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arXiv · 2412.10223

Non-Local to Local Eigenbasis Permutations of Pauli Product Diagonal Operators

Benjamin Commeau·Kevin Player
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Abstract

This paper investigates the feasibility of mapping non-local, sparse, diagonal forms of quantum Hamiltonians to local forms via eigenbasis permutations. We prove that such a mapping is not always possible, definitively refuting the "Quasiparticle Locality Conjecture." This refutation is achieved by establishing a lower bound, denoted $G_m$, on the number of non-zero terms in a localized diagonal form. Remarkably, $G_m$ reaches cosmologically large values, comparable to the entropy of the observable universe for certain localities $m$. While this theoretically guarantees the conjecture's falsity, the immense scale of $G_m$ motivates us to explore the implications for practically sized systems through a probabilistic approach. We construct a set of random, non-local, sparse, diagonal forms and hypothesize their probability of finding a local representation. Our hypothesize suggests a sharp transition in this probability, linked to the Hamiltonian's sparsity relative to the Bekenstein-Hawking entropy of neutron stars to black holes transition. This observation hints at a potential connection between Hamiltonian sparsity, localizability, critical phenomena warranting further investigation into their interplay in both theoretical and astrophysical contexts.

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Benjamin Commeau, Kevin Player. 2024-12-13. Non-Local to Local Eigenbasis Permutations of Pauli Product Diagonal Operators. https://arxiv.org/abs/2412.10223

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