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(alpha,Delta,h), where alpha is the dissipation strength, Delta is the tunneling amplitude between qubit states, and h is the level asymmetry. For 1-alpha >> Delta/omega(c) and (Delta,h) << omega(c), we show that the Kondo energy scale T-K controls the entanglement between the qubit and the bosonic environment (omega(c) is a high-energy cutoff). For h << T-K, the disentanglement proceeds as (h/T-K)(2); for h > T-K, E vanishes as (T-K/h)(2-2 alpha), up to a logarithmic correction. For a given h, the maximum entanglement occurs at a value of alpha which lies in the crossover regime h similar to T-K. We emphasize the possibility of measuring this entanglement using charge qubits subject to electromagnetic noise.