Entirety of Quantum Uncertainty and Its Experimental Verification

TL;DR

This paper introduces a geometric method to characterize quantum uncertainty relations as areas of variances over all states, providing analytical descriptions and experimental verification, especially for position-momentum and finite-dimensional systems.

Contribution

It presents a novel geometric framework for uncertainty relations, including analytical characterizations and the first experimental observation of variance areas in photonic systems.

Findings

Heisenberg's uncertainty points to the variance area for position and momentum.

The variance area for finite-dimensional systems is semialgebraic.

First experimental observation of variance areas in a photonic system.

Abstract

As a foundation of modern physics, uncertainty relations describe an ultimate limit for the measurement uncertainty of incompatible observables. Traditionally, uncertain relations are formulated by mathematical bounds for a specific state. Here we present a method for geometrically characterizing uncertainty relations as an entire area of variances of the observables, ranging over all possible input states. We find that for the pair of position $x$ and momentum $p$ operators, Heisenberg's uncertainty principle points exactly to the area of the variances of $x$ and $p$. Moreover, for finite-dimensional systems, we prove that the corresponding area is necessarily semialgebraic; in other words, this set can be represented via finite polynomial equations and inequalities, or any finite union of such sets. In particular, we give the analytical characterization of the areas of variances of (a) a pair of one-qubit observables, (b) a pair of projective observables for arbitrary dimension, and give the first experimental observation of such areas in a photonic system.