On the complexity of approximating the diamond norm

Abstract

The diamond norm is a norm defined over the space of quantum transformations. This norm has a natural operational interpretation: it measures how well one can distinguish between two transformations by applying them to a state of arbitrarily large dimension. This interpretation makes this norm useful in the study of quantum interactive proof systems. In this note we exhibit an efficient algorithm for computing this norm using convex programming. Independently of us, Watrous recently showed a different algorithm to compute this norm. An immediate corollary of this algorithm is a slight simplification of the argument of Kitaev and Watrous [STOC 2000] that QIP is contained in EXP.