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We study decoherence of one, two, and $n$ non-interacting qubits.Decoherence, measured in terms of purity, is calculated in linear responseapproximation, making use of the spectator configuration. The environment andits interaction with the qubits are modelled by random matrices. For twoqubits, numerical studies reveal a simple one to one correspondence between itsdecoherence and its internal entanglement decay. Using this relation we areable to give a formula for concurrence decay. For large environments theevolution induces a unital channel in the two qubits, providing a partialexplanation for the relation above. Using a kicked Ising spin network, we studythe exact evolution of two non-interacting qubits in the presence of a spinbath. We find that the entanglement (as measured by concurrence) of the twoqubits has a close relation to the purity of the pair, and closely follows ananalytic relation derived for Werner states. As a collateral result we findthat an integrable environment causes quadratic decay of concurrence as well asof purity, while a chaotic environment causes linear decay. Both quantitiesdisplay recurrences in some integrable environments. Good agreement with theresults found using random matrix theory is obtained. Finally, we analyzedecoherence of a quantum register in the absence of non-local operations. Theproblem is solved in terms of a sum rule which implies linear scaling in thenumber of qubits. Each term involves a single qubit and its entanglement withthe remaining ones. Two conditions are essential: decoherence must be small andthe coupling of different qubits must be uncorrelated in the interactionpicture. We apply the result to the random matrix model, and illustrate itsreach considering a GHZ state coupled to a spin bath.

One, Two, and $n$ Qubit Decoherence