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/ω_c and (Δ,h)≪ω_c, we show that the Kondo energy scale T_K controls the entanglement between the qubit and the bosonic environment (ω_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)^2-2α, 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∼T_K. We emphasize the possibility of measuring this entanglement using charge qubits subject to electromagnetic noise.