Entropy

33 pages, 3062 KB

Open AccessArticle

by Karl Friston, Thomas Parr, Conor Heins, Lancelot Da Costa, Tommaso Salvatori, Alexander Tschantz, Magnus Koudahl, Toon Van de Maele, Christopher Buckley and Tim Verbelen

Entropy 2025, 27(9), 992; https://doi.org/10.3390/e27090992 - 22 Sep 2025

Viewed by 282

Abstract

This technical note applies active inference to the problem of learning goal-directed behaviour from scratch, namely, de novo learning. By de novo learning, we mean discovering, directly from observations, the structure and parameters of a discrete generative model for sequential policy optimisation. Concretely, [...] Read more.

This technical note applies active inference to the problem of learning goal-directed behaviour from scratch, namely, de novo learning. By de novo learning, we mean discovering, directly from observations, the structure and parameters of a discrete generative model for sequential policy optimisation. Concretely, our procedure grows and then reduces a model until it discovers a pullback attractor over (generalised) states; this attracting set supplies paths of least action among goal states while avoiding costly states. The implicit efficiency rests upon reframing the learning problem through the lens of the free energy principle, under which it is sufficient to learn a generative model whose dynamics feature such an attracting set. For context, we briefly relate this perspective to value-based formulations (e.g., Bellman optimality) and then apply the active inference formulation to a small arcade game to illustrate de novo structure learning and ensuing agency. Full article

(This article belongs to the Special Issue Active Inference in Cognitive Neuroscience)

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26 pages, 1333 KB

Open AccessArticle

Category Name Expansion and an Enhanced Multimodal Fusion Framework for Few-Shot Learning

by Tianlei Gao, Lei Lyu, Xiaoyun Xie, Nuo Wei, Yushui Geng and Minglei Shu

Entropy 2025, 27(9), 991; https://doi.org/10.3390/e27090991 - 22 Sep 2025

Viewed by 202

Abstract

With the advancement of image processing techniques, few-shot learning (FSL) has gradually become a key approach to addressing the problem of data scarcity. However, existing FSL methods often rely on unimodal information under limited sample conditions, making it difficult to capture fine-grained differences [...] Read more.

With the advancement of image processing techniques, few-shot learning (FSL) has gradually become a key approach to addressing the problem of data scarcity. However, existing FSL methods often rely on unimodal information under limited sample conditions, making it difficult to capture fine-grained differences between categories. To address this issue, we propose a multimodal few-shot learning method based on category name expansion and image feature enhancement. By integrating the expanded category text with image features, the proposed method enriches the semantic representation of categories and enhances the model’s sensitivity to detailed features. To further improve the quality of cross-modal information transfer, we introduce a cross-modal residual connection strategy that aligns features across layers through progressive fusion. This approach enables the fused representations to maximize mutual information while reducing redundancy, effectively alleviating the information bottleneck caused by uneven entropy distribution between modalities and enhancing the model’s generalization ability. Experimental results demonstrate that our method achieves superior performance on both natural image datasets (CIFAR-FS and FC100) and a medical image dataset. Full article

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11 pages, 552 KB

Open AccessArticle

Continuous-Variable Quantum Key Distribution Based on N-APSK Modulation over Seawater Channel

by Lei Mao, Zhangtao Liang, Zhiyue Zuo, Hang Zhang and Yijun Wang

Entropy 2025, 27(9), 990; https://doi.org/10.3390/e27090990 - 22 Sep 2025

Viewed by 136

Abstract

A continuous-variable quantum key distribution (CVQKD) can be realized over the seawater channel, but the transmission of quantum signals in seawater media exhibits significant attenuation effects. Therefore, we propose an N-symbol amplitude and phase shift keying (N-APSK) modulation scheme to [...] Read more.

A continuous-variable quantum key distribution (CVQKD) can be realized over the seawater channel, but the transmission of quantum signals in seawater media exhibits significant attenuation effects. Therefore, we propose an N-symbol amplitude and phase shift keying (N-APSK) modulation scheme to enhance the transmission performance of the CVQKD over the seawater channel. Specifically, an optimal N-APSK modulation scheme is designed based on the principle of maximizing the minimum Euclidean distance (MED). The simulation results show that the CVQKD protocol based on N-APSK modulation achieves a longer transmission distance over the seawater channel compared to the Gaussian modulation protocol. Additionally, increasing the value of N simultaneously expands the number of rings in the constellation diagram, further enhancing the communication distance. This study transfers modulation methods from the field of classical communications to the field of quantum communications, achieving a substantial improvement in communication distance and thereby promoting the integration of quantum communications and classical communications. Full article

(This article belongs to the Special Issue Recent Advances in Continuous-Variable Quantum Key Distribution)

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22 pages, 1426 KB

Open AccessArticle

Dataset-Learning Duality and Emergent Criticality

by Ekaterina Kukleva and Vitaly Vanchurin

Entropy 2025, 27(9), 989; https://doi.org/10.3390/e27090989 - 22 Sep 2025

Viewed by 157

Abstract

In artificial neural networks, the activation dynamics of non-trainable variables are strongly coupled to the learning dynamics of trainable variables. During the activation pass, the boundary neurons (e.g., input neurons) are mapped to the bulk neurons (e.g., hidden neurons), and during the learning [...] Read more.

In artificial neural networks, the activation dynamics of non-trainable variables are strongly coupled to the learning dynamics of trainable variables. During the activation pass, the boundary neurons (e.g., input neurons) are mapped to the bulk neurons (e.g., hidden neurons), and during the learning pass, both bulk and boundary neurons are mapped to changes in trainable variables (e.g., weights and biases). For example, in feedforward neural networks, forward propagation is the activation pass and backward propagation is the learning pass. We show that a composition of the two maps establishes a duality map between a subspace of non-trainable boundary variables (e.g., dataset) and a tangent subspace of trainable variables (i.e., learning). In general, the dataset-learning duality is a complex nonlinear map between high-dimensional spaces. We use duality to study the emergence of criticality, or the power-law distribution of fluctuations of the trainable variables, using a toy and large models at learning equilibrium. In particular, we show that criticality can emerge in the learning system even from the dataset in a non-critical state, and that the power-law distribution can be modified by changing either the activation function or the loss function. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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15 pages, 342 KB

Open AccessArticle

On the Application of a Hybrid Incomplete Exponential Sum to Aperiodic Hamming Correlation of Some Frequency-Hopping Sequences

by Peihua Li and Hongyu Han

Entropy 2025, 27(9), 988; https://doi.org/10.3390/e27090988 - 21 Sep 2025

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Abstract

Frequency-hopping sequences are essential in frequency-hopping spread spectrum communication systems due to their strong anti-interference capabilities, low probability of interception, and high confidentiality. Existing research has predominantly focused on the periodic Hamming correlation properties of sequences, whereas the aperiodic Hamming correlation performance more [...] Read more.

