Note on the saturation of the norm inequalities between diamond and   nuclear norm
  The diamond norm plays an important role in quantum information and operator theory. Recently, it has also been proposed as a regularizer for low-rank matrix recovery. The norm constants that bound the diamond norm in terms of the nuclear norm (also known as trace norm) are explicitly known. This note provides a simple characterization of all operators saturating the upper and the lower bound. 
Ulrich Michel, Martin Kliesch, Richard Kueng, and David Gross
cs.IT math.IT quant-ph