  We analyze decoherence of a quantum register in the absence of non-local
operations i.e. of $n$ non-interacting qubits coupled to an environment. The
problem is solved in terms of a sum rule which implies linear scaling in the
number of qubits. Each term involves a single qubit and its entanglement with
the remaining ones. Two conditions are essential: first decoherence must be
small and second the coupling of different qubits must be uncorrelated in the
interaction picture. We apply the result to a random matrix model, and
illustrate its reach considering a GHZ state coupled to a spin bath.
