Gaussian-Probe Connected-Correlation Tomography Interaction-Order Certificates, Local Normal Completeness,and Global Fermionic Hamiltonian Identifiability

A Short-Time Tomographic Framework for Interaction-Order Certification and Global Nonquadratic Hamiltonian Identification

抄録

We develop a short-time tomography framework for parity-even fermionic Hamiltonians using pure Gaussian probe states and connected correlation functions.
The framework separates three logically distinct tasks. First, it establishes a Hamiltonian-complexity diagnostic based on the short-time growth of connected correlations. For Gaussian initial states, a Hamiltonian whose Majorana operator structure has bounded polynomial degree cannot generate connected correlations of arbitrarily high order at first order in time. Consequently, the observation of a nonzero initial growth rate for a sufficiently high-order connected correlation provides a one-sided certificate that the Hamiltonian contains interactions of at least the corresponding Majorana degree.
Second, we characterize the infinitesimal directions that take a pure Gaussian state away from the Gaussian manifold. These non-Gaussian directions decompose into even quasiparticle sectors, and connected correlations of increasing order provide nondegenerate coordinates on the corresponding sectors. Complete connected-correlation data therefore give an injective, though generally redundant, real-valued readout of infinitesimal non-Gaussian dynamics.
Third, using the geometric description of pure Gaussian states in terms of spinor varieties, we show that a Hermitian operator whose infinitesimal evolution remains tangent to the Gaussian manifold at every Gaussian state must reduce to a scalar contribution together with a quadratic Majorana Hamiltonian. For systems with at least four fermionic modes, this implies that Gaussian-probe connected-correlation responses can globally distinguish the genuinely nonquadratic part of the Hamiltonian.
As a consequence, finite families of Gaussian probe states are sufficient to separate the nonquadratic Hamiltonian sector for any fixed finite system size. Explicit finite-mode calculations are consistent with corresponding lower bounds on the required information and probe span, while the optimal number of probes in the general case remains conjectural.
The contribution is therefore a tomography and model-falsification framework assembled from established structures in Gaussian-state theory, cumulant methods, and spinor geometry, rather than a claim of a new underlying theory of Gaussian geometry.