Entanglement between a qubit and the environment in the spin-boson model

The quantitative description of the quantum entanglement between a qubit and its environment is considered. Specifically, for the ground state of the spin-boson model, the entropy of entanglement of the spin is calculated as a function of $\alpha ,$ the strength of the ohmic coupling to the environment, and $\varepsilon ,$ the level asymmetry. This is done by a numerical renormalization group treatment of the related anisotropic Kondo model. For $\varepsilon =0,$ the entanglement increases monotonically with $\alpha ,$ until it becomes maximal for $\overrightarrow{\alpha }{1}^{-}.$ For fixed $\varepsilon >0,$ the entanglement is a maximum as a function of $\alpha$ for a value, $\alpha ={\alpha }_M<1.\mathrm{}$