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

Abstract

A qubit (containing two quantum states, 1 and 2), is coupled to a control register (state 3), which is subject to telegraph noise. We study the time evolution of the density matrix $ρ$ of an electron which starts in some coherent state on the qubit. At infinite time, $ρ$ usually approaches the fully decoherent state, with $ρ^{}_{nm}=δ^{}_{nm}/3$. However, when the Hamiltonian is symmetric under $1\leftrightarrow 2$, the element $ρ^{}_{12}$ approaches a non-zero real value, implying a partial coherence of the asymptotic state. The asymptotic density matrix depends only on ${\rm Re}[ρ^{}_{12}(t=0)]$. In several cases, the information stored on the qubit is protected from the noise.