Quantum Systems’ Past and Future Become Linked Via New Mathematical Framework

Pengfei Zhu, of the Federal Institute for Materials Research and Testing (BAM), and collaborators investigate the fundamental limits of reconstructing quantum dynamics from imaginary-time evolution using a new spectral-semigroup framework. Evolution can be understood as reweighting spectral components via a single operator, unifying unitary and dissipative processes. A quantifiable relationship between imaginary-time evolution and recoverable information is revealed, showing that irreversibility arises from the geometry and scale of the system’s spectrum. By recasting analytic continuation as a structured filtering process, the research offers a unified perspective on the interplay between quantum dynamics, spectral geometry, and information recovery, with implications for diverse systems including those with continuous and discrete spectra, and non-Hermitian generators.

Spectral geometry governs reconstruction accuracy of quantum dynamics

On May 12, 2026, a quantifiable improvement in reconstructing quantum dynamics was achieved, enabling stable recovery of low-energy features where previous methods failed beyond a scale-dependent spectral threshold. This threshold, defining recoverable modes, governs the inverse reconstruction problem inherent in analytic continuation, the process of calculating real-time behaviour from imaginary-time data. Prior to this work, the limits of this reconstruction remained poorly defined, leading to uncontrolled noise amplification and unreliable results.

The new spectral-semigroup framework recasts analytic continuation not as a problematic inversion, but as a structured filtering process with predictable resolution, linking quantum dynamics to spectral geometry. Imaginary-time evolution acts as an effective low-pass filter, suppressing high-frequency components, as revealed by analysis across systems exhibiting both continuous and discrete spectra, including those with few-level coherence and non-Hermitian generators. Irreversibility isn’t simply a property of the system, but arises from the geometry and scale of its spectrum, linked to both damping and the non-orthogonality of its eigenstates. However, modelling the complex interaction of many-body interactions remains a key hurdle to practical application.

Spectral Geometry Dictates Reconstruction Fidelity in Open Quantum Systems

Recoverable information and its relation to bandwidth-resolved asymmetry between forward propagation and inverse recovery are increasingly the focus of research. In non-Hermitian and open-system settings, irreversibility emerges as a geometry- and scale-dependent feature of the spectrum, linked to both damping and eigenstate non-orthogonality. This finding highlights the key role of spectral characteristics in determining the limits of information recovery.

Analytic continuation is now recast as a structured, scale-dependent filtering process with quantifiable and systematically accessible reconstruction limits, providing a unified perspective on the interaction between dynamics, spectral geometry, and information recovery. Despite its widespread use in quantum physics, underpinning equilibrium formulations, ground-state projection, and a wide range of numerical methods, reconstructing real-time dynamics from imaginary-time data remains a challenging ill-posed problem. Analytic continuation leads to exponential noise amplification, but a precise characterisation of which dynamical information can be stably recovered was still lacking.

Quantum evolution in real time is generated by anti-Hermitian operators, yielding unitary propagation with phase-coherent oscillations. Commonly implemented via Wick rotation t →−iτ, analytic continuation maps the unitary propagator e−iHt/ħ to the contractive semigroup e−Hτ/ħ. This transformation converts oscillatory dynamics into a dissipative spectral filtering process, exponentially suppressing high-energy components. Imaginary-time evolution inherently limits the accessible dynamical information from this perspective.

This work identifies a universal structure underlying analytic continuation, showing that the inverse reconstruction problem is governed by a sharp recoverability bound, which defines a scale-dependent spectral threshold separating stably reconstructable modes from those with rapidly growing sensitivity. Consequently, analytic continuation is established as a transformation with quantitatively predictable resolution, in which information accessibility follows a universal scaling law than system-specific details. To uncover the mechanism behind this behaviour, a unified spectral, semigroup framework for quantum dynamics was developed, formulating analytic continuation at the operator level.

Within this framework, the relation between real-time and imaginary-time evolution is recast as a nonlocal spectral flow generated by a fractional operator. A square-root spectral deformation emerges as a minimal and natural transformation interpolating between unitary and contractive dynamics, defining a continuous boundary between reversible and effectively irreversible regimes. These results demonstrate that analytic continuation is not merely a formal mapping in the complex time plane, but a structured spectral transformation with well-defined and quantitatively accessible reconstruction limits.

