On a measure of distance for quantum strategies

The present paper studies an operator norm that captures the distinguishability of quantum strategies in the same sense that the trace norm captures the distinguishability of quantum states or the diamond norm captures the distinguishability of quantum channels. Characterizations of its unit ball and dual norm are established via strong duality of a semidefinite optimization problem. A full, formal proof of strong duality is presented for the semidefinite optimization problem in question. This norm and its properties are employed to generalize a state discrimination result of Gutoski and Watrous [In Proceedings of the 22nd Symposium on Theoretical Aspects of Computer Science (STACS'05), Lecture Notes in Computer Science, Vol. 3404 (Springer, 2005), pp. 605-616. The generalized result states that for any two convex sets S-0, S-1 of strategies there exists a fixed interactive measurement scheme that successfully distinguishes any choice of S-0 is an element of S-0 from any choice of S-1 is an element of S-1 with bias proportional to the minimal distance between the sets S-0 and S-1 as measured by this norm. A similar discrimination result for channels then follows as a special case. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3693621]