Injective norm of real and complex random tensors I: From spin glasses to geometric entanglement

• URL: http://arxiv.org/abs/2404.03627v1
• Date: Thu, 4 Apr 2024 17:49:23 GMT
• Title: Injective norm of real and complex random tensors I: From spin glasses to geometric entanglement
• Authors: Stephane Dartois, Benjamin McKenna,
• Abstract summary: In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states. In this paper, we give a high-probability upper bound on the injective norm of real and complex Gaussian random tensors.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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