  Two electronic channels competing to screen a single impurity spin, as in the
two-channel Kondo model, are expected to generate a ground state with
nontrivial entanglement structure. We exploit a spin-chain representation of
the two-channel Kondo model to probe the ground-state block entropy,
negativity, tangle, and Schmidt gap, using a density matrix renormalization
group approach. In the presence of symmetric coupling to the two channels we
confirm field-theory predictions for the boundary entropy difference, $\ln
(g_{UV}/g_{IR})=\ln(2)/2$, between the ultraviolet and infrared limits and the
leading $\ln(x)/x$ impurity correction to the block entropy. The impurity
entanglement, $S_{\text{imp}}$, is shown to scale with the characteristic
length $\xi_{2CK}$. We show that both the Schmidt gap and the entanglement of
the impurity with one of the channels $-$ as measured by the negativity$-$
faithfully serve as order parameters for the impurity quantum phase transition
appearing as a function of channel asymmetry, allowing for explicit
determination of critical exponents, $\nu\!\approx\! 2$ and $\beta \!\approx\!
0.2$. Remarkably, we find the emergence of tripartite entanglement only in the
vicinity of the critical channel-symmetric point.
