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Mathematics
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Mathematics Methods in Quantum Physics and Its Applications
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Mathematics Methods in Quantum Physics and Its Applications

A Special Issue of Mathematics (ISSN 2227-7390) belonging to the section "E4: Mathematical Physics".

Deadline for manuscript submissions: 31 October 2026 | Viewed by 5781

Editor

Dr. Lebin Ho

E-Mail Website
Guest Editor
Institute for Interdisciplinary Science (FRIS), Tohoku University, Sendai 980-0845, Japan
Interests: quantum mechanics; quantum physics; quantum measurement; weak values and weak measurement; quantum state tomography; quantum metrology and sensing; quantum estimation theory; quantum computing; quantum algorithms; variational quantum algorithms; quantum machine learning; quantum neural networks

Special Issue Information

Dear Colleagues,

The year 2025 has been designated the Quantum Year, marking 100 years of significant advances in quantum science and technology. Worldwide celebrations highlight the profound impact of quantum theory on modern physics and emerging technologies. However, this brilliant progress would not have been possible without the essential foundation provided by mathematics. Key tools such as linear algebra, functional analysis, group theory, operator algebras, differential geometry, and complex analysis have enabled breakthroughs and continue to drive innovation. Overlooking the vital role of mathematics in this quantum revolution is a serious oversight. To acknowledge this, we are launching this Special Issue to celebrate and further explore the deep connection between mathematics and quantum physics.

This Special Issue invites high-quality submissions on developing, analyzing, and applying mathematical methods in quantum physics, including quantum computing and information theory. We aim to highlight theoretical insights and practical tools that enhance the understanding and control of quantum systems. We welcome original research, review papers, and methodological contributions from all areas of mathematical physics and quantum theory.

The topics include, but are not limited to:

  1. Mathematical foundations of quantum mechanics: Hilbert space theory, operator algebras, spectral theory, and formal axiomatic approaches;
  2. Quantum dynamics and open systems: Unitary and non-unitary evolution, Lindblad equations, decoherence, and quantum noise modeling;
  3. Quantum estimation theory and metrology: Quantum Fisher information, Cramér–Rao bounds, optimal measurements, and multiparameter estimation;
  4. Quantum control and optimization: Time-optimal control, variational control methods, control landscapes, and feedback systems;
  5. Quantum optics: Field quantization and light–matter interaction;
  6. Quantum thermodynamics: Quantum work and heat, fluctuation theorems, entropy production, and resource–theoretic approaches, quantum battery, and ergotropy;
  7. Many-body quantum systems: Integrable models, tensor network methods, entanglement structure, and emergent phenomena;
  8. Non-Hermitian and PT-symmetric quantum systems: Complex eigenvalue problems, exceptional points, and pseudo-Hermitian formulations;
  9. Topological and geometrical methods: Berry phase, fiber bundles, topological order, and geometrical quantization;
  10. Group theory and representation theory: Lie groups and algebras, symmetry classifications, and their applications in quantum theory;
  11. Numerical and computational methods: Quantum simulation algorithms, matrix product states, spectral solvers, and numerical optimization;
  12. Stochastic and probabilistic approaches: Quantum trajectories, path integral formulations, and noise-driven quantum systems;
  13. Variational quantum algorithms and hybrid methods: VQE, QAOA, and variational principles applied to quantum simulation and optimization;
  14. Quantum machine learning: Quantum neural networks, quantum kernel methods, generative models, and data-driven approaches to quantum systems;
  15. Quantum information and computation: Entanglement theory, quantum circuits, quantum algorithms, error correction, and complexity theory.

Dr. Lebin Ho
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mathematical methods in quantum physics
  • quantum information theory
  • quantum computing
  • quantum control
  • quantum estimation and metrology
  • open quantum systems
  • quantum thermodynamics
  • quantum field theory
  • many-body systems
  • variational quantum algorithms
  • quantum machine learning
  • operator theory
  • functional analysis
  • group theory
  • topological methods
  • numerical methods

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (5 papers)

