Uncertainty relations from state polynomial optimization

TL;DR

This paper introduces a semidefinite programming hierarchy based on state polynomial optimization to systematically find tight additive uncertainty relations in quantum mechanics, improving bounds for many cases.

Contribution

It develops a complete hierarchy for additive uncertainty relations using state polynomial optimization, applicable to various operator scenarios and higher moments.

Findings

Improved bounds for 1292 additive uncertainty relations

Hierarchy converges to tight uncertainty bounds

Applicable to multiple operator types and higher moments

Abstract

Uncertainty relations are a fundamental feature of quantum mechanics. How can these relations be found systematically? Here we develop a semidefinite programming hierarchy for additive uncertainty relations in the variances of non-commuting observables. Our hierarchy is built on the state polynomial optimization framework, also known as scalar extension. The hierarchy is complete, in the sense that it converges to tight uncertainty relations. We improve upon upper bounds for all 1292 additive uncertainty relations on up to nine operators for which a tight bound is not known. The bounds are dimension-free and depend entirely on the algebraic relations among the operators. The techniques apply to a range of scenarios, including Pauli, Heisenberg-Weyl, and fermionic operators, and generalize to higher order moments and multiplicative uncertainty relations.