Operator E-norms and their use

TL;DR

This paper introduces operator E-norms on bounded operators in Hilbert spaces, explores their topological properties, and extends the Kretschmann-Schlingemann-Werner theorem to energy-constrained norms, with applications in quantum information.

Contribution

It defines a family of operator E-norms induced by positive operators, analyzes their topologies, and generalizes key theorems to energy-constrained settings in quantum information theory.

Findings

Operator E-norms produce consistent topologies on B(H).

Generalized Kretschmann-Schlingemann-Werner theorem for energy-constrained norms.

Explicit relations and bounds for G-bounded operators.

Abstract

We consider a family of norms (called operator E-norms) on the algebra $B(H)$ of all bounded operators on a separable Hilbert space $H$ induced by a positive densely defined operator $G$ on $H$. Each norm of this family produces the same topology on $B(H)$ depending on $G$. By choosing different generating operator $G$ one can obtain operator E-norms producing different topologies, in particular, the strong operator topology on bounded subsets of $B(H)$. We obtain a generalised version of the Kretschmann-Schlingemann-Werner theorem, which shows continuity of the Stinespring representation of CP linear maps w.r.t. the energy-constrained $cb$-norm (diamond norm) on the set of CP linear maps and the operator E-norm on the set of Stinespring operators. The operator E-norms induced by a positive operator $G$ are well defined for linear operators relatively bounded w.r.t. the operator $\sqrt{G}$ and the linear space of such operators equipped with any of these norms is a Banach space. We obtain explicit relations between the operator E-norms and the standard characteristics of $\sqrt{G}$-bounded operators. The operator E-norms allow to obtain simple upper bounds and continuity bounds for some functions depending on $\sqrt{G}$-bounded operators used in applications.