arXiv subjects

Lionel Martellini

Publications and source records attributed to Lionel Martellini.

8 recordsLinked to original sources

arXiv2026

The Physics of Unresolved Uncertainty: Quantum Mechanics as a Theory of Potentiality

We propose a reformulation of quantum mechanics as a theory of unresolved uncertainty. This theory of potentiality is formulated in the language of complex-valued measure theory, regarded as a pre-probabilistic counterpart of ordinary probability theory. In this formulation, additivity, conditioning, independence, mixtures, transition kernels, and temporal divisibility retain natural linear forms at the potentiality level, while non-classical probability-level features such as interference arise from the nonlinear Born map. Measurement is described as Bayesian-type conditioning of potentialities on actualized information, and non-selective measurement as the replacement of coherent potentiality by statistical mixtures of conditional potentiality branches. Mixed states, decoherence, composite systems, entanglement, and Bell-type correlations are also given a unified potentiality-level interpretation. The density matrix is interpreted as a coherence kernel whose off-diagonal blocks encode retained phase relations. For pure bipartite states, potentiality independence is shown to be equivalent to factorization of the Born distribution in every pair of local contexts. The resulting formulation is empirically equivalent to standard quantum mechanics, but it makes explicit a pre-probabilistic description of physical reality that is usually implicit in the Hilbert-space formalism.

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arXiv2026

The Emergence of Time from Quantum Records: Actualization and the Surprisal Clock

We develop a record-based account of internal time in quantum mechanics. Persistent records first define an ordinal chronology through inclusion of their accumulated Boolean algebras. On a specified record filtration, we prove that, under three minimal consistency requirements, namely dependence only on conditional Born weight, invariance under sequential refinement of the same recorded fact, and continuity, the actualization of each outcome contributes an internal time proportional to its \textit{surprisal}, defined as the negative logarithm of that probability. A certain outcome therefore contributes no duration, whereas less probable outcomes contribute larger increments. The mean and variance of the accumulated clock are governed by the Shannon entropy and varentropy of the record process, its moment-generating function is related to the R'enyi entropy spectrum, and its pathwise fluctuations admit a Doob decomposition into a predictable entropic compensator and a martingale. Two further results provide independent consistency checks. Within the stated class of finite-dimensional bipartite states, universal additivity of local clock readings across all admissible local record contexts is equivalent to the absence of entanglement. In the quantum Zeno regime, the number of monitoring rounds may diverge while the expected accumulated surprisal tends to zero. The construction therefore supplies a canonical internal-time functional for a specified quantum record process without presupposing an external clock. It does not by itself identify which physical record processes constitute material clocks; determining their production rates and dynamical realization remains a separate problem.

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arXiv2026

Quantum stroboscopy for time measurements

Mielnik's cannonball argument uses the Zeno effect to argue that projective measurements for time of arrival are impossible. If one repeatedly measures the position of a particle (or a cannonball!) that has yet to arrive at a detector, the Zeno effect will repeatedly collapse its wavefunction away from it: the particle never arrives. Here we introduce quantum stroboscopic measurements where we accumulate statistics of projective position measurements, performed on different copies of the system at different times, to obtain a time-of-arrival distribution. We show that, under appropriate limits, this gives the same statistics as time measurements of conventional ``always on'' particle detectors, that bypass Mielnik's argument using non-projective, weak continuous measurements. In addition to time of arrival, quantum stroboscopy can describe distributions of general time measurements. It can also be adapted to obtain the conditional probability distribution of arrival times, given that the particle was not previously detected at the detector.

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arXiv2025

Quantum arrival times in free fall

The probability distribution of a time measurement $T_x$ at position $x$ can be inferred from the probability distribution of a position measurement $X_t$ at time $t$ as given by the Born rule [Time-of-arrival distributions for continuous quantum systems and application to quantum backflow, Phys. Rev. A 110, 052217 (2024)]. In an application to free-fall, this finding has been used to predict the existence of a mass-dependent positive relative shift with respect to the classical time-of-arrival in the long time-of-flight regime for dropped quantum particles [M. Beau and L. Martellini, Quantum delay in the time of arrival of free-falling atoms, Phys. Rev. A 109, 012216 (2024).]. The present paper extends these results in two important directions. We first show that for a Gaussian quantum particle of mass $m$ dropped in a uniform gravitational field $g$, the uncertainties about time and position measurements are related by the relation $ ΔT_x ΔX_t \geq \frac{\hbar}{2mg} . $ This novel form of uncertainty relation suggests that choosing the initial state so as to obtain a lower uncertainty in the measured position leads to a higher uncertainty in the measured arrival time. Secondly, we examine the case of a free-falling particle starting from a non-Gaussian initial superposed state, for which we predict the presence of gravitationally induced interferences and oscillations in the mean time-of-arrival as a function of the detector's position that can be interpreted as the signature of a Zitterbewegung-like effect.

