Universal and measurable entanglement entropy in the spin-boson model

We study the entanglement between a qubit and its environment from the spin-boson model with Ohmic dissipation. Through a mapping to the anisotropic Kondo model, we derive the entropy of entanglement of the spin E(α,Δ,h), where α is the dissipation strength, Δ is the tunneling amplitude between qubit states, and h is the level asymmetry. For 1-α/Δ/ωc and (Δ,h)/ωc, we show that the Kondo energy scale TK controls the entanglement between the qubit and the bosonic environment (ωc is a high-energy cutoff). For h/TK, the disentanglement proceeds as (h/TK)2; for h/TK, E vanishes as (TK/h)2-2α, up to a logarithmic correction. For a given h, the maximum entanglement occurs at a value of α which lies in the crossover regime h∼TK. We emphasize the possibility of measuring this entanglement using charge qubits subject to electromagnetic noise.