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A qubit (containing two quantum states, 1 and 2), is coupled to a controlregister (state 3), which is subject to telegraph noise. We study the timeevolution of the density matrix $\rho$ of an electron which starts in somecoherent state on the qubit. At infinite time, $\rho$ usually approaches thefully decoherent state, with $\rho^{}_{nm}=\delta^{}_{nm}/3$. However, when theHamiltonian is symmetric under $1\leftrightarrow 2$, the element $\rho^{}_{12}$approaches a non-zero real value, implying a partial coherence of theasymptotic state. The asymptotic density matrix depends only on ${\rmRe}[\rho^{}_{12}(t=0)]$. In several cases, the information stored on the qubitis protected from the noise.

Partial decoherence in a qubit coupled to a control register subject to telegraph noise