Local numerical range for a class of $2\otimes d$ hermitian operators

TL;DR

This paper investigates the local numerical range of circulant observables in $2 imes d$ quantum systems, simplifying the problem to real-valued optimization and providing explicit results for the case when d=2.

Contribution

It demonstrates that for any $2 imes d$ circulant operator, a basis exists where the problem reduces to real non-negative off-diagonal elements, enabling analytical solutions.

Findings

Existence of a basis with real non-negative off-diagonal elements for $2 imes d$ circulant operators

Reduction of the extremum problem to real vector spaces

Analytical solution provided for the case when d=2

Abstract

A local numerical range is analyzed for a family of circulant observables and states of composite $2 \otimes d$ systems. It is shown that for any $2\otimes d$ circulant operator $\cal O$ there exists a basis giving rise to the matrix representation with real non-negative off-diagonal elements. In this basis the problem of finding extremum of $\cal O$ on product vectors $\ket{x}\otimes \ket{y} \in \mathbb{C}^2\otimes \mathbb{C}^d$ reduces to the corresponding problem in $\mathbb{R}^2\otimes \mathbb{R}^d$. The final analytical result for $d=2$ is presented.