State-independent preparation uncertainty relations

TL;DR

This paper introduces tight, state-independent uncertainty relations based on Lie algebraic properties, providing more accurate bounds for observable incompatibility and extending entanglement detection methods.

Contribution

It develops a class of tight variance-based sum-uncertainty relations that depend only on the algebraic structure of observables, overcoming limitations of traditional state-dependent bounds.

Findings

Derived tight uncertainty relations for various Lie algebras.

Lower bounds depend solely on the irreducible representation.

Extended entanglement detection criteria using these bounds.

Abstract

The standard state-dependent Heisenberg-Robertson uncertainly-relation lower bound fails to capture the quintessential incompatibility of observables as the bound can be zero for some states. To remedy this problem, we establish a class of tight (i.e., inequalities are saturated)variance-based sum-uncertainty relations derived from the Lie algebraic properties of observables and show that our lower bounds depend only on the irreducible representation assumed carried by the Hilbert space of state of the system. We illustrate our result for the cases of the Weyl-Heisenberg algebra, special unitary algebras up to rank 4, and any semisimple compact algebra. We also prove the usefulness of our results by extending a known variance-based entanglement detection criterion.