Entanglement Complexity in Many-body Systems from Positivity Scaling Laws
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911624489598976 |
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| author | Schouten, Anna O. Mazziotti, David A. |
| author_facet | Schouten, Anna O. Mazziotti, David A. |
| contents | Area laws describe how entanglement entropy scales and thus provide important necessary conditions for efficient quantum many-body simulation, but they do not, by themselves, yield a direct measure of computational complexity. Here we introduce a complementary framework based on $p$-particle positivity conditions from reduced density matrix (RDM) theory. These conditions form a hierarchy of $N$-representability constraints for an RDM to correspond to a valid $N$-particle quantum system, becoming exact when the Hamiltonian can be expressed as a convex combination of positive semidefinite $p$-particle operators. We prove a general complexity bound: if a quantum system is solvable with level-$p$ positivity independent of its size, then its entanglement complexity scales polynomially with order $p$. This theorem connects structural constraints on RDMs with computational tractability and provides a rigorous framework for certifying when many-body methods including RDM methods can efficiently simulate correlated quantum matter and materials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02944 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Entanglement Complexity in Many-body Systems from Positivity Scaling Laws Schouten, Anna O. Mazziotti, David A. Quantum Physics Chemical Physics Computational Physics Area laws describe how entanglement entropy scales and thus provide important necessary conditions for efficient quantum many-body simulation, but they do not, by themselves, yield a direct measure of computational complexity. Here we introduce a complementary framework based on $p$-particle positivity conditions from reduced density matrix (RDM) theory. These conditions form a hierarchy of $N$-representability constraints for an RDM to correspond to a valid $N$-particle quantum system, becoming exact when the Hamiltonian can be expressed as a convex combination of positive semidefinite $p$-particle operators. We prove a general complexity bound: if a quantum system is solvable with level-$p$ positivity independent of its size, then its entanglement complexity scales polynomially with order $p$. This theorem connects structural constraints on RDMs with computational tractability and provides a rigorous framework for certifying when many-body methods including RDM methods can efficiently simulate correlated quantum matter and materials. |
| title | Entanglement Complexity in Many-body Systems from Positivity Scaling Laws |
| topic | Quantum Physics Chemical Physics Computational Physics |
| url | https://arxiv.org/abs/2509.02944 |