Scalable programmable quantum gates and a new aspect of the additivity problem for the classical capacity of quantum channels

Abstract

We consider two apparently separated problems: in the first part of the paper we study the concept of a scalable (approximate) programmable quantum gate (SPQG). These are special (approximate) programmable quantum gates, with nice properties that could have implications on the theory of universal computation. Unfortunately, as we prove, such objects do not exist in the domain of usual quantum theory. In the second part the problem of noisy dense coding (and generalizations) is addressed. We observe that the additivity problem for the classical capacity obtained is of apparently greater generality than for the usual quantum channel (completely positive maps): i.e., the latter occurs as a special case of the former, but, as we shall argue with the help of the non-existence result of the first part, the former cannot be reduced to an instance of the latter. We conclude by suggesting that the additivity problem for the classical capacity of quantum channels, as posed until now, may conceptually not be in its appropriate generality.