General method for obtaining the energy minimum of spin Hamiltonians for separable states
Abstract
We introduce a highly general method to determine the energy minimum of spin Hamiltonians over separable states when the single-particle reduced density matrices are fixed. We present efficient numerical methods that reduce the problem to calculations in a two-particle system. For a broad class of very general spin models, we present analytic formulas for the minimum. For ferromagnetic Ising and Ising-like models of spin-j particles, the minimum over separable states on two-colorable lattices, and over symmetric separable states on fully connected graphs, is given by a compact analytic formula involving the quantum Fisher information. For antiferromagnetic Ising-like and XY-like models of spin j-particles living on a complete graph, the energy minimum for symmetric separable states is given with the expectation value of single-particle operators. For the ferromagnetic Heisenberg chain of spin-1/2 particles, the minimum is expressed via the Uhlmann-Jozsa fidelity. These relations enable the extraction of the quantum Fisher information and quantum fidelity of the corresponding single-particle reduced states from correlation measurements on ground-state ensembles of suitably engineered spin models. For a number of other models, we obtain not necessarily tight lower bounds. We also present bounds for various levels of multiparticle entanglement.
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BibTeX
@misc{toth2026general,
title = {{General method for obtaining the energy minimum of spin Hamiltonians for separable states}},
author = {Géza Tóth and József Pitrik},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.03022},
eprint = {2605.03022},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.03022},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - General method for obtaining the energy minimum of spin Hamiltonians for separable states
AU - Géza Tóth
AU - József Pitrik
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2605.03022
UR - https://arxiv.org/abs/2605.03022
AB - We introduce a highly general method to determine the energy minimum of spin Hamiltonians over separable states when the single-particle reduced density matrices are fixed. We present efficient numerical methods that reduce the problem to calculations in a two-particle system. For a broad class of very general spin models, we present analytic formulas for the minimum. For ferromagnetic Ising and Ising-like models of spin-j particles, the minimum over separable states on two-colorable lattices, and over symmetric separable states on fully connected graphs, is given by a compact analytic formula involving the quantum Fisher information. For antiferromagnetic Ising-like and XY-like models of spin j-particles living on a complete graph, the energy minimum for symmetric separable states is given with the expectation value of single-particle operators. For the ferromagnetic Heisenberg chain of spin-1/2 particles, the minimum is expressed via the Uhlmann-Jozsa fidelity. These relations enable the extraction of the quantum Fisher information and quantum fidelity of the corresponding single-particle reduced states from correlation measurements on ground-state ensembles of suitably engineered spin models. For a number of other models, we obtain not necessarily tight lower bounds. We also present bounds for various levels of multiparticle entanglement.
ER -