Hamiltonian and Symplectic Tensors in the T-product Algebra
Abstract
We study Hamiltonian and symplectic tensor structures in the T-product algebra. We define T-Hamiltonian and T-symplectic tensors and characterize them through their Fourier-domain slices. For T-Hamiltonian tensors we establish the standard block form and the spectral symmetry of T-eigenvalues, while for T-symplectic tensors we derive the inverse and exponential-map properties. Our main result is a constructive T-Williamson normal form for tensors whose Fourier-domain slices are real symmetric positive-definite matrices. We also show that, under the Hermitian symplectic convention adopted here, this decomposition does not extend directly to arbitrary Hermitian positive-definite Fourier-domain slices, and we derive a real-valued recovery criterion under Fourier conjugate symmetry. Numerical experiments verify the construction, exhibit runtime trends consistent with the slice-wise complexity O(pn3), and illustrate the framework on a Fourier-domain encoding of covariance-matrix families arising in continuous-variable quantum dynamics.
Versions
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BibTeX
@misc{lopezmoreno2026hamiltonian,
title = {{Hamiltonian and Symplectic Tensors in the T-product Algebra}},
author = {Susana Lopez-Moreno and Taehyeong Kim},
howpublished = {arXiv},
year = {2026},
month = may,
doi = {10.48550/arxiv.2605.20829},
eprint = {2605.20829},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.20829},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hamiltonian and Symplectic Tensors in the T-product Algebra
AU - Susana Lopez-Moreno
AU - Taehyeong Kim
PY - 2026
DA - 2026/05/20
PB - arXiv
DO - 10.48550/arxiv.2605.20829
UR - https://arxiv.org/abs/2605.20829
AB - We study Hamiltonian and symplectic tensor structures in the T-product algebra. We define T-Hamiltonian and T-symplectic tensors and characterize them through their Fourier-domain slices. For T-Hamiltonian tensors we establish the standard block form and the spectral symmetry of T-eigenvalues, while for T-symplectic tensors we derive the inverse and exponential-map properties. Our main result is a constructive T-Williamson normal form for tensors whose Fourier-domain slices are real symmetric positive-definite matrices. We also show that, under the Hermitian symplectic convention adopted here, this decomposition does not extend directly to arbitrary Hermitian positive-definite Fourier-domain slices, and we derive a real-valued recovery criterion under Fourier conjugate symmetry. Numerical experiments verify the construction, exhibit runtime trends consistent with the slice-wise complexity O(pn^3), and illustrate the framework on a Fourier-domain encoding of covariance-matrix families arising in continuous-variable quantum dynamics.
ER -