Frequency-hopping sequences are essential in frequency-hopping spread spectrum communication systems due to their strong anti-interference capabilities, low probability of interception, and high confidentiality. Existing research has predominantly focused on the periodic Hamming correlation properties of sequences, whereas the aperiodic Hamming correlation performance more accurately reflects the actual system performance. Owing to the complexity of its application scenarios and considerable research challenges, results in this area remain scarce. In this paper, we utilize exponential sums over finite fields to derive an upper bound on a hybrid incomplete exponential sum. Then, based on this upper bound, we derive bounds on the aperiodic Hamming correlation of some frequency-hopping sequence sets constructed by trace functions. Finally, by analyzing the maximum estimation error between the average and actual frequency collision numbers of such sequence sets, the validity of the derived bound is demonstrated. Full article

(This article belongs to the Special Issue Coding Theory and Its Applications)

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30 pages, 3270 KB

Open AccessArticle

Tree–Hillclimb Search: An Efficient and Interpretable Threat Assessment Method for Uncertain Battlefield Environments

by Zuoxin Zeng, Jinye Peng and Qi Feng

Entropy 2025, 27(9), 987; https://doi.org/10.3390/e27090987 - 21 Sep 2025

Viewed by 168

Abstract

In uncertain battlefield environments, rapid and accurate detection, identification of hostile targets, and assessment of threat levels are crucial for supporting effective decision-making. Despite offering the advantage of structural transparency, traditional analytical methods rely on expert knowledge to construct models and often fail [...] Read more.

In uncertain battlefield environments, rapid and accurate detection, identification of hostile targets, and assessment of threat levels are crucial for supporting effective decision-making. Despite offering the advantage of structural transparency, traditional analytical methods rely on expert knowledge to construct models and often fail to comprehensively capture the non-linear causal relationships among complex threat factors. In contrast, data-driven methods excel at uncovering patterns in data but suffer from limited interpretability due to their black-box nature. Owing to probabilistic graphical modeling capabilities, Bayesian networks possess unique advantages in threat assessment. However, existing models are either constrained by the limitation of expert experience or suffer from excessively high complexity due to structure learning algorithms, making it difficult to meet the stringent real-time requirements of uncertain battlefield environments. To address these issues, this paper proposes a new method, the Tree–Hillclimb Search method—an efficient and interpretable threat assessment method specifically designed for uncertain battlefield environments. The core of the method is a structure learning algorithm constrained by expert knowledge—the initial network structure constructed from expert knowledge serves as a constraint, enabling the discovery of hidden causal dependencies among variables through structure learning. The model is then refined under these expert knowledge constraints and can effectively balance accuracy and complexity. Sensitivity analysis further validates the consistency between the model structure and the influence degree of threat factors, providing a theoretical basis for formulating hierarchical threat assessment strategies under resource-constrained conditions, which can effectively optimize sensor resource allocation. The Tree–Hillclimb Search method features (1) enhanced interpretability; (2) high predictive accuracy; (3) high efficiency and real-time performance; (4) actual impact on battlefield decision-making; and (5) good generality and broad applicability. Full article

(This article belongs to the Special Issue Bayesian Networks and Causal Discovery)

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16 pages, 2013 KB

Open AccessArticle

Cross-Subject EEG Emotion Recognition Using SSA-EMS Algorithm for Feature Extraction

by Yuan Lu and Jingying Chen

Entropy 2025, 27(9), 986; https://doi.org/10.3390/e27090986 - 21 Sep 2025

Viewed by 236

Abstract

This study proposes a novel SSA-EMS framework that integrates Singular Spectrum Analysis (SSA) with Effect-Matched Spatial Filtering (EMS), combining the noise-reduction capability of SSA with the dynamic feature extraction advantages of EMS to optimize cross-subject EEG-based emotion feature extraction. Experiments were conducted using [...] Read more.

This study proposes a novel SSA-EMS framework that integrates Singular Spectrum Analysis (SSA) with Effect-Matched Spatial Filtering (EMS), combining the noise-reduction capability of SSA with the dynamic feature extraction advantages of EMS to optimize cross-subject EEG-based emotion feature extraction. Experiments were conducted using the SEED dataset under two evaluation paradigms: “cross-subject sample combination” and “subject-independent” assessment. Random Forest (RF) and SVM classifiers were employed to perform pairwise classification of three emotional states—positive, neutral, and negative. Results demonstrate that the SSA-EMS framework achieves RF classification accuracies exceeding 98% across the full frequency band, significantly outperforming single frequency bands. Notably, in the subject-independent evaluation, model accuracy remains above 96%, confirming the algorithm’s strong cross-subject generalization capability. Experimental results validate that the SSA-EMS framework effectively captures dynamic neural differences associated with emotions. Nevertheless, limitations in binary classification and the potential for multimodal extension remain important directions for future research. Full article

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15 pages, 260 KB

Open AccessArticle

Entropy Methods on Finding Optimal Linear Combinations with an Application to Biomarkers

by Mehmet Sinan İyisoy and Pınar Özdemir

Entropy 2025, 27(9), 985; https://doi.org/10.3390/e27090985 - 21 Sep 2025

Viewed by 202

Abstract

Identifying an optimal linear combination of continuous variables is a key objective in various fields of research, such as medicine. This manuscript explores the use of information-theoretical approaches used to establish these linear combinations. Coefficients obtained from logistic regression can be used to [...] Read more.

