The diamond norm of a linear transformation 
between finite-dimensional matrix
algebras is its trace-norm operator norm after adjoining
an auxiliary identity map of sufficient dimension.
For input dimension 
, it can be written
 |
where 
ranges over matrices on the input and auxiliary tensor
factors and 
is the trace norm. The auxiliary factor permits entanglement
with a reference system. The diamond-norm distance between quantum
channels is at most 2. Some error conventions use half this distance; the unitary synthesis problem uses the full
norm.
