Refined uncertainty relation for q-commutator

TL;DR

This paper refines the Robertson uncertainty relation specifically for q-commutators, incorporating eigenvalues of the state to provide a more precise bound.

Contribution

It introduces a refined uncertainty relation for q-commutators that depends on the eigenvalues of the quantum state, extending traditional uncertainty principles.

Findings

Refined uncertainty relation for q-commutator derived

Eigenvalues of the state influence the uncertainty bounds

Potential applications in quantum information and operator theory

Abstract

We show that Robertson uncertainty relation can be refined for q-commutator which is defined by $[A,B]_q = AB-qBA$, where $A, B$ are self-adjoint operators and real number $q\in \mathbb{R}$. The coefficient is represented by the eigenvalues of state $\rho$.