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arXiv:2210.14548 · Springer INdAM series

Asymptotic Dynamics of Open Quantum Systems and Modular Theory

Daniele Amato, Paolo Facchi A, Arturo Konderak

Published Jan 1, 2023• 6 citations
arXivHTMLPDF
DOI
10.1007/978-981-99-5884-9_5

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We develop a mathematically rigorous and experimentally testable framework for the classification of physical regimes based on the modular structure of quantum states. The central object of the theory is the modular generator K = −log ρ, which provides a unified language for spectral, geometric, and information-theoretic diagnostics. We introduce a universal instrument panel (MSRO: Modular–Spectral RG Observables) that integrates spectral quantiles, commutator-based response functions, partition-based locality diagnostics, and information-geometric structures. A key requirement of the framework is strong portability: the same diagnostic protocol applies across multiple domains—including open quantum systems, information geometry, entanglement-based models, and RG-like flows—without domain-specific retuning. The main result of the work is the derivation of a universal and experimentally testable scaling law for the modular response signal: ν(λ) ∼ 1 log λ, which characterizes critical regimes in a wide class of quantum systems. We prove that this scaling emerges from spectral asymptotics of the modular operator and establish a strict equivalence between spectral, geometric, and information-theoretic descriptions of criticality: ν ↔ k(q) ↔ I(λ) ↔ K ↔ R ↔ Δ. The framework is supported by functional analytic results (including Fr´echet differentiability of log ρ), operator inequalities (Golden–Thompson, monotonicity of relative entropy), and stability estimates. We further provide a concrete experimental protocol for quantum simulators, including statistical error bounds and Fisher information constraints, making the prediction directly testable. Beyond regime classification, the results suggest a new perspective on criticality as a geometric phenomenon associated with vanishing modular curvature. The approach opens pathways toward large-N limits, quantum field extensions, and the discovery of new classes of non-standard critical behavior.
We report the discovery of a new class of critical phenomena in open quantum systems that is not governed by divergence of correlation length. In contrast to the standard paradigm, where criticality is identified by ξ → ∞, we demonstrate the existence of regimes characterized by: Δ → 0, Γ > 0, ξ = O(1), where Δ is the partition gap and Γ is the switching rate of optimal decompositions. We show that such systems exhibit a universal modular scaling law: ν(λ) ∼ 1 log λ, derived from the spectral structure of the modular generator K = −log ρ. This scaling defines a new diagnostic observable independent of spatial correlations. We formulate a hard, experimentally testable prediction: Ξ(λ) = ν(λ) · log λ → 1, which provides a falsifiable signature of modular criticality accessible in quantum simulators. The results establish a new notion of criticality rooted in spectral redistribution, commutator dynamics, and information geometry, rather than correlation length. This reveals a previously unrecognized universality class applicable to open quantum systems, non-equilibrium dynamics, and information-theoretic regimes.
Metastability in open system dynamics describes the phenomena of initial relaxation to long-lived metastable states before decaying to the asymptotic stable states. It has been found in continuous-time stochastic dynamics of both classical and quantum systems. However, many cases of open quantum system dynamics are intrinsically discrete, and the evolution within each discrete time interval is described by an arbitrary quantum channel, which often cannot be generated by continuous-time master equations. Here we develop a general theory of metastability in discrete-time open quantum dynamics, described by sequential repetitive quantum channels. We apply the general metastability theory to a typical class of quantum channels on a target system, induced by an ancilla qubit with a pure-dephasing coupling to the target system and under Ramsey sequences. Interesting metastable behaviors are predicted and numerically demonstrated by decomposing the average dynamics of sequential quantum channels into stochastic trajectories. We also present examples of applications in quantum state and dynamics engineering of a target quantum system with an ancilla qubit.