Non-Local to Local Eigenbasis Permutations of Pauli Product Diagonal Operators
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910743930077184 |
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| author | Commeau, Benjamin Player, Kevin |
| author_facet | Commeau, Benjamin Player, Kevin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10223 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Local to Local Eigenbasis Permutations of Pauli Product Diagonal Operators Commeau, Benjamin Player, Kevin Quantum Physics 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. |
| title | Non-Local to Local Eigenbasis Permutations of Pauli Product Diagonal Operators |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2412.10223 |