  Quantum entanglement between an impurity and its environment is expected to
be central in quantum impurity problems. We develop a method to compute the
entanglement in spin-1/2 impurity problems, based on the entanglement
negativity and the boundary conformal field theory (BCFT). Using the method, we
study the thermal decay of the entanglement in the multichannel Kondo effects.
At zero temperature, the entanglement has the maximal value independent of the
number of the screening channels. At low temperature, the entanglement exhibits
a power-law thermal decay. The power-law exponent equals two times of the
scaling dimension of the BCFT boundary operator describing the impurity spin,
and it is attributed to the energy-dependent scaling behavior of the
entanglement in energy eigenstates. These agree with numerical renormalization
group results, unveiling quantum coherence inside the Kondo screening length.
