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We derive universal entanglement entropy and Schmidt eigenvalue behaviors forthe eigenstates of two quantum chaotic systems coupled with a weak interaction.The progression from a lack of entanglement in the noninteracting limit to theentanglement expected of fully randomized states in the opposite limit isgoverned by the single scaling transition parameter, $\Lambda$. The behaviorsapply equally well to few- and many-body systems, e.g.\ interacting particlesin quantum dots, spin chains, coupled quantum maps, and Floquet systems as longas their subsystems are quantum chaotic, and not localized in some manner. Tocalculate the generalized moments of the Schmidt eigenvalues in theperturbative regime, a regularized theory is applied, whose leading orderbehaviors depend on $\sqrt{\Lambda}$. The marginal case of the $1/2$ moment,which is related to the distance to closest maximally entangled state, is anexception having a $\sqrt{\Lambda}\ln \Lambda$ leading order and a logarithmicdependence on subsystem size. A recursive embedding of the regularizedperturbation theory gives a simple exponential behavior for the von Neumannentropy and the Havrda-Charv{\' a}t-Tsallis entropies for increasinginteraction strength, demonstrating a universal transition to nearly maximalentanglement. Moreover, the full probability densities of the Schmidteigenvalues, i.e.\ the entanglement spectrum, show a transition from power lawsand L\'evy distribution in the weakly interacting regime to random matrixresults for the strongly interacting regime. The predicted behaviors are testedon a pair of weakly interacting kicked rotors, which follow the universalbehaviors extremely well.

Eigenstate entanglement between quantum chaotic subsystems: universal transitions and power laws in the entanglement spectrum