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Editor’s Choice Articles

Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.

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15 pages, 5530 KB  
Article
Color Recurrence Plots from Uniform Delay Embeddings for Bearing Degradation Tracking and Prognostics
by Algirdas Kazlauskas, Rita Baublienė, Mantas Landauskas and Minvydas Ragulskis
Entropy 2026, 28(6), 668; https://doi.org/10.3390/e28060668 - 11 Jun 2026
Viewed by 413
Abstract
Prognostic health management of rolling element bearings requires feature representations that reliably track degradation while remaining tractable for real-time deployment. This paper investigates whether uniform time-delay embedding can serve as a near-optimal substitute for computationally expensive non-uniform embedding in recurrence-based vibration analysis. We [...] Read more.
Prognostic health management of rolling element bearings requires feature representations that reliably track degradation while remaining tractable for real-time deployment. This paper investigates whether uniform time-delay embedding can serve as a near-optimal substitute for computationally expensive non-uniform embedding in recurrence-based vibration analysis. We show empirically that optimally chosen uniform delay vectors yield phase-space reconstructions of bearing vibration signals not significantly inferior to those produced by globally optimized non-uniform delay vectors, compressing the parameter search from a combinatorial optimization to a single scalar selection. Building on this near-optimality result, we construct color recurrence plots from uniformly embedded phase spaces and apply them to remaining useful life (RUL) prediction on the Intelligent Maintenance Systems (IMS) bearing dataset. We further demonstrate that standard binary recurrence plots are poorly suited for RUL estimation: their dense and erratically varying local patterns obscure the degradation trends required for reliable prognostics. Color recurrence plots, by contrast, suppress these local instabilities by averaging recurrence structures across multiple phase-space projections, exposing a globally evolving intensity that tracks bearing health throughout its degradation trajectory. This work establishes uniform delay embedding combined with color recurrence representation as an efficient, principled, and practically deployable approach to recurrence-based condition monitoring in industrial predictive maintenance. Full article
(This article belongs to the Special Issue Advances in Fault Diagnosis and Fault-Tolerant Control: From Classical Approaches to AI-Driven Solutions)
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18 pages, 12188 KB  
Review
The Simplest Complexity: The Story of the Three-Body Problem
by Barak Kol
Entropy 2026, 28(6), 610; https://doi.org/10.3390/e28060610 - 29 May 2026
Viewed by 1001
Abstract
This article offers a broad-brush account of the Newtonian three-body problem, from its origins with Newton to its vibrant present, emphasizing its enduring influence on theoretical physics. It unfolds through a series of self-contained episodes that illuminate the scientific fields and the paradigm [...] Read more.
This article offers a broad-brush account of the Newtonian three-body problem, from its origins with Newton to its vibrant present, emphasizing its enduring influence on theoretical physics. It unfolds through a series of self-contained episodes that illuminate the scientific fields and the paradigm shift that have grown out of this problem. Full article
(This article belongs to the Special Issue Statistical Physics and Nonlinear Dynamics for Complex Systems)
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12 pages, 473 KB  
Article
Phase Transitions of the Majority-Vote Model with Inertia on Directed Erdös–Rényi Networks
by Talia Costa Rodrigues, David Santana Alencar, Tayroni Alencar Alves, Gladstone Alencar Alves, Francisco Welington Lima and João Antônio Plascak
Entropy 2026, 28(6), 591; https://doi.org/10.3390/e28060591 - 26 May 2026
Viewed by 264
Abstract
The phase transition of the majority vote model with inertia has been investigated by means of extensive Monte Carlo simulations on directed Erdös–Rényi networks. Besides the usual average connectivity and local field that adds the opinion of the site itself, an additional term [...] Read more.
The phase transition of the majority vote model with inertia has been investigated by means of extensive Monte Carlo simulations on directed Erdös–Rényi networks. Besides the usual average connectivity and local field that adds the opinion of the site itself, an additional term of inertia is considered. The relaxation time of the average opinion state of the network, together with the average opinion state fourth-order Binder cumulant and the corresponding opinion state susceptibility, have been analyzed for several different network sizes and local field and inertia parameter values, for average connectivity of 20 connections. The present results show that the phase transition of this model strongly depends on the inertia parameter, being quite different and richer than previous results of the same model on other regular networks. For inertia parameters between zero and 0.1 the system undergoes a continuous phase transition; for values in the range 0.1 and 0.2 no transition can be seen; for still larger values of inertia up to 0.5 a first-order phase transition takes place; finally, for values larger than 0.5 the dynamics is fully dominated by the inertia, and again no phase transition is observed. Full article
(This article belongs to the Special Issue Complexity of Social Networks)
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25 pages, 3056 KB  
Article
On Intention and Fluctuations in the Coordination Dynamics of Animate Movement
by Amaury Dechaux, Aliza T. Sloan and J. A. Scott Kelso
Entropy 2026, 28(5), 556; https://doi.org/10.3390/e28050556 - 15 May 2026
Viewed by 496
Abstract
Many of life’s biggest dilemmas can be summed up as a tension between holding on and letting go. The very language evokes a notion of intentionality which, for the most part, has evaded scientific understanding. How might we even get a window into [...] Read more.
Many of life’s biggest dilemmas can be summed up as a tension between holding on and letting go. The very language evokes a notion of intentionality which, for the most part, has evaded scientific understanding. How might we even get a window into it? Important insights have come from a seemingly simple task: wiggling one’s fingers to and fro to the beat of a metronome. As the metronome pace increases to some critical frequency, one coordinative pattern becomes unstable and switches spontaneously to another. Such transitions are typically preceded by critical fluctuations, a predicted feature of self-organization in complex, dynamical systems. Here we address the nature and source of these fluctuations, usually assumed to be: (1) random; (2) of external origin