  The concept of entanglement entropy appears in multiple contexts, from black
hole physics to quantum information theory, where it measures the entanglement
of quantum states. We investigate the entanglement entropy in a simple model,
the spin-boson model, which describes a qubit (two-level system) interacting
with a collection of harmonic oscillators that models the environment
responsible for decoherence and dissipation. The entanglement entropy allows to
make a precise unification between entanglement of the spin with its
environment, decoherence, and quantum phase transitions. We derive exact
analytical results which are confirmed by Numerical Renormalization Group
arguments both for an ohmic and a subohmic bosonic bath. Those demonstrate that
the entanglement entropy obeys universal scalings. We make comparisons with
entanglement properties in the quantum Ising model and in the Dicke model. We
also emphasize the possibility of measuring this entanglement entropy using
charge qubits subject to electromagnetic noise; such measurements would provide
an empirical proof of the existence of entanglement entropy.
