Hamiltonian and Symplectic Tensors in the T-product Algebra

TL;DR

This paper introduces a framework for Hamiltonian and symplectic tensors within the T-product algebra, providing new normal forms and properties, with applications in quantum dynamics.

Contribution

It defines T-Hamiltonian and T-symplectic tensors, characterizes them via Fourier slices, and develops a constructive T-Williamson normal form with practical numerical validation.

Findings

Established the spectral symmetry of T-eigenvalues for T-Hamiltonian tensors.

Derived inverse and exponential-map properties for T-symplectic tensors.

Numerical experiments confirm the construction and analyze runtime complexity.

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 $O(pn^3)$, and illustrate the framework on a Fourier-domain encoding of covariance-matrix families arising in continuous-variable quantum dynamics.