $$N$$ 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.

DOI
10.1007/s10701-014-9829-6

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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 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.
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.
Abstract Collapse models including some external noise of unknown origin are routinely used to describe phenomena on the quantum-classical border; in particular, quantum measurement. Although containing nonlinear dynamics and thereby exposed to the possibility of superluminal signaling in individual events, such models are widely accepted on the basis of fully reproducing the non-signaling statistical predictions of quantum mechanics. Here we present a deterministic nonlinear model without any external noise, in which randomness—instead of being universally present—emerges in the measurement process, from deterministic irregular dynamics of the detectors. The treatment is based on a minimally nonlinear von Neumann equation for a Stern–Gerlach or Bell-type measuring setup, containing coordinate and momentum operators in a self-adjoint skew-symmetric, split scalar product structure over the configuration space. The microscopic states of the detectors act as a nonlocal set of hidden parameters, controlling individual outcomes. The model is shown to display pumping of weights between setup-defined basis states, with a single winner randomly selected and the rest collapsing to zero. Environmental decoherence has no role in the scenario. Through stochastic modelling, based on Pearle’s ‘gambler’s ruin’ scheme, outcome probabilities are shown to obey Born’s rule under a no-drift or ‘fair-game’ condition. This fully reproduces quantum statistical predictions, implying that the proposed non-linear deterministic model satisfies the non-signaling requirement. Our treatment is still vulnerable to hidden signaling in individual events, which remains to be handled by future research.
Collapse models including some external noise of unknown origin are routinely used to describe phenomena on the quantum-classical border; in particular, quantum measurement. Although containing nonlinear dynamics and thereby exposed to the possibility of superluminal signaling in individual events, such models are widely accepted on the basis of fully reproducing the non-signaling statistical predictions of quantum mechanics. Here we present a deterministic nonlinear model without any external noise, in which randomness - instead of being universally present - emerges in the measurement process, from deterministic irregular dynamics of the detectors. The treatment is based on a minimally nonlinear von Neumann equation for a Stern-Gerlach or Bell-type measuring setup, containing coordinate and momentum operators in a self-adjoint skew-symmetric, split scalar product structure over the configuration space. The microscopic states of the detectors act as a nonlocal set of hidden parameters, controlling individual outcomes. The model is shown to display pumping of weights between setup-defined basis states, with a single winner randomly selected and the rest collapsing to zero. Environmental decoherence has no role in the scenario. Through stochastic modelling, based on Pearle's gambler's ruin scheme, outcome probabilities are shown to obey Born's rule under a no-drift or fair-game condition. This fully reproduces quantum statistical predictions, implying that the proposed non-linear deterministic model satisfies the non-signaling requirement. Our treatment is still vulnerable to hidden signaling in individual events, which remains to be handled by future research.
We present a simple classical (random) signal model reproducing Born's rule. The crucial point of our approach is that the presence of detector's threshold and calibration procedure have to be treated not as simply experimental technicalities, but as the basic counterparts of the theoretical model. We call this approach threshold signal detection model (TSD). The experiment on coincidence detection which was done by Grangier in 1986 [22] played a crucial role in rejection of (semi-)classical field models in favour of quantum mechanics (QM): impossibility to resolve the wave-particle duality in favour of a purely wave model. QM predicts that the relative probability of coincidence detection, the coefficient g (2) (0), is zero (for one photon states), but in (semi-)classical models g (2) (0) ≥ 1. In TSD the coefficient g (2) (0) decreases as 1/ ε 2 d , where ε d > 0 is the detection threshold. Hence, by increasing this threshold an experimenter can make the coefficient g (2) (0) essentially less than 1. The TSD-prediction can be tested experimentally in new Grangier type experiments presenting a detailed monitoring of dependence of the coefficient g (2) (0) on the detection threshold. Structurally our model has some similarity with the prequantum model of Grossing et al. Subquantum stochasticity is composed of the two counterparts: a stationary process in the space of internal degrees of freedom and the random walk type motion describing the temporal dynamics.
Collapse models including some external noise of unknown origin are routinely used to describe phenomena on the quantum-classical border; in particular, quantum measurement. Although containing nonlinear dynamics and thereby exposed to the possibility of superluminal signaling in individual events, such models are widely accepted on the basis of fully reproducing the non-signaling statistical predictions of quantum mechanics. Here we present a deterministic nonlinear model without any external noise, with randomness emerging from deterministic irregular dynamics of the detectors, shown to be just as non-signaling as noise-based collapse models are: it fully reproduces quantum statistical predictions, while being still vulnerable to signaling in individual events. The model is based on a minimally nonlinear von Neumann equation for a Stern-Gerlach or Bell-type measuring setup, containing coordinate and momentum operators in a skew-symmetric, split scalar product structure over the configuration space. The microscopic states of the detectors act as a nonlocal set of hidden parameters, controlling individual outcomes. Following Pearle's gambler's ruin scheme, the model is shown to display pumping of weights between setup-defined basis states, with a single winner randomly selected and the rest collapsing to zero. Outcome probabilities obey Born's rule under a no-drift or fair-game condition, implying that collapse must run to conclusion during the metastable pre-firing period of the detector system. Hidden signaling in individual events remains to be handled by future research; what is reached here is to put deterministic nonlinear models and noise-based collapse models on equal footing.