Identifying an optimal linear combination of continuous variables is a key objective in various fields of research, such as medicine. This manuscript explores the use of information-theoretical approaches used to establish these linear combinations. Coefficients obtained from logistic regression can be used to construct such a linear combination, and this approach has been commonly adopted in the literature for comparison purposes. The main contribution of this work is to propose novel ways of determining these linear combination coefficients by optimizing information-theoretical objective functions. Biomarkers are usually continuous measurements utilized to diagnose if a patient has the underlying disease. Certain disease contexts may lack high diagnostic power biomarkers, making their optimal combination a critical area of interest. We apply the above-mentioned novel methods to the problem of a combination of biomarkers. We assess the performance of our proposed methods against combinations derived from logistic regression coefficients, by comparing area under the ROC curve (AUC) values and other metrics in a broad simulation and a real life data application. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

21 pages, 4638 KB

Open AccessArticle

Symbolic Analysis of the Quality of Texts Translated into a Language Preserving Vowel Harmony

by Kazuya Hayata

Entropy 2025, 27(9), 984; https://doi.org/10.3390/e27090984 - 20 Sep 2025

Viewed by 333

Abstract

To date, the ordinal pattern-based method has been applied to problems in natural and social sciences. We report, for the first time to our knowledge, an attempt to apply this methodology to a topic in the humanities. Specifically, in an effort to investigate [...] Read more.

To date, the ordinal pattern-based method has been applied to problems in natural and social sciences. We report, for the first time to our knowledge, an attempt to apply this methodology to a topic in the humanities. Specifically, in an effort to investigate the applicability of the methodology in analyzing the quality of texts that are translated into a language preserving the so-called vowel harmony, computed results are presented for the metrics of divergence between the back-translated and the original texts. As a specific language we focus on Japanese, and as metrics the Hellinger distance as well as the chi-square statistic are employed. Here, the former is a typical information-theoretical measure that can be quantified in natural unit, nat for short, while the latter is useful for performing a non-parametric testing of a null hypothesis with a significance level. The methods are applied to three cases: a Japanese novel along with a translated version available, the Preamble to the Constitution of Japan, and seventeen translations of an opening paragraph of a famous American detective story, which include thirteen human and four machine translations using DeepL and Google Translate. Numerical results aptly show unexpectedly high scores of the machine translations, but it still might be too soon to speculate on their unconditional potentialities. Both our attempt and results are not only novel but are also expected to make a contribution toward an interdisciplinary study between physics and linguistics. Full article

(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)

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23 pages, 878 KB

Open AccessReview

Review of Recent (2015–2024) Popular Entropy Definitions Applied to Physiological Signals

by Dimitrios Platakis and George Manis

Entropy 2025, 27(9), 983; https://doi.org/10.3390/e27090983 - 20 Sep 2025

Viewed by 144

Abstract

Entropy estimation is widely used in time series analysis, particularly in the field of Biomedical Engineering. It plays a key role in analyzing a wide range of physiological signals and serves as a measure of signal complexity, which reflects the complexity of the [...] Read more.

Entropy estimation is widely used in time series analysis, particularly in the field of Biomedical Engineering. It plays a key role in analyzing a wide range of physiological signals and serves as a measure of signal complexity, which reflects the complexity of the underlying system. The widespread adoption of entropy in research has led to numerous entropy definitions, with Approximate Entropy and Sample Entropy being among the most widely used. Over the past decade, the field has remained highly active, with a significant number of new entropy definitions being proposed, some inspired by Approximate and Sample Entropy, some by Permutation entropy, while others followed their own course of thought. In this paper, we review and compare the most prominent entropy definitions that have appeared in the last decade (2015–2024). We performed the search on 20 December 2024. We adopt the PRISMA methodology for this purpose, a widely accepted standard for conducting systematic literature reviews. With the included articles, we present statistical results on the number of citations for each method and the application domains in which they have been used. We also conducted a thorough review of the selected articles, documenting for each paper which definition has been employed and on which physiological signal it has been applied. Full article

(This article belongs to the Special Issue Entropy in Biomedical Engineering, 3rd Edition)

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23 pages, 1016 KB

Open AccessArticle

Tsallis Entropy in Consecutive k-out-of-n Good Systems: Bounds, Characterization, and Testing for Exponentiality

by Anfal A. Alqefari, Ghadah Alomani and Mohamed Kayid

Entropy 2025, 27(9), 982; https://doi.org/10.3390/e27090982 - 20 Sep 2025

Viewed by 137

Abstract

This study explores the application of Tsallis entropy in evaluating uncertainty within the framework of consecutive k-out-of-n good systems, which are widely utilized in various reliability and engineering contexts. We derive new analytical expressions and meaningful bounds for the Tsallis entropy [...] Read more.

This study explores the application of Tsallis entropy in evaluating uncertainty within the framework of consecutive k-out-of-n good systems, which are widely utilized in various reliability and engineering contexts. We derive new analytical expressions and meaningful bounds for the Tsallis entropy under various lifetime distributions, offering fresh insight into the structural behavior of system-level uncertainty. The approach establishes theoretical connections with classical entropy measures, such as Shannon and Rényi entropies, and provides a foundation for comparing systems under different stochastic orders. A nonparametric estimator is proposed to estimate the Tsallis entropy in this setting, and its performance is evaluated through Monte Carlo simulations. In addition, we develop a new entropy-based test for exponentiality, building on the distinctive properties of system lifetimes. So, Tsallis entropy serves as a flexible tool in both reliability characterization and statistical inference. Full article

(This article belongs to the Special Issue Entropy, Statistical Evidence, and Scientific Inference: Evidence Functions in Theory and Applications, 2nd Edition)

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31 pages, 9207 KB

Open AccessArticle

A Model Framework for Ion Channels with Selectivity Filters Based on Non-Equilibrium Thermodynamics

by Christine Keller, Manuel Landstorfer, Jürgen Fuhrmann and Barbara Wagner

Entropy 2025, 27(9), 981; https://doi.org/10.3390/e27090981 - 20 Sep 2025

Viewed by 176

Abstract

A thermodynamically consistent model framework to describe ion transport in nanopores is presented. The continuum model unifies electro-diffusion and selective ion transport and extends the classical Poisson–Nernst–Planck (PNP) system for an idealized incompressible mixture by including finite ion size and solvation effects. Special [...] Read more.

