Energy-Time Uncertainty Relations in Quantum Measurements

TL;DR

This paper establishes a fundamental energy-time uncertainty relation for quantum measurements, linking the apparatus's energy fluctuation with measurement duration, and explores implications for spacetime structure.

Contribution

It introduces a quantum-mechanical model of measurement that derives a new energy-time uncertainty relation involving the apparatus's energy fluctuation and measurement time.

Findings

Proves the energy-time uncertainty relation $ au\,\Delta H_A \geq \frac{\pi \hbar}{4}$.

Shows the necessity of a larger quantum apparatus acting as a timing device.

Derives a trade-off between measurement time and interaction strength.

Abstract

Quantum measurement is a physical process. A system and an apparatus interact for a certain time period (measurement time), and during this interaction, information about an observable is transferred from the system to the apparatus. In this study, we quantify the energy fluctuation of the quantum apparatus required for this physical process to occur autonomously. We first examine the so-called standard model of measurement, which is free from any non-trivial energy-time uncertainty relation, to find that it needs an external system that switches on the interaction between the system and the apparatus. In such a sense this model is not closed. Therefore to treat a measurement process in a fully quantum manner we need to consider a "larger" quantum apparatus which works also as a timing device switching on the interaction. In this setting we prove that a trade-off relation (energy-time uncertainty relation), $\tau\cdot \Delta H_A \geq \frac{\pi \hbar}{4}$, holds between the energy fluctuation $\Delta H_A$ of the quantum apparatus and the measurement time $\tau$. We use this trade-off relation to discuss the spacetime uncertainty relation concerning the operational meaning of the microscopic structure of spacetime. In addition, we derive another trade-off inequality between the measurement time and the strength of interaction between the system and the apparatus.