Abstract

The diamond norm on superoperators (aka completely bounded norm’ in the Heisenberg picture) provides a natural, operationally well-motivated, and computable metric on quantum channels (cptp maps). In particular, it quantifies the optimal bias in hypothesis testing between two channels, it is given by a semidefinite programme, and it enjoys a number of good mathematical properties. It furthermore provides the natural setting when discussing continuity of channel capacities. In some situations, however, notably in infinite dimension, the diamond norm is too strong to be reasonably applicable. This can be illustrated with simple Bosonic channels. Inspired by communication theory, i propose a definition of diamond norm with an energy constraint (with respect to a given Hamiltonian), and show that this allows for a resolution of most of the issues of the diamond norm, while retaining its good properties. As an application, i show how to prove the continuity of Bosonic quantum channel capacities (C, Q, P, etc) with an energy constraint at the input.

[Based on arXiv:1712.10267]