Energy-time and frequency-time uncertainty relations: exact inequalities

TL;DR

This paper reviews exact inequalities related to energy-time and frequency-time uncertainty principles, exploring minimal uncertainty signals, quantum state stability, and implications for quantum evolution and measurement.

Contribution

It provides a precise form of uncertainty inequalities, analyzes stationarity times for Gaussian states, and discusses quantum measurement and evolution speed limits.

Findings

Pure quantum states are more fragile than mixed states with same energy dispersion.

Explicit calculation of stationarity time for Gaussian states.

Discussion of quantum speed limits and measurement issues.

Abstract

We give a short review of known exact inequalities that can be interpreted as "energy-time" and "frequency-time" uncertainty relations. In particular we discuss a precise form of signals minimizing the physical frequency-time uncertainty product. Also, we calculate the "stationarity time" for mixed Gaussian states of a quantum harmonic oscillator, showing explicitly that pure quantum states are "more fragile" than mixed ones with the same value of the energy dispersion. The problems of quantum evolution speed limits, time operators and measurements of energy and time are briefly discussed, too.