A thermodynamically consistent model framework to describe ion transport in nanopores is presented. The continuum model unifies electro-diffusion and selective ion transport and extends the classical Poisson–Nernst–Planck (PNP) system for an idealized incompressible mixture by including finite ion size and solvation effects. Special emphasis is placed on the consistent modeling of the selectivity filter within the pore. It is treated as an embedded domain in which the constituents can change their chemical properties and mobility. Using this framework, we achieve good agreement with an experimentally observed current–voltage (IV) characteristic for an L-type selective calcium ion channel for a range of ion concentrations. In particular, we show that the model captures the experimentally observed anomalous mole fraction effect (AMFE). As a result, we find that calcium and sodium currents depend on the surface charge in the selectivity filter, the mobility of ions and the available space in the channel. Our results show that negative charges within the pore have a decisive influence on the selectivity of divalent over monovalent ions, supporting the view that AMFE can emerge from competition and binding effects in a multi-ion environment. Furthermore, the flexibility of the model allows its application in a wide range of channel types and environmental conditions, including both biological ion channels and synthetic nanopores, such as engineered membrane systems with selective ion transport. Full article

(This article belongs to the Special Issue Mathematical Modeling for Ion Channels)

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16 pages, 5654 KB

Open AccessArticle

Target Recognition for Ultra-Wideband Radio Fuzes Using 1D-CGAN-Augmented 1D-CNN

by Kaiwei Wu, Shijun Hao, Yanbin Liang, Bing Yang and Zhonghua Huang

Entropy 2025, 27(9), 980; https://doi.org/10.3390/e27090980 - 19 Sep 2025

Viewed by 286

Abstract

In ultra-wideband (UWB) radio fuzes, the signal processing unit’s capability to rapidly and accurately extract target characteristics under battlefield conditions directly determines detonation precision and reliability. Escalating electronic warfare creates complex electromagnetic environments that compromise UWB fuze reliability through false alarms and missed [...] Read more.

In ultra-wideband (UWB) radio fuzes, the signal processing unit’s capability to rapidly and accurately extract target characteristics under battlefield conditions directly determines detonation precision and reliability. Escalating electronic warfare creates complex electromagnetic environments that compromise UWB fuze reliability through false alarms and missed detections. This study pioneers a novel signal processing architecture. The framework integrates: (1) fixed-parameter Least Mean Squares (LMS) front-end filtering for interference suppression; (2) One-Dimensional Convnlutional Neural Network (1D-CNN) recognition trained on One-Dimensional Conditional Generative Adversarial Network (1D-CGAN)-augmented datasets. Validated on test samples, the system achieves 0% false alarm/miss detection rates and 97.66% segment recognition accuracy—representing a 5.32% improvement over the baseline 1D-CNN model trained solely on original data. This breakthrough resolves energy-threshold detection’s critical vulnerability to deliberate jamming while establishing a new technical framework for UWB fuze operation in contested spectra. Full article

(This article belongs to the Section Multidisciplinary Applications)

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22 pages, 12628 KB

Open AccessArticle

Physical and Statistical Pattern of the Thiva (Greece) 2020–2022 Seismic Swarm

by Filippos Vallianatos, Eirini Sardeli, Kyriaki Pavlou and Andreas Karakonstantis

Entropy 2025, 27(9), 979; https://doi.org/10.3390/e27090979 - 19 Sep 2025

Viewed by 249

Abstract

On 2 December 2020, an earthquake with a magnitude of M_w 4.5 occurred near the city of Thiva (Greece). The aftershock sequence, triggered by ruptures on or near the Kallithea fault, continued until January 2021. Seven months later, new seismic activity began [...] Read more.

On 2 December 2020, an earthquake with a magnitude of M_w 4.5 occurred near the city of Thiva (Greece). The aftershock sequence, triggered by ruptures on or near the Kallithea fault, continued until January 2021. Seven months later, new seismic activity began a few kilometers west of the initial events, with the swarm displaying a general trend of spatiotemporal migration toward the east–southeast until the middle of 2022. In order to understand the physical and statistical pattern of the swarm, the seismicity was relocated using HypoDD, and the magnitude of completeness was determined using the frequency–magnitude distribution. In order to define the existence of spatiotemporal seismicity clusters in an objective way, the DBSCAN clustering algorithm was applied to the 2020–2022 Thiva earthquake sequence. The extracted clusters permit the analysis of the spatiotemporal scaling properties of the main clusters using the Non-Extensive Statistical Physics (NESP) approach, providing detailed insights into the nature of the long-term correlation of the seismic swarm. The statistical pattern observed aligns with a Q-exponential distribution, with q_D values ranging from 0.7 to 0.8 and q_T values from 1.44 to 1.50. Furthermore, the frequency–magnitude distributions were analyzed using the fragment–asperity model proposed within the NESP framework, providing the non-additive entropic parameter (q_M). The results suggest that the statistical characteristics of earthquake clusters can be effectively interpreted using NESP, highlighting the complexity and non-additive nature of the spatiotemporal evolution of seismicity. In addition, the analysis of the properties of the seismicity clusters extracted using the DBSCAN algorithm permits the suggestion of possible physical mechanisms that drive the evolution of the two main and larger clusters. For the cluster that activated first and is located in the west–northwest part, an afterslip mechanism activated after the 2 September 2021, M 4.0 events seems to predominately control its evolution, while for the second activated cluster located in the east–southeast part, a normal diffusion mechanism is proposed to describe its migration pattern. Concluding, we can state that in the present work the application of the DBSCAN algorithm to recognize the existence of any possible spatiotemporal clustering of seismicity could be helping to provide detailed insight into the statistical and physical patterns in earthquake swarms. Full article

(This article belongs to the Special Issue Time Series Analysis in Earthquake Complex Networks)

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24 pages, 3542 KB

Open AccessArticle

Multilayer Network Analysis of European Regional Flows

by Emanuele Calò and Angelo Facchini

Entropy 2025, 27(9), 978; https://doi.org/10.3390/e27090978 - 19 Sep 2025

Viewed by 314

Abstract

In Regional Economics, the attractiveness of regions for capital, migrants, tourists, and other kinds of flows is a relevant topic. Usually, studies in this field explore single flows, characterizing the dimensions of territorial attractiveness separately, rarely considering the interwoven effect of flows. Here, [...] Read more.

