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arXiv:2507.23059 · Physical Review A

Context-dependent time-energy uncertainty relations from projective quantum measurements

Mathieu Beau

Published Nov 13, 2025• 3 citations
arXivHTMLPDF

Abstract

We introduce a general framework for defining context-dependent time distributions in quantum systems using projective measurements. The time-of-flow (TF) distribution, derived from population transfer rates into a measurement subspace, yields a time-energy uncertainty relation of the form $\mathrm{\ensuremath{\Delta}}\mathcal{T}\mathrm{\ensuremath{\Delta}}H\ensuremath{\ge}\ensuremath{\hbar}/(6\sqrt{3})\ensuremath{\delta}\ensuremath{\theta}$, where $\ensuremath{\delta}\ensuremath{\theta}$ quantifies the net population transfer measured by the projector. This bound applies to arbitrary projectors under unitary dynamics and reveals that time uncertainty is inherently measurement dependent. We demonstrate the framework with two applications: a general time-of-arrival--energy uncertainty relation and a driven three-level system under detuned coherent driving. The TF framework unifies timing observables across spin, atomic, and matter-wave systems and offers an experimentally accessible route to probing quantum timing in controlled measurements.

DOI
10.1103/f9s1-32kz

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We introduce a general framework for defining context-dependent time distributions in quantum systems using projective measurements. The time-of-flow (TF) distribution, derived from population transfer rates into a measurement subspace, yields a time-energy uncertainty relation of the form \ensuremathΔT\ensuremathΔH\ensuremath≥\ensuremathℏ/(63)\ensuremathδ\ensuremathθ\mathrm{\ensuremath{\Delta}}\mathcal{T}\mathrm{\ensuremath{\Delta}}H\ensuremath{\ge}\ensuremath{\hbar}/(6\sqrt{3})\ensuremath{\delta}\ensuremath{\theta}\ensuremathΔT\ensuremathΔH\ensuremath≥\ensuremathℏ/(63​)\ensuremathδ\ensuremathθ, where \ensuremathδ\ensuremathθ\ensuremath{\delta}\ensuremath{\theta}\ensuremathδ\ensuremathθ quantifies the net population transfer measured by the projector. This bound applies to arbitrary projectors under unitary dynamics and reveals that time uncertainty is inherently measurement dependent. We demonstrate the framework with two applications: a general time-of-arrival--energy uncertainty relation and a driven three-level system under detuned coherent driving. The TF framework unifies timing observables across spin, atomic, and matter-wave systems and offers an experimentally accessible route to probing quantum timing in controlled measurements.
We propose a general and experimentally accessible framework to quantify transition timing in discrete quantum systems via the time-of-flow (TF) distribution. Defined from the rate of population change in a target state, the TF distribution can be reconstructed through repeated projective measurements at discrete times on independently prepared systems, thus avoiding Zeno inhibition. In monotonic regimes, it admits a clear interpretation as a time-of-arrival (TOA) or time-of-departure (TOD) distribution. We apply this approach to optimize time-dependent Hamiltonians, analyze shortcut-to-adiabaticity (STA) protocols, study non-adiabatic features in the dynamics of a three-level time-dependent detuning model, and derive a transition-based quantum speed limit (TF-QSL) for both closed and open quantum systems. We also establish a lower bound on temporal uncertainty and examine decoherence effects, demonstrating the versatility of the TF framework for quantum control and diagnostics. This method provides both a conceptual tool and an experimental protocol for probing and engineering quantum dynamics in discrete-state platforms.
Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.
A precise form of the quantum-mechanical time-energy uncertainty relation is derived. For any given initial state (density operator), time-dependent Hamiltonian, and subspace of reference states, it gives upper and lower bounds for the probability of finding the system in a state in that subspace at a later or earlier time. The bounds involve only the initial data, the energy uncertainty in the initial state, and the energy uncertainty in the reference subspace. They describe how fast the state enters or leaves the reference subspace. They are exact if, but not only if, the initial state or the projection onto the reference subspace commutes with the Hamiltonian. The basic tool used in the proof is a simple inequality for expectation values of commutators, which generalizes the usual uncertainty relation. By introducing suitable comparison dynamics (trial propagators), the bounds can be made arbitrarily tight. They represent a time-dependent variational principle, in terms of trial propagators, which provides explicit error estimates and reproduces the exact time evolution when one varies over all trial propagators. As illustrations, we derive accurate lower bounds on the escape time of a particle out of a potential well modeling a quantum dot, and the total time before which a He+{\mathrm{He}}^{+}He+ ion moving in a uniform magnetic field loses its electron.
We present a unified energy-ontological interpretation of non-relativistic quantum mechanics within the framework of Energy-Efficiency Theory (EET). Starting from three physically grounded axioms, we define the constrained potential U(r,t) as the spatial density of localized bound-state energy, and derive the continuity equation for the normalized constrained potential ρ(r,t)=U(r,t)/E_c. We prove that ρ(r,t) satisfies the quantum probability continuity equation, establishing the exact correspondence ρ(r,t)=|ψ|^2. We introduce the dimensionless energy ratio η = Ė_resp / Ė_main, which quantifies the balance between energy allocated to spatial exploration (response) and constraint maintenance. From first principles, we derive a modified Schrödinger equation with a real effective potential that preserves probability conservation for all values of η. Using an exact analytical solution for Gaussian wave packets, we derive the scaling relations Δx ∝ η^{-1/2} and Δp ∝ η^{1/2}, which imply the Heisenberg uncertainty principle ΔxΔp ≥ ħ/2 independent of η. Wave-particle duality is interpreted as the continuous sliding of η between exploration-dominated (wave-like, η>1) and maintenance-dominated (particle-like, η<1) regimes. We extend the framework to quantum measurement, deriving the scaling of η from first-principles system-apparatus energy coupling, and unify projective measurements, weak measurements, the quantum eraser, and decoherence within a single framework. We present four experimentally falsifiable predictions with strict statistical criteria, along with a protocol for independent experimental control of η. This work provides a self-consistent energy-ontological foundation for non-relativistic quantum mechanics, unifying probability density, wave-particle duality, the uncertainty principle, and quantum measurement under a single energy allocation framework.
We develop a general framework for the construction of probabilities for the time of arrival in quantum systems. The time of arrival is identified with the time instant when a transition in the detector's degrees of freedom takes place. Thus, its definition is embedded within the larger issue of defining probabilities with respect to time for general quantum transitions. The key point in our analysis is that we manage to reduce the problem of defining a quantum time observable to a mathematical model where time is associated to a transition from a subspace of the Hilbert space of the total system to its complementary subspace. This property makes it possible to derive a general expression for the probability for the time of transition, valid for any quantum system, with the only requirement that the time of transition is correlated with a definite macroscopic record. The framework developed here allows for the consideration of any experimental configuration for the measurement of the time of arrival, and it also applies to relativistic systems with interactions described by quantum field theory. We use the method in order to describe time-of-arrival measurements in high-energy particle reactions and for a rigorous derivation of the time-integrated probabilities in particle oscillations.