This perspective establishes a direct connection between quantum dynamics, inverse problems, and fractional evolution, providing a unified and predictive framework for understanding dynamical reconstruction across a broad class of systems. The mathematical and physical ingredients underlying this work have a long history across quantum statistical mechanics, inverse problems, semigroup theory, and quantum field theory. The contractive evolution operator e−τH and its interpretation as a ground-state projector or spectral filter are standard in quantum Monte Carlo and Euclidean formulations, where high-energy suppression is well understood.

Similarly, the instability of analytic continuation is closely related to the ill-posedness of the inverse Laplace transform, where exponential damping leads to severe noise amplification. Rooted in the Hille, Yosida theory of dissipative evolution, the semigroup formulation employed here provides the standard bridge between real- and imaginary-time dynamics in Euclidean quantum field theory and the Osterwalder, Schrader framework. Fractional powers of generators and their associated semigroups have also been extensively studied in subordination theory and nonlocal dynamics.

In parallel, analytic continuation in complex analysis is governed by Hardy space methods and dispersion relations such as the Kramers, Kronig relations, which emphasize analyticity and boundary-value reconstruction. The contribution of the present work lies not in revisiting these well-established results individually, but in unifying them within a common operator-theoretic framework and introducing a quantitative notion of recoverability that applies across these domains. Within this perspective, analytic continuation is recast as a structured spectral compression process characterised by a scale-dependent bandwidth, rather than solely as an ill-posed inversion problem.

The resulting recoverability bound provides a precise criterion for which dynamical information can be stably reconstructed, establishing a quantitative bridge between spectral structure and reconstruction fidelity. This complements existing analyticity-based formulations by introducing a spectral-semigroup perspective that makes the limits of reconstruction explicit, systematic, and predictive. Throughout this work, the generator G is assumed to be a densely defined, closed linear operator on a Hilbert space H that generates a strongly continuous (C0) semigroup {K(τ)}τ≥0 in the sense of the Hille, Yosida theorem. Therefore, the evolution operator K(τ) = e−τG is defined via semigroup theory rather than spectral decomposition in full generality. For self-adjoint or, more generally, normal operators, G admits a spectral representation in terms of a projection-valued measure.

Unifying semigroup theory, fractional dynamics and analytic continuation to define quantum

Addressing a longstanding challenge in fields ranging from materials science to fundamental physics, this work establishes a predictable boundary for recoverable information in quantum systems. While this work elegantly unifies seemingly disparate approaches, semigroup theory, fractional dynamics, and analytic continuation, it presently focuses on theoretical foundations, leaving open the question of computational cost for complex simulations. Professor [Name] of [Institution] acknowledges that many-body interactions, prevalent in realistic materials, introduce significant hurdles not yet addressed by this framework.

This work offers a valuable, unified language for understanding how information is lost and recovered in quantum systems, despite the computational difficulties of applying this theoretical framework to complex, real-world materials. By linking diverse mathematical tools, scientists gain a more complete picture of quantum behaviour. This advancement promises to refine modelling across multiple scientific disciplines, even if immediate practical application requires further development of efficient algorithms.

A new approach has been devised, unifying how quantum systems evolve over time and recover information. This research, utilising concepts like semigroup theory and analytic continuation, a technique to extend functions beyond their initial definition, reveals a fundamental link between seemingly opposing processes. Establishing a new understanding of how accurately quantum behaviour can be predicted from imaginary-time data, this research moves beyond simply identifying recoverable information to quantifying its limits. By uniting concepts from semigroup theory, the study of how operators evolve over time, with analytic continuation, scientists have revealed a direct link between a system’s spectral structure and the fidelity of its reconstruction.

This research demonstrates a unified framework connecting real-time and imaginary-time quantum dynamics through analytic continuation and semigroup theory. It establishes a quantitative relationship between the evolution of a quantum system and the limits of recoverable information, revealing that spectral structure governs reconstruction fidelity. The findings indicate that irreversibility in open quantum systems arises from the geometry and scale of the system’s spectrum, rather than being an inherent property. Researchers acknowledge that applying this theoretical framework to complex materials requires further development of efficient computational algorithms.