In Regional Economics, the attractiveness of regions for capital, migrants, tourists, and other kinds of flows is a relevant topic. Usually, studies in this field explore single flows, characterizing the dimensions of territorial attractiveness separately, rarely considering the interwoven effect of flows. Here, we investigate attractiveness from a multi-dimensional perspective (i.e., dealing with different flows), asking how various types of regional flows collectively shape the attractiveness dynamics of European regions. We analyze eight distinct flow types across NUTS2 regions from 2010 to 2018, employing a multilayer network approach. Notably, the multilayer approach unveils insights that would be missed in single-layer analyses. Community detection reveals complex structures that demonstrate the cohesive power of national borders and the existence of strong cross-border ties in specific regions. Our study contributes to a more nuanced understanding of regional attractiveness, with implications for targeted policy interventions in regional development and European cohesion. Full article

(This article belongs to the Special Issue Entropy, Econophysics, and Complexity)

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22 pages, 3553 KB

Open AccessArticle

An Extended Epistemic Framework Beyond Probability for Quantum Information Processing with Applications in Security, Artificial Intelligence, and Financial Computing

by Gerardo Iovane

Entropy 2025, 27(9), 977; https://doi.org/10.3390/e27090977 - 18 Sep 2025

Viewed by 199

Abstract

In this work, we propose a novel quantum-informed epistemic framework that extends the classical notion of probability by integrating plausibility, credibility, and possibility as distinct yet complementary measures of uncertainty. This enriched quadruple (P, Pl, Cr, Ps) enables a deeper characterization of quantum [...] Read more.

In this work, we propose a novel quantum-informed epistemic framework that extends the classical notion of probability by integrating plausibility, credibility, and possibility as distinct yet complementary measures of uncertainty. This enriched quadruple (P, Pl, Cr, Ps) enables a deeper characterization of quantum systems and decision-making processes under partial, noisy, or ambiguous information. Our formalism generalizes the Born rule within a multi-valued logic structure, linking Positive Operator-Valued Measures (POVMs) with data-driven plausibility estimators, agent-based credibility priors, and fuzzy-theoretic possibility functions. We develop a hybrid classical–quantum inference engine that computes a vectorial aggregation of the quadruples, enhancing robustness and semantic expressivity in contexts where classical probability fails to capture non-Kolmogorovian phenomena such as entanglement, contextuality, or decoherence. The approach is validated through three real-world application domains—quantum cybersecurity, quantum AI, and financial computing—where the proposed model outperforms standard probabilistic reasoning in terms of accuracy, resilience to noise, interpretability, and decision stability. Comparative analysis against QBism, Dempster–Shafer, and fuzzy quantum logic further demonstrates the uniqueness of architecture in both operational semantics and practical outcomes. This contribution lays the groundwork for a new theory of epistemic quantum computing capable of modelling and acting under uncertainty beyond traditional paradigms. Full article

(This article belongs to the Special Issue Probability Theory and Quantum Information)

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15 pages, 292 KB

Open AccessArticle

On the Coupling Between Cosmological Dynamics and Quantum Behavior: A Multiscale Thermodynamic Framework

by Andreas Warkentin

Entropy 2025, 27(9), 976; https://doi.org/10.3390/e27090976 - 18 Sep 2025

Viewed by 233

Abstract

A multiscale thermodynamic model is considered, in which cosmological dynamics enforce persistent non-equilibrium conditions through recursive energy exchange across hierarchically ordered subsystems. The internal energy of each subsystem is recursively determined by energetic interactions with its subcomponents, forming a nested hierarchy extending up [...] Read more.

A multiscale thermodynamic model is considered, in which cosmological dynamics enforce persistent non-equilibrium conditions through recursive energy exchange across hierarchically ordered subsystems. The internal energy of each subsystem is recursively determined by energetic interactions with its subcomponents, forming a nested hierarchy extending up to cosmological scales. The total energy of the universe is assumed to be constant, imposing global consistency conditions on local dynamics. On the quantum scale, subsystems remain thermodynamically constrained in their accessible state space due to the unresolved energetic embedding imposed by higher-order couplings. As a result, quantum behavior is interpreted as an effective projection of unresolved thermodynamic interactions. In this view, the wave function serves as a mathematical representation of a subsystem’s thermodynamic embedding, summarizing the unresolved energetic couplings with its environment, as shaped by recursive interactions across cosmological and microscopic scales. Phenomena such as zero-point energy and vacuum fluctuations are thereby understood as residual effects of structural energy constraints. Classical mechanics arises as a limiting case under full energetic resolution, while the quantum formalism reflects thermodynamic incompleteness. This formulation bridges statistical mechanics and quantum theory without metaphysical assumptions. It remains fully compatible with standard formalism, offering a thermodynamic interpretation based solely on energy conservation and hierarchical organization. All effects arise from scale-dependent resolution, not from violations of established physics. Full article

(This article belongs to the Special Issue Non-Equilibrium Thermodynamics and Quantum Information Theory)

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20 pages, 378 KB

Open AccessArticle

On the Storage–Communication Trade-Off in Graph-Based X-Secure T-Private Linear Computation

by Yueyang Liu, Haobo Jia and Zhuqing Jia

Entropy 2025, 27(9), 975; https://doi.org/10.3390/e27090975 - 18 Sep 2025

Viewed by 198

Abstract

The problem of graph-based X-secure T-private linear computation (GXSTPLC) is to allow a user to retrieve a linear combination of K messages from a set of N distributed servers that store the messages in a graph-based fashion, i.e., each message is [...] Read more.

