This paper establishes a rigorous, unified geometric framework for quantum state spaces by constructing smooth, regular embeddings into higher-order dual number algebras $\mathcal{Q}_N \equiv \mathbb{R}[\varepsilon]/(\varepsilon^{N^2-1})$, wherein every quantum state is faithfully represented as a non-reduced scheme-theoretic point. We show that under this unified family of truncated rings, the non-linear matrix commutators governing the Liouville-von Neumann dynamics map globally onto flat, linear, and rigid algebraic flows, establishing nilpotent dual algebras as a pristine geometric landscape for higher-dimensional quantum kinematics. As $N \to \infty$, this family converges to a Cauchy-complete power series ring $\mathcal{Q}_\infty$, where non-Archimedean completion linearizes the phase space, aligning the Fubini-Study geometry with the classical Fisher-Rao manifold.
We introduce the Quantum Information Space at Infinity (Quinfinity) $\mathcal{Q}_\infty$ as the inverse limit of the symbolic Quantum $N$-Spaces $\mathcal{Q}_N$, identified with the complete local real algebra of formal power series under the adic topology. The pure state pro-variety is formalized as the inverse limit...
On the admissible cone $\mathcal C_\Omega=\{\Sigma\in\mathrm{Sym}^+(2n):\Sigma+i\Omega\ge0\}$ of Gaussian covariance matrices, the classical Bures--Wasserstein transport metric, the Fisher--Rao information metric, and the quantum Bures metric are individually well understood, but their mutual relationship is obscured b...
We study the quantum geometry of doubly degenerate energy levels in a four-dimensional parameter space. We find that the scalar quantum metric $g$ and the Berry curvature $F$ obey $(\operatorname{tr} g)^2/16\geq\sqrt{\det g}\geq |2\Tr(F\wedge F)-\Tr F\wedge \Tr F|/24$. The first inequality characterizes the anisotropy...
Junwen Zhao, Zhi-Ming Pan, Kang Yang et al.· 2 citations· ⚡1
We introduce \emph{latent geometry} in BFSS/BMN models. Before an emergent geometry evaporates at the geometric transition, large-$d$ Gaussianization encodes its information in the exceptional $N=2$ sector selected by the baby fuzzy sphere, a single active spin-$1/2$ Pauli block. Dimensional reduction, finite-$N$ Gauss...
We consider a collection of pairwise commuting quantum observables in the setting of Berezin--Toeplitz quantization of a closed K\"{a}hler manifold and assume that the Arnold--Liouville theorem applies to their principal symbols. We use joint eigensections of these observables to define isometric embeddings of the quan...
We deform the Carlen--Maas--Wirth framework for noncommutative dynamical optimal transport by an Arnold--Nielsen type complexity operator. A positive state-independent operator $G$ compatible with the Hilbert bimodule structure of a noncommutative differential calculus $\partial\colon\M\to\Hcal$ can be absorbed into th...
Alberto Acevedo, Antonio Falcó· 0 citations
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