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Research

22 pages, 4191 KB  
Article
Regression-Based Machine Learning Prediction of Electronic and Nonlinear Optical Properties in Coupled GaN/AlN Quantum Dots
by Tesnim Brahim, Adel Bouazra, Beriham Ibrahim Basha and Fatma Aouaini
Mathematics 2026, 14(13), 2298; https://doi.org/10.3390/math14132298 - 28 Jun 2026
Cited by 1 | Viewed by 344
Abstract
This study investigates the electronic and nonlinear optical properties of coupled GaN/AlN quantum dots using a numerical approach based on coordinate transformation combined with the finite difference method (FDM). The Schrödinger equation is solved to determine the electronic energy levels and wave functions [...] Read more.
This study investigates the electronic and nonlinear optical properties of coupled GaN/AlN quantum dots using a numerical approach based on coordinate transformation combined with the finite difference method (FDM). The Schrödinger equation is solved to determine the electronic energy levels and wave functions of the system, which are subsequently used to evaluate the nonlinear optical rectification (NOR) response. Since numerical simulations become computationally expensive for large quantum dot systems, several regression-based models, including Polynomial Regression, Ridge Regression, LASSO, and Elastic Net, are trained on high-fidelity numerical data. These models learn the relationship between structural parameters and the resulting electronic and optical properties, enabling fast and reliable predictions for larger quantum dot configurations. The predictive performance of the ML models is assessed by comparing their results with the numerical simulations, showing excellent agreement while significantly reducing computational effort. The proposed hybrid physics–machine learning framework therefore provides an efficient and reliable approach for predicting the electronic and nonlinear optical behavior of coupled GaN/AlN quantum dots. Full article
(This article belongs to the Special Issue Mathematics Methods in Quantum Physics and Its Applications)
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14 pages, 294 KB  
Article
Foundations of Quantum Mechanics: Generalizations of the Mathematical Axiomatic Derivation of the Schrödinger Equation
by Olavo L. Silva Filho
Mathematics 2026, 14(11), 1961; https://doi.org/10.3390/math14111961 - 3 Jun 2026
Viewed by 326
Abstract
An axiomatic approach to quantum mechanics that has, as a theorem, the Schrödinger equation may be of enormous value to cope with interpretation issues, since all the interpretation constructs must be present in the axioms, or directly derived by them from their mathematical [...] Read more.
An axiomatic approach to quantum mechanics that has, as a theorem, the Schrödinger equation may be of enormous value to cope with interpretation issues, since all the interpretation constructs must be present in the axioms, or directly derived by them from their mathematical unfolding. Thus, it is critical to show that this axiomatic derivation is reliable beyond any possible doubt. To show this, it is possible to make generalizations and extensions of the axioms to derive the Schrödinger equation in the underlying generalized or extended formats. In previous papers, we have shown that the axiomatic approach we propose can be used to derive the Schrödinger equation as a direct axiom. Since then, we have also shown that it was possible to generalize that derivation to coordinate systems other than the Cartesian, as well as its relativistic extensions that lead to the relativistic wave equations. An extension to dissipative systems was also performed, allowing us to mathematically derive the Caldirola–Kanai equation from first principles. All these derivations were performed using pure states and in the absence of the electromagnetic field. This means that we can further generalize the approach to embrace these two possibilities. Being an axiomatic approach, we show that we need only to slightly modify the axioms to derive the Schrödinger equation for these two contexts. Despite being quite direct, the algebraic complexity of these derivations should give the reader the desired confidence in the proposed axioms. Full article
(This article belongs to the Special Issue Mathematics Methods in Quantum Physics and Its Applications)
20 pages, 1730 KB  
Article
Zeno and Anti-Zeno Effects in Dark-State Dynamics Under Thermal Dephasing: A Numerical Study
by Ran Chen, Jiangchuan You, Alexey Vladimirovich Kulagin, Hui-hui Miao and Yuri Igorevich Ozhigov
Mathematics 2026, 14(11), 1836; https://doi.org/10.3390/math14111836 - 25 May 2026
Viewed by 597
Abstract
The quantum Zeno and anti-Zeno effects describe how frequent measurements can either suppress or accelerate quantum dynamics. While extensively studied in various platforms, their manifestation in dark-state dynamics remains largely unexplored. Here we investigate the stability of dark states in a cavity quantum [...] Read more.