The problem of graph-based X-secure T-private linear computation (GXSTPLC) is to allow a user to retrieve a linear combination of K messages from a set of N distributed servers that store the messages in a graph-based fashion, i.e., each message is restricted to be distributed among a subset of servers. T-privacy requires that the coefficients of the linear combination are not revealed to any group of up to T colluding servers, and X-security guarantees that any set of up to X colluding servers learns nothing about the messages. In this paper, we propose an achievability scheme for GXSTPLC that enables a storage–communication trade-off by exploiting non-replicated storage codes. Novel aspects of our achievability scheme include the usage of the idea of cross-subspace alignment null shaper that addresses various challenges posed by the graph-based storage structure. In addition, unlike previous works, our scheme allows a direct transformation into a quantum one to achieve a superdense coding gain by leveraging the idea of N-Sum Box abstraction of quantum “over-the-air” computing. Full article

(This article belongs to the Special Issue Information-Theoretic Security and Privacy)

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7 pages, 246 KB

Open AccessArticle

Bounded Variation Separates Weak and Strong Average Lipschitz

by Ariel Elperin and Aryeh Kontorovich

Entropy 2025, 27(9), 974; https://doi.org/10.3390/e27090974 - 18 Sep 2025

Viewed by 175

Abstract

We closely examine a recently introduced notion of average smoothness. The latter defined a weak and strong average-Lipschitz seminorm for real-valued functions on general metric spaces. Specializing to the standard metric on the real line, we compare these notions to bounded variation (BV) [...] Read more.

We closely examine a recently introduced notion of average smoothness. The latter defined a weak and strong average-Lipschitz seminorm for real-valued functions on general metric spaces. Specializing to the standard metric on the real line, we compare these notions to bounded variation (BV) and discover that the weak notion is strictly weaker than BV while the strong notion is strictly stronger. Along the way, we discover that the weak average smooth class is also considerably larger in a certain combinatorial sense, which is made precise by the fat-shattering dimension. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

16 pages, 350 KB

Open AccessFeature PaperArticle

Symplectic QSD, LCD, and ACD Codes over a Non-Commutative Non-Unitary Ring of Order Nine

by Sarra Manseri, Patrick Solé, Adel Alahmadi and Widyan Basaffar

Entropy 2025, 27(9), 973; https://doi.org/10.3390/e27090973 - 18 Sep 2025

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Abstract

We introduce quasi self-dual (QSD), linear complementary dual (LCD), and additive complementary dual (ACD) codes for the symplectic inner product over a non-commutative non-unitary ring of order 9. We establish connections with symplectic–self-orthogonal and LCD ternary codes. We characterize right-symplectic ACD codes. Full article

(This article belongs to the Special Issue Discrete Math in Coding Theory, 2nd Edition)

29 pages, 642 KB

Open AccessArticle

Timely Updating on Ber/Geo/1/2 Queue Modeled Status Updating System with Eavesdropper

by Jixiang Zhang, Han Xu, Anqi Zheng, Daming Cao, Yinfei Xu and Chengyu Lin

Entropy 2025, 27(9), 972; https://doi.org/10.3390/e27090972 - 18 Sep 2025

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Abstract

We consider that the source sends packets to the receiver through a Ber/Geo/1/2 queue modeled status updating system, where the transmitted packets are subject to potential eavesdropping. Time is discretized into identical time slots. This paper studies the tradeoffs between the information freshness [...] Read more.

We consider that the source sends packets to the receiver through a Ber/Geo/1/2 queue modeled status updating system, where the transmitted packets are subject to potential eavesdropping. Time is discretized into identical time slots. This paper studies the tradeoffs between the information freshness and transmission security of a system, where freshness is characterized by the age of information (AoI) metric and transmission security is represented by the proportion of obtained insecure packets over a long period of time. We assume that in a time slot, the source generates a new packet with probability p, and a packet arrives at the receiver with probability

γ_{d}

. With probability

γ_{E}

, a transmitted packet is eavesdropped. At the receiver, AoI is defined as the elapsed time since the generation instant of the latest obtained packet. A packet is defined to be insecure if it is obtained by the eavesdropper earlier than the receiver. To control the proportion of insecure packets obtained in the receiver, we propose using the probabilistic deletion/retaining scheme. More specifically, when a packet is eavesdropped before arriving at the receiver, this packet is deleted with probability

δ

or retained with probability

1 - δ

. Under this transmission policy, we derive the system’s average AoI which we call the average

δ -

secure AoI, and investigate its relations with the insecure packet proportion, which is denoted as

η (δ)

. The obtained formulas are then calculated in three special cases, including

γ_{E} = 0

,

γ_{E} = 1

, and

δ = 1

. We explain that these cases correspond to the average AoI of a basic status updating system with Ber/Geo/1/2 queue, packet with random geometric deadline in service process, and average age of secure information (AoSI), respectively. Numerical simulations of obtained results are provided. In particular, the tradeoffs between average

δ -

secure AoI and

η (δ)

are analyzed in detail. We demonstrate that depending on the value of the eavesdropping probability

γ_{E}

, average

δ -

secure AoI varies in different trends with

η (δ)

, and in most cases the average

δ -

secure AoI and

η (δ)

can be minimized simultaneously. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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18 pages, 939 KB

Open AccessArticle

A DREM-Based Approach for the Identification of Chaotic Systems

by Carlos Aguilar-Ibanez, Miguel S. Suarez-Castanon, Belem Saldivar, José E. Valdez-Rodríguez and Eloísa García-Canseco

Entropy 2025, 27(9), 971; https://doi.org/10.3390/e27090971 - 18 Sep 2025

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Abstract

A straightforward methodology for identifying certain classes of chaotic systems based on a novel version of the least-squares method, assuming they are algebraically observable and identifiable with respect to a measurable output, is introduced. This output allows us to express the original system [...] Read more.

