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Recoverability Geometry from Algebra Filtrations:A Coarse-Grained Framework for Operator Dynamics

Jeffrey Satinover

Published Jun 17, 2026• 0 citations
Publisher

Abstract

Many diagnostics of quantum dynamics focus on operator growth, scrambling, complexity generation, or information spreading. Examples include out-of-time-order correlators, Krylov complexity, operator entanglement, and related algebraic formulations of scrambling. These diagnostics provide complementary perspectives on how operators evolve under unitary dynamics, but they generally emphasize expansion into progressively larger regions of operator space. In this work we develop a filtration-relative viewpoint. Given a nested sequence of operator algebras\[\mathcal A_0\subset \mathcal A_1\subset\cdots\subset\mathcal A_n=B(\mathcal H),\]the Hilbert--Schmidt structure of operator space induces an orthogonal decomposition into filtration sectors. Hamiltonian evolution generates transport among these sectors. We define transport coefficients that quantify the strength of sector-to-sector coupling and show that they induce a weighted operator-flow network. Recoverability is then described geometrically relative to a chosen coarse-graining. Recoverable regions, boundary transport, persistence, leakage pathways, and bottlenecks arise naturally from the network structure generated by the filtration. The resulting framework does not attempt to replace existing scrambling diagnostics. Rather, it provides a complementary coarse-grained description adapted to physically meaningful operator-algebra structures. A central structural result is a reciprocity relation for transport coefficients generated by Hermitian Hamiltonians,\[d_i T_{ji}=d_j T_{ij},\]where \(d_i\) denotes the dimension of the \(i\)-th filtration sector. Consequently, transport support graphs induced by Hamiltonian evolution are intrinsically undirected, although normalized transport weights may remain asymmetric. This reciprocity theorem clarifies the geometric structure underlying filtration-induced transport. Several finite-dimensional examples are developed explicitly, including diagonal and block-algebra filtrations. We further analyze an associated effective leakage model illustrating how network bottlenecks can generate long persistence times. This model is presented as an illustrative coarse-grained approximation rather than as a consequence of the underlying Hamiltonian framework. The principal contribution of the paper is therefore not a new theory of recoverability, but a filtration-based geometry that organizes persistence, leakage, and recoverability properties relative to nested operator-algebra structures.

DOI
10.5281/zenodo.20737920

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Many diagnostics of quantum dynamics focus on operator growth, scrambling, complexity generation, or information spreading. Examples include out-of-time-order correlators, Krylov complexity, operator entanglement, and related algebraic formulations of scrambling. These diagnostics provide complementary perspectives on how operators evolve under unitary dynamics, but they generally emphasize expansion into progressively larger regions of operator space. In this work we develop a filtration-relative viewpoint. Given a nested sequence of operator algebras[\mathcal A_0\subset \mathcal A_1\subset\cdots\subset\mathcal A_n=B(\mathcal H),]the Hilbert--Schmidt structure of operator space induces an orthogonal decomposition into filtration sectors. Hamiltonian evolution generates transport among these sectors. We define transport coefficients that quantify the strength of sector-to-sector coupling and show that they induce a weighted operator-flow network. Recoverability is then described geometrically relative to a chosen coarse-graining. Recoverable regions, boundary transport, persistence, leakage pathways, and bottlenecks arise naturally from the network structure generated by the filtration. The resulting framework does not attempt to replace existing scrambling diagnostics. Rather, it provides a complementary coarse-grained description adapted to physically meaningful operator-algebra structures. A central structural result is a reciprocity relation for transport coefficients generated by Hermitian Hamiltonians,[d_i T_{ji}=d_j T_{ij},]where (d_i) denotes the dimension of the (i)-th filtration sector. Consequently, transport support graphs induced by Hamiltonian evolution are intrinsically undirected, although normalized transport weights may remain asymmetric. This reciprocity theorem clarifies the geometric structure underlying filtration-induced transport. Several finite-dimensional examples are developed explicitly, including diagonal and block-algebra filtrations. We further analyze an associated effective leakage model illustrating how network bottlenecks can generate long persistence times. This model is presented as an illustrative coarse-grained approximation rather than as a consequence of the underlying Hamiltonian framework. The principal contribution of the paper is therefore not a new theory of recoverability, but a filtration-based geometry that organizes persistence, leakage, and recoverability properties relative to nested operator-algebra structures.
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