Uncertainty relation and the constrained quadratic programming

TL;DR

This paper investigates the additive uncertainty relation in qudit systems, formulating the problem as a quadratic programming challenge to find tight lower bounds and extremal states, with analytical and numerical solutions.

Contribution

It introduces a quadratic programming framework for uncertainty relations in qudit systems, providing analytical solutions and an efficient numerical algorithm for tight bounds.

Findings

Derived analytical solutions for lower-dimensional systems.

Developed a numerical algorithm for solving quadratic programming problems.

Established tight state-independent lower bounds for variance sums.

Abstract

The uncertainty relation is a fundamental concept in quantum theory, plays a pivotal role in various quantum information processing tasks. In this study, we explore the additive uncertainty relation pertaining to two or more observables, in terms of their variance,by utilizing the generalized Gell-Mann representation in qudit systems. We find that the tight state-independent lower bound of the variance sum can be characterized as a quadratic programming problem with nonlinear constraints in optimization theory. As illustrative examples, we derive analytical solutions for these quadratic programming problems in lower-dimensional systems, which align with the state-independent lower bounds. Additionally, we introduce a numerical algorithm tailored for solving these quadratic programming instances, highlighting its efficiency and accuracy. The advantage of our approach lies in its potential ability to simultaneously achieve the optimal value of the quadratic programming problem with nonlinear constraints but also precisely identify the extremal state where this optimal value is attained. This enables us to establish a tight state-independent lower bound for the sum of variances, and further identify the extremal state at which this lower bound is realized.