A straightforward methodology for identifying certain classes of chaotic systems based on a novel version of the least-squares method, assuming they are algebraically observable and identifiable with respect to a measurable output, is introduced. This output allows us to express the original system as a chain of integrators, where the last term, which depends on the output and its corresponding time derivatives, lumps the system’s non-linearities. We can factorize this term into a regressor function multiplied by an unknown-parameter vector, suggesting that a high-gain observer can be used to simultaneously and approximately estimate the states of the pure integrator and the evolution of the lumped nonlinear term. This allows us to rewrite the original system as a linear regression equation. This configuration enables the above-mentioned least-squares method to recover the chaotic-system parameters. Full article

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14 pages, 670 KB

Open AccessReview

Disorder at the Synapse: How the Active Inference Framework Unifies Competing Perspectives on Depression

by Christopher G. Davey and Paul B. Badcock

Entropy 2025, 27(9), 970; https://doi.org/10.3390/e27090970 - 18 Sep 2025

Viewed by 383

Abstract

Depression is one of the most disabling of all disorders across the community, yet many aspects of the disorder remain contentious. Psychosocial and biological perspectives are often placed in opposition to one another, which in part reflects a failure of our explanatory frameworks. [...] Read more.

Depression is one of the most disabling of all disorders across the community, yet many aspects of the disorder remain contentious. Psychosocial and biological perspectives are often placed in opposition to one another, which in part reflects a failure of our explanatory frameworks. The active inference account of brain function breaks down this dualism, demonstrating that bodily processes are deeply integrated with the social world. It shows us that there is no contradiction in understanding depression as a product of the social environment at the same time as having a brain basis and manifesting in biological symptoms. From an active inference perspective, depression can be thought of as a synaptopathy: a disorder that arises from alterations to the excitatory-inhibitory balance enacted at the synapse, reflecting the interoceptive precision-weightings that have changed in the context of psychosocial instability. Therapies that alleviate depressive symptoms act at different levels of the active inference framework to re-weight precision estimates and the confidence we have in our predictions: this is true for psychotherapies, lifestyle interventions and antidepressant medications. Their effectiveness is often only partial, and while different treatment modalities can complement one another, there is a need for continued development of new and better treatment options. Full article

(This article belongs to the Special Issue Bayesian Inference for Psychology and Psychiatry)

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18 pages, 3602 KB

Open AccessArticle

Information Dynamics of the Mother–Fetus System Using Kolmogorov–Sinai Entropy Derived from Heart Sounds: A Longitudinal Study from Early Pregnancy to Postpartum

by Sayuri Ishiyama, Takashi Tahara, Hiroaki Iwanaga and Kazutomo Ohashi

Entropy 2025, 27(9), 969; https://doi.org/10.3390/e27090969 - 17 Sep 2025

Viewed by 255

Abstract

Kolmogorov–Sinai (KS) entropy is an indicator of the chaotic behavior of entire systems from an information-theoretic viewpoint. Here, we used KS entropy values derived from the heart sounds of four fetus–mother pairs to identify the changes in fetal and maternal informational patterns during [...] Read more.

Kolmogorov–Sinai (KS) entropy is an indicator of the chaotic behavior of entire systems from an information-theoretic viewpoint. Here, we used KS entropy values derived from the heart sounds of four fetus–mother pairs to identify the changes in fetal and maternal informational patterns during the four phases of pregnancy (early, mid, late, and postnatal). Time-series data of the heart sounds were reconstructed in a five-dimensional phase space to obtain the Lyapunov spectrum, and KS entropy was calculated. Statistical analyses were then conducted separately for the fetus and mother for the four phases of pregnancy. The fetal KS entropy significantly increased from early pregnancy to the postnatal period (0.054 ± 0.007 vs. 0.097 ± 0.007; p < 0.001), whereas the maternal KS entropy decreased in late pregnancy and then significantly increased after birth (0.098 ± 0.002 vs. 0.133 ± 0.003; p < 0.001). The increase in KS entropy with the course of fetal gestation reflects an increase in information generation and adaptive capacity during the developmental process. Thus, changes in maternal KS entropy play a dual role, temporarily enhancing physiological stability to support fetal development and helping to rebuild the mother’s own adaptive capacity in the postpartum period. Full article

(This article belongs to the Special Issue Synchronization and Information Patterns in Human Dynamics)

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11 pages, 301 KB

Open AccessArticle

Thermodynamics of Observations

by Arno Keppens and Jean-Christopher Lambert

Entropy 2025, 27(9), 968; https://doi.org/10.3390/e27090968 - 17 Sep 2025

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Abstract

This work demonstrates that the four laws of classical thermodynamics apply to the statistics of symmetric observation distributions, and provides examples of how this can be exploited in uncertainty assessments. First, an expression for the partition function Z is derived. In contrast with [...] Read more.

This work demonstrates that the four laws of classical thermodynamics apply to the statistics of symmetric observation distributions, and provides examples of how this can be exploited in uncertainty assessments. First, an expression for the partition function Z is derived. In contrast with general classical thermodynamics, however, this can be performed without the need for variational calculus, while Z also equals the number of observations N directly. Apart from the partition function

Z \equiv N

as a scaling factor, three state variables m, n, and

ϵ

fully statistically characterize the observation distribution, corresponding to its expectation value, degrees of freedom, and random error, respectively. Each term in the first law of thermodynamics is then shown to be a variation on

δ m^{2} = δ {(n ϵ)}^{2}

for both canonical (constant n and

ϵ

) and macro-canonical (constant

ϵ

) observation ensembles, while micro-canonical ensembles correspond to a single observation result bin having

δ m^{2} = 0

. This view enables the improved fitting and combining of observation distributions, capturing both measurand variability and measurement precision. Full article

(This article belongs to the Section Multidisciplinary Applications)

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24 pages, 1224 KB

Open AccessArticle

Multi-UAV-Assisted ISAC System: Joint User Association, Trajectory Design, and Resource Allocation

by Jinwei Wang, Renhui Xu, Laixian Peng and Xianglin Wei

Entropy 2025, 27(9), 967; https://doi.org/10.3390/e27090967 - 17 Sep 2025

Viewed by 264

Abstract

Unmanned aerial vehicle (UAV)-assisted integrated sensing and communication (ISAC) systems have developed rapidly in the sixth generation (6G) era. However, factors such as the mobility of ground users and malicious jamming pose significant challenges to systems’ performance and reliability. Against this backdrop, this [...] Read more.