The quantum Zeno and anti-Zeno effects describe how frequent measurements can either suppress or accelerate quantum dynamics. While extensively studied in various platforms, their manifestation in dark-state dynamics remains largely unexplored. Here we investigate the stability of dark states in a cavity quantum electrodynamics (QED) system consisting of two atoms coupled to a single-mode cavity, subject to thermal dephasing that models continuous quantum non-demolition monitoring. Using the Tavis–Cummings model within a Lindblad master equation framework, we perform numerical simulations to investigate how measurement-induced dephasing affects dark-state retention and stabilization time. Through systematic numerical scans, we identify distinct parameter regimes corresponding to Zeno and anti-Zeno behavior: at low dephasing intensities, increasing the measurement strength accelerates the loss of dark-state coherence (anti-Zeno regime), while at higher intensities, it slows down the dynamics and partially recovers dark-state weight (Zeno regime). The transition between these regimes is controlled by the dephasing rates, the cavity photon exchange, and the asymmetry in atom–field couplings. We show that even under strong dephasing, a finite dark-state component persists, demonstrating remarkable robustness. Our results provide insights into the interplay between measurement back-action and decoherence in open quantum systems, with implications for quantum control and information storage. Full article
(This article belongs to the Special Issue Mathematics Methods in Quantum Physics and Its Applications)
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32 pages, 373 KB  
Article
Semiotics and Epistemology of Physics: Reflections on Language and the Interpretation of Quantum Mechanics
by Olavo L. Silva Filho, Samuel J. Simon and Marcello Ferreira
Mathematics 2026, 14(3), 550; https://doi.org/10.3390/math14030550 - 3 Feb 2026
Cited by 1 | Viewed by 1701
Abstract
When a physical theory is in its early stages of development, it presents many concepts and constructs that may not be necessary for its interpretation, or may simply be equivocal. As the theory is developed by the physics community, it hopefully passes through [...] Read more.
When a physical theory is in its early stages of development, it presents many concepts and constructs that may not be necessary for its interpretation, or may simply be equivocal. As the theory is developed by the physics community, it hopefully passes through a depuration process that washes away many of these constructs and introduces others not initially devised. This process can be viewed, and modeled, as a semiotic process by which the suggested interpretations of the theory, initially dispersed in the space of concepts, taper off in a way that leaves only a small number of possibilities, ideally only one. However, to qualify this process and impose semiotic and epistemological constraints on the depuration process, it seems natural to consider a physical theory as an excerpt of a language and its suggested interpretations as texts, endowed with syntactics and semantics. In this paper we present this framing of general physical theories and apply the resulting semiotic and epistemological constraints we uphold to the special case of quantum mechanics, which shows particular resistance to interpretation tapering. We then show that the findings of this paper are especially important for allowing one to form a hierarchy of interpretations of the same formal structure of a physical theory, even in the case of an experimental underdetermination of these interpretations, which is precisely the case for quantum mechanics. This result is particularly important for more modern physical theories, which are becoming increasingly more abstract and difficult to interpret. Full article
(This article belongs to the Special Issue Mathematics Methods in Quantum Physics and Its Applications)
10 pages, 5326 KB  
Article
Probing Chirality of the Quantum Hall Effect via the Landauer–Büttiker Formalism with Two Current Sources
by Kyung Ho Kim
Mathematics 2025, 13(18), 2981; https://doi.org/10.3390/math13182981 - 15 Sep 2025
Viewed by 1886
Abstract
The quantum Hall effect is a paradigmatic example of topological order, characterized by precisely quantized Hall resistance and dissipationless edge transport. These edge states are chiral, propagating unidirectionally along the boundary, and their directionality is determined by the external magnetic field. While chirality [...] Read more.
The quantum Hall effect is a paradigmatic example of topological order, characterized by precisely quantized Hall resistance and dissipationless edge transport. These edge states are chiral, propagating unidirectionally along the boundary, and their directionality is determined by the external magnetic field. While chirality is a central feature of the quantum Hall effect, directly probing it remains experimentally nontrivial. In this study, we introduce a simple and effective method to probe the chirality of edge transport using two independently controlled current sources in a Hall bar geometry. The system under investigation is monolayer epitaxial graphene grown on a silicon carbide substrate, exhibiting robust quantum Hall states. By varying the configurations of the two current sources, we measure terminal voltages and analyze the transport characteristics. Our results demonstrate that the observed behavior can be understood as a linear superposition of chiral contributions to the edge transport. This superposition enables tunable combinations of longitudinal and Hall resistances and enables additive or canceling behavior of Hall voltages depending on current source configuration. The Landauer–Büttiker formalism provides a quantitative framework to describe these observations, capturing the interplay between edge state chirality and the measurement configuration. This research offers a simple yet effective experimental and analytical approach for probing chiral edge currents and highlights the linear superposition principle in the quantum Hall effect. Full article
(This article belongs to the Special Issue Mathematics Methods in Quantum Physics and Its Applications)
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