We construct entangled states of gluons that scatter exactly as if they were gravitons. Operationally, these objects implement the double copy at the level of the wave function... Our analysis begins with a general ansatz for a wave function characterizing gluons in two copies of ${\rm SU}(N)$ gauge theory. Given relatively minimal assumptions following from permutation invariance and dimensional analysis, the three- and four-particle wave functions generate scattering amplitudes that automatically coincide exactly with gravity, modulo normalization. For five-particle scattering the match is not automatic but imposing certain known selection rules on the amplitude is sufficient to uniquely reproduce gravity. The resulting amplitudes exhibit a color-dressed and permutation-invariant form of the usual double copy relations. We compute the entanglement entropy between the two gauge theory copies and learn that these states are maximally-entangled at large $N$. Moreover, this approach extends immediately to effective field theories, where Born-Infeld photons and Galileons can be similarly recast as entangled gluons and pions. (read more)