Unmanned aerial vehicle (UAV)-assisted integrated sensing and communication (ISAC) systems have developed rapidly in the sixth generation (6G) era. However, factors such as the mobility of ground users and malicious jamming pose significant challenges to systems’ performance and reliability. Against this backdrop, this paper designs a multi-UAV-assisted ISAC system model under malicious jamming environments. Under the constraint of sensing accuracy, the total communication rate of the system is maximized through joint optimization of user association, UAV trajectory, and transmit power. The problem is then decomposed into three subproblems, which are solved using the improved auction algorithm (IAA), dream optimization algorithm (DOA), and rapidly-exploring random trees-based optimizer algorithm (RRTOA). The global optimal solution is approached through the alternating optimization-based predictive scheduling algorithm (AOPSA). Meanwhile, this paper also introduces a long short-term memory (LSTM) network to predict users’ dynamic positions, addressing the impact of user mobility and enhancing the system’s real-time performance. Simulation results show that compared with the baseline scheme, the proposed algorithm achieves a 188% improvement in communication rate, which verifies its effectiveness and superiority. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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20 pages, 281 KB

Open AccessArticle

Digital Economy and Green Development: Mechanisms of Action, Spillover Effects and Transmission Mechanisms

by Xin Tong, Ke Li and Xuesen Li

Entropy 2025, 27(9), 966; https://doi.org/10.3390/e27090966 - 17 Sep 2025

Viewed by 293

Abstract

The digital economy plays an important role in promoting green economic growth. This study evaluates the degree of green economic development generated by green innovation and green sharing based on data from 30 provinces in China from 2011 to 2022. An empirical analysis [...] Read more.

The digital economy plays an important role in promoting green economic growth. This study evaluates the degree of green economic development generated by green innovation and green sharing based on data from 30 provinces in China from 2011 to 2022. An empirical analysis of the digital economy’s influence on the growth of the green economy and its transmission mechanisms is performed. The analysis results demonstrate that the digital economy can significantly promote green economic development, encompassing improvements in both green innovation and green sharing, and exhibits a nonlinear “increasing marginal effect”. The analysis of transmission channels reveals that, on one hand, the digital economy can promote green economic development by optimizing the allocation of data elements, while on the other, its impact is also influenced by the intensity of environmental regulations, exhibiting a threshold effect. Further heterogeneity analysis suggests that the promotional effect of the digital economy on green economic development is more pronounced in regions with high levels of economic development, a robust infrastructure, and strong policy support. Full article

(This article belongs to the Section Complexity)

12 pages, 442 KB

Open AccessArticle

Fundamental Solutions to Fractional Heat Conduction in Two Joint Half-Lines Under Conditions of Nonperfect Thermal Contact

by Yuriy Povstenko, Tamara Kyrylych, Viktor Dashkiiev and Andrzej Yatsko

Entropy 2025, 27(9), 965; https://doi.org/10.3390/e27090965 - 16 Sep 2025

Viewed by 189

Abstract

At the interface dividing two media, an area appears that has its own physical characteristics which differ from the properties of the bulk materials. The small width of the interface area permits considering this area as a two-dimensional median surface with the specified [...] Read more.

At the interface dividing two media, an area appears that has its own physical characteristics which differ from the properties of the bulk materials. The small width of the interface area permits considering this area as a two-dimensional median surface with the specified physical characteristics. The fundamental solutions to the Cauchy problem as well as to the source problem are considered for fractional heat conduction in two joint half-lines under conditions of nonperfect thermal contact. The specific example of classical heat conduction is also investigated. Full article

(This article belongs to the Special Issue Random Walks and Stochastic Processes in Complex Systems: From Physics to Socio-Economic Phenomena, Second Edition)

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26 pages, 457 KB

Open AccessArticle

Statistical Inference for High-Dimensional Heteroscedastic Partially Single-Index Models

by Jianglin Fang and Zhikun Tian

Entropy 2025, 27(9), 964; https://doi.org/10.3390/e27090964 - 16 Sep 2025

Viewed by 201

Abstract

In this study, we propose a novel penalized empirical likelihood approach that simultaneously performs parameter estimation and variable selection in heteroscedastic partially linear single-index models with a diverging number of parameters. It is rigorously proved that the proposed method possesses the oracle property: [...] Read more.

In this study, we propose a novel penalized empirical likelihood approach that simultaneously performs parameter estimation and variable selection in heteroscedastic partially linear single-index models with a diverging number of parameters. It is rigorously proved that the proposed method possesses the oracle property: (i) with probability tending to 1, the zero components are consistently estimated as zero; (ii) the estimators for nonzero coefficients achieve asymptotic efficiency. Furthermore, the penalized empirical log-likelihood ratio statistic is shown to asymptotically follow a standard chi-squared distribution under the null hypothesis. This methodology can be naturally applied to pure partially linear models and single-index models in high-dimensional settings. Simulation studies and real-world data analysis are conducted to examine the properties of the presented approach. Full article

(This article belongs to the Special Issue Statistical Inference: Theory and Methods)

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15 pages, 17666 KB

Open AccessArticle

Multi-Dimensional Quantum-like Resources from Complex Synchronized Networks

by Debadrita Saha and Gregory D. Scholes

Entropy 2025, 27(9), 963; https://doi.org/10.3390/e27090963 - 16 Sep 2025

Viewed by 236

Abstract

Recent publications have introduced the concept of quantum-like (QL) bits, along with their associated QL states and QL gate operations, which emerge from the dynamics of complex, synchronized networks. The present work extends these ideas to multi-level QL resources, referred to as QL [...] Read more.

Recent publications have introduced the concept of quantum-like (QL) bits, along with their associated QL states and QL gate operations, which emerge from the dynamics of complex, synchronized networks. The present work extends these ideas to multi-level QL resources, referred to as QL dits, as higher-dimensional analogs of QL bits. We employ systems of k-regular graphs to construct QL-dits for arbitrary dimensions, where the emergent eigenspectrum of their adjacency matrices defines the QL-state space. The tensor product structure of multi-QL dit systems is realized through the Cartesian product of graphs. Furthermore, we examine the potential computational advantages of employing d-nary QL systems over two-level QL bit systems, particularly in terms of classical resource efficiency. Overall, this study generalizes the paradigm of using synchronized network dynamics for QL information processing to include higher-dimensional QL resources. Full article

(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering IV)

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