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

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 $\mathcal{O}$ 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 $\mathcal{O}$ on product vectors $|x⟩\otimes |y⟩\in {\mathbb{C}}^2\otimes {\mathbb{C}}^d$ reduces to the corresponding problem in ${\mathbb{R}}^2\otimes {\mathbb{R}}^d$. The final analytical result for $d=2$ is presented.

References in corpus (3)

• On circulant states with positive partial transpose
• Generalized Circulant Densities and a Sufficient Condition for Separability
• Relative $C$"-Numerical Ranges for Applications in Quantum Control and Quantum Information