Induced trace norm ($\Vert T{\Vert }_1$)

The induced trace norm of a linear map $T$ between matrix spaces isIt measures the largest trace-norm amplification available without an auxiliary tensor factor.

The diamond norm is the stabilized induced trace normFor a map on $d$-dimensional inputs, an auxiliary space of dimension $d$ suffices. The tensor factor lets the norm detect how $T$ acts on one half of an entangled input.