Tasks, references, and claim contracts

All Hilbert spaces are finite dimensional. A channel is a completely positive trace-preserving linear map between full matrix algebras. We use Schrödinger-picture channels and Hilbert–Schmidt adjoints.

For channels with the same input and output spaces define

For channel differences, the diamond norm can be evaluated on states with an ancillary reference of dimension equal to the input dimension. Thus $d_{\diamond }$ includes arbitrary input-reference correlations. Its range is $[0,1]$. We use the normalized Choi convention throughout; see Section sec:choi. These conventions agree with the channel-distance discussion in WallmanFlammia.