We demonstrate in straightforward calculations that even under ideally weak noise the relaxation of bipartite open quantum systems contains elements not previously encountered in quantum noise physics. While additivity of decay rates is known to be generic for decoherence of a single system, we demonstrate that it breaks down for bipartite coherence of even the simplest composite systems.PACS numbers: 03.65. Yz, 03.65.Ud, 42.50.Lc In this note we will establish results that contradict the long-standing belief that additivity of coherence decay rates is a natural consequence of weak noises. This belief means that the relaxation rate of any system exposed to a collection of weak noises is the sum of the relaxation rates associated with the noises separately. Although implied in many textbook discussions, an actual proof of additivity may be difficult to locate. We supply here a proof of additivity for a single qubit coupled to two independent weak noises (here amplitude noise and phase noise), but our main message is the demonstration of violations of additivity in the case of entanglement decay when two or more qubits are involved. That is, we will show that a quantum system with the most elementary composite structure (e.g., simply made of two distinct parts) need not and generally does not exhibit relaxation-rate additivity even though the separate parts do. This result is purely quantum mechanical and extends our understanding of the power of quantum coherence in an unexpected direction.We now present an additivity proof that is quantum mechanical in order to eliminate from concern the possibility that quantum systems are intrinsically different from classical ones in their response to weak noise. We will consider ideal noise sources where "ideal" means that the noise is sufficiently weak and random that the noisesystem interaction is both reliably linear and without significant back action on the noise source. Each noise source can then be treated as a reservoir made of an infinite collection of random and very broadband harmonic oscillators at zero temperature.Of course the relaxing system need not be linear, so we choose the simplest nonlinear system, a qubit (two-level atom, spin one-half, etc.), for our example. The total Hamiltonian for a qubit coupled to two noise sources (two "environments") can be written as follows:(1) * Electronic address: ting@pas.rochester.edu † Electronic address: eberly@pas.rochester.edu where (with = 1)are the Hamiltonians of the qubit system and two local environments. As an example of the types of relaxation that will be relevant, we suppose the two environments couple in the one case longitudinally and in the other case transversely to the qubit. Thus we have for the interaction of the qubit with its two different noise sources:We naturally assume that the two noise sources are not cross-correlated, and in the ideal case under consideration they can be treated in the familiar Born-Markov limit. Thus we write the longitudinal and transverse selfcorrelation functions in the...