Injective norm of real and complex random tensors I: From spin glasses to geometric entanglement
arxiv(2024)
摘要
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
ϵ-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.
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