Tighter quantum uncertainty relations follow from a general probabilistic bound

TL;DR

This paper derives quantum uncertainty relations from a classical estimation theory bound, providing a clearer interpretation and tighter bounds for mixed states, with applications to thermal, Gaussian, and many-qubit states.

Contribution

It introduces a simple derivation of quantum URs from a classical inequality, separating probabilistic aspects from quantum structure, and extends to tighter bounds for mixed states.

Findings

Thermal states saturate the tighter bound for natural operators.

Gaussian states of harmonic oscillators reach the bound.

Entanglement influences the structure of UR-saturating operators.

Abstract

Uncertainty relations (URs) like the Heisenberg-Robertson or the time-energy UR are often considered to be hallmarks of quantum theory. Here, a simple derivation of these URs is presented based on a single classical inequality from estimation theory, a Cram\'er-Rao-like bound. The Heisenberg-Robertson UR is then obtained by using the Born rule and the Schr\"odinger equation. This allows a clear separtion of the probabilistic nature of quantum mechanics from the Hilbert space structure and the dynamical law. It also simplifies the interpretation of the bound. In addition, the Heisenberg-Robertson UR is tightened for mixed states by replacing one variance by the so-called quantum Fisher information. Thermal states of Hamiltonians with evenly-gapped energy levels are shown to saturate the tighter bound for natural choices of the operators. This example is further extended to Gaussian states of a harmonic oscillator. For many-qubit systems, we illustrate the interplay between entanglement and the structure of the operators that saturate the UR with spin-squeezed states and Dicke states.