arXiv:1712.10267

Energy-constrained diamond norm with applications to the uniform continuity of continuous variable channel capacities

Andreas Winter

Published Dec 29, 2017• 41 citations

Abstract

The channels, and more generally superoperators acting on the trace class operators of a quantum system naturally form a Banach space under the completely bounded trace norm (aka diamond norm). However, it is well-known that in infinite dimension, the norm topology is often "too strong" for reasonable applications. Here, we explore a recently introduced energy-constrained diamond norm on superoperators (subject to an energy bound on the input states). Our main motivation is the continuity of capacities and other entropic quantities of quantum channels, but we also present an application to the continuity of one-parameter unitary groups and certain one-parameter semigroups of quantum channels.

DOI: 10.48550/arxiv.1712.10267

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arXiv:1712.10267

Energy-constrained diamond norm with applications to the uniform continuity of continuous variable channel capacities

Andreas Winter

Published Dec 29, 2017• 41 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.1712.10267

Related papers

On the Energy-Constrained Diamond Norm and Its Application in Quantum Information Theory

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quant-pharXiv:2610.16270Oct 13, 20260 citations

For a given positive operator G we consider the cones of linear maps between Banach spaces of trace class operators characterized by the Stinespring-like representation with

\sqrt G

-bounded and

\sqrt G

-infinitesimal operators correspondingly. We prove the completeness of both cones w.r.t. the energy-constrained diamond norm induced by G (as an energy observable) and the coincidence of the second cone with the completion of the cone of completely positive linear maps w.r.t. this norm. We show that the sets of quantum channels and quantum operations are complete w.r.t. the energy-constrained diamond norm for any energy observable. Some properties of the maps belonging to the introduced cones are described. In particular, the corresponding generalization of the Kretschmann-Schlingemann-Werner theorem is obtained. We also give a nonconstructive description of the completion of the set of all Hermitian-preserving completely bounded linear maps w.r.t. the energy-constrained diamond norm.

0

and

1

. The remedy is the Arveson--Radon--Nikodym derivative

T

of one map with respect to the sum

Ω

, whose transpose is the environment effect selecting that map in a Stinespring dilation of

Ω

. In

T

, these quantities and all maximal

f

-divergences of operator convex

f

are continuous, with explicit dimension-free moduli; for two pairs of channels with the same sum the fidelities differ by at most

2\|T_1-T_2\|_\infty^{1/2}

, which gives Hölder continuity in the diamond norm on such splittings. The fidelity is continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes: noisy channels of full Kraus rank are continuity points, unitary gates are not.