Relative $C$"-Numerical Ranges for Applications in Quantum Control and Quantum Information

TL;DR

This paper introduces the relative $C$-numerical range, a generalization of the classical $C$-numerical range using subgroups of the unitary group, and explores its geometric properties and applications in quantum information.

Contribution

It defines the relative $C$-numerical range for subgroups of the unitary group, analyzes its geometric properties, and applies Lie theory to extend classical results, with focus on $SU_{ m loc}(2^n)$.

Findings

$W_K(C,A)$ is not necessarily star-shaped or simply-connected.

Rotational symmetry results extend to $W_K(C,A)$ via Lie theory.

Conditions for $W_K(C,A)$ to be a centered circular disc are derived.

Abstract

Motivated by applications in quantum information and quantum control, a new type of $C$"-numerical range, the relative $C$"-numerical range denoted $W_K(C,A)$, is introduced. It arises upon replacing the unitary group U(N) in the definition of the classical $C$"-numerical range by any of its compact and connected subgroups $K \subset U(N)$. The geometric properties of the relative $C$"-numerical range are analysed in detail. Counterexamples prove its geometry is more intricate than in the classical case: e.g. $W_K(C,A)$ is neither star-shaped nor simply-connected. Yet, a well-known result on the rotational symmetry of the classical $C$"-numerical range extends to $W_K(C,A)$, as shown by a new approach based on Lie theory. Furthermore, we concentrate on the subgroup $SU_{\rm loc}(2^n) := SU(2)\otimes ... \otimes SU(2)$, i.e. the $n$-fold tensor product of SU(2), which is of particular interest in applications. In this case, sufficient conditions are derived for $W_{K}(C,A)$ being a circular disc centered at origin of the complex plane. Finally, the previous results are illustrated in detail for $SU(2) \otimes SU(2)$.