  The injective norm is a natural generalization to tensors of the operator
norm of a matrix. In quantum information, the injective norm is one important
measure of genuine multipartite entanglement of quantum states, where it is
known as the geometric entanglement. In this paper, we give a high-probability
upper bound on the injective norm of real and complex Gaussian random tensors,
corresponding to a lower bound on the geometric entanglement of random quantum
states, and to a bound on the ground-state energy of a particular multispecies
spherical spin glass model. For some cases of our model, previous work used
$\epsilon$-net techniques to identify the correct order of magnitude; in the
present work, we use the Kac--Rice formula to give a one-sided bound on the
constant which we believe to be tight.
