Эта статья является препринтом и не была отрецензирована.
О результатах, изложенных в препринтах, не следует сообщать в СМИ как о проверенной информации.
Testable Prediction of Modular Criticality in Quantum Simulators
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.
1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge
2. University Press (2000).
3. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
4. M. M. Wilde, Quantum Information Theory, Cambridge University Press (2013).
5. D. Petz, Quantum Information Theory and Quantum Statistics, Springer (2008).
6. S. Amari and H. Nagaoka, Methods of Information Geometry, AMS and Oxford University Press (2000).
7. F. Nielsen, An Elementary Introduction to Information Geometry, Entropy 22, 1100 (2020).
8. R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, American Mathematical Society (1997).
9. B. Simon, Trace Ideals and Their Applications, American Mathematical Society (2005).
10. M. Takesaki, Theory of Operator Algebras I, Springer (2001).
11. E. H. Lieb and M. B. Ruskai, Proof of the Strong Subadditivity of Quantum-Mechanical Entropy, J. Math. Phys. 14, 1938 (1973).
12. H. Araki, Relative Entropy of States of von Neumann Algebras, Publ. RIMS Kyoto Univ. 11, 809 (1976).
13. E. Witten, Notes on Some Entanglement Properties of Quantum Field Theory, Rev. Mod. Phys. 90, 045003 (2018).
14. R. Haag, Local Quantum Physics, Springer (1996).
15. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Springer (1987).
16. K. G. Wilson, The Renormalization Group and Critical Phenomena, Rev. Mod. Phys. 47, 773 (1975).
17. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford University Press (2002).
18. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-Enhanced Measurements, Science 306, 1330 (2004).
19. M. G. A. Paris, Quantum Estimation for Quantum Technology, Int. J. Quantum Inf. 7, 125 (2009).
20. A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum Entanglement Growth under Random Unitary Dynamics, Phys. Rev. X 7, 031016 (2017).
21. V. Khemani, A. Vishwanath, and D. A. Huse, Signatures of Localization in Many-Body Systems, Phys. Rev. Lett. 119, 075702 (2017).
22. M. Van Raamsdonk, Building up Spacetime with Quantum Entanglement, Gen. Relativ. Gravit. 42, 2323 (2010).
23. T. Jacobson, Thermodynamics of Spacetime: The Einstein Equation of State, Phys. Rev. Lett. 75, 1260 (1995).
24. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, Springer (2011).
25. F. Hiai and D. Petz, Introduction to Matrix Analysis and Applications, Springer (2014).
