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

• URL: http://arxiv.org/abs/2412.10223v1
• Date: Fri, 13 Dec 2024 15:48:01 GMT
• Title: Non-Local to Local Eigenbasis Permutations of Pauli Product Diagonal Operators
• Authors: Benjamin Commeau, Kevin Player,
• Abstract summary: This paper investigates the feasibility of mapping non-local, sparse, diagonal forms of quantum Hamiltonians to local forms. We prove that such a mapping is not always possible, definitively refuting the "Quasiparticle Locality Conjecture" 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.
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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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