Kraus-Like Decompositions

We introduce a new decomposition of quantum channels acting on group algebras, which we term Kraus-like (operator) decompositions. We motivate this decomposition with a general nonexistence result for Kraus operator decompositions in this setting. Given a length function which is a class function on a finite group, we construct a corresponding Kraus-like decomposition. We prove that this Kraus-like decomposition is \textit{convex} (meaning its coefficients are nonnegative and satisfy a sum rule) if and only if the length is conditionally negative definite. For a general finite group, we prove a stability condition which shows that the existence of a convex Kraus-like decomposition for all $t>0$ small enough necessarily implies existence for all time $t>0$. Using the stability condition, we show that for a general finite group, conditional negativity of the length function is equivalent to a set of semidefinite linear constraints on the length function. Our result implies that in the group algebra setting, a semigroup $P_t$ induced by a length function which is a class function is a quantum channel for all $t\geq 0$ if and only if it possesses a convex Kraus-like decomposition for all $t>0$.