Product numerical range in a space with tensor product structure

TL;DR

This paper investigates the properties of the product numerical range for operators on tensor product Hilbert spaces, deriving bounds and exploring geometric features, with implications for quantum information theory.

Contribution

It introduces new bounds for the product numerical range of Hermitian operators and characterizes the product numerical range for non-Hermitian operators in tensor spaces.

Findings

Product numerical range of Hermitian operators has specific bounds.

Product numerical range of non-Hermitian operators contains the spectrum barycenter.

The product numerical range equals the Minkowski product of individual ranges.

Abstract

We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.