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arXiv2024

Time-of-arrival distributions for continuous quantum systems and application to quantum backflow

Using standard results from statistics, we show that for any continuous quantum system (Gaussian or otherwise) and any observable $\widehat{A}$ (position or otherwise), the distribution $π_{a}\left(t\right)$ of time measurement at a fixed state $a$ can be inferred from the distribution $ρ_{t}\left( a\right)$ of a state measurement at a fixed time $t$ via the transformation $π_{a}(t) \propto \left\vert \frac{\partial }{\partial t} \int_{-\infty }^a ρ_t(u) du \right\vert$. This finding suggests that the answer to the long-lasting time-of-arrival problem is in fact secretly hidden within the Born rule, and therefore does not require the introduction of a time operator or a commitment to a specific (e.g., Bohmian) ontology. The generality and versatility of the result are illustrated by applications to the time-of-arrival at a given location for a free particle in a superposed state and to the time required to reach a given velocity for a free-falling quantum particle. Our approach also offers a potentially promising new avenue toward the design of an experimental protocol for the yet-to-be-performed observation of the phenomenon of quantum backflow.

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arXiv2024

On the mutual exclusiveness of time and position in quantum physics and the corresponding uncertainty relation for free falling particles

The uncertainty principle is one of the characteristic properties of quantum theory, where it signals the incompatibility of two types of measurements. In this paper, we argue that measurements of time-of-arrival $T_x$ at position $x$ and position $X_t$ at time $t$ are mutually exclusive for a quantum system, each providing complementary information about the state of that system. For a quantum particle of mass $m$ falling in a uniform gravitational field $g$, we show that the corresponding uncertainty relation can be expressed as $ΔT_x ΔX_t \geq \frac{\hbar}{2mg}$. This uncertainty relationship can be taken as evidence of the presence of a form of epistemic incompatibility in the sense that preparing the initial state of the system so as to decrease the measured position uncertainty will lead to an increase in the measured time-of-arrival uncertainty. These findings can be empirically tested in the context of ongoing or forthcoming experiments on measurements of time-of-arrival for free-falling quantum particles.

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arXiv2024

Quantum delay in the time of arrival of free-falling atoms

Using standard results from statistics, we show that for Gaussian quantum systems the distribution of a time measurement at a fixed position can be directly inferred from the distribution of a position measurement at a fixed time as given by the Born rule. In an application to a quantum particle of mass $m$ falling in a uniform gravitational field $g$, we use this approach to obtain an exact explicit expression for the probability density of the time-of-arrival (TOA). In the long time-of-flight approximation, we predict that the average positive relative shift with respect to the classical TOA in case of a zero initial mean velocity is asymptotically given by $δ= \frac{q^2}{2} $ when the factor $q\equiv \frac{\hbar}{2mσ\sqrt{2gx}} \ll 1$ (semi-classical regime), and by $δ= \sqrt{\frac{2}π}q $ when $q\gg 1$ (quantum regime), where $σ$ is the width of the initial Gaussian wavepacket and $x$ is the mean distance to the detector. We also discuss experimental conditions under which these predictions can be tested.

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arXiv2014

A Semi-Parametric Approach to the Detection of Non-Gaussian Gravitational Wave Stochastic Backgrounds

Using a semi-parametric approach based on the fourth-order Edgeworth expansion for the unknown signal distribution, we derive an explicit expression for the likelihood detection statistic in the presence of non-normally distributed gravitational wave stochastic backgrounds. Numerical likelihood maximization exercises based on Monte-Carlo simulations for a set of large tail symmetric non-Gaussian distributions suggest that the fourth cumulant of the signal distribution can be estimated with reasonable precision when the ratio between the signal and the noise variances is larger than 0.01. The estimation of higher-order cumulants of the observed gravitational wave signal distribution is expected to provide additional constraints on astrophysical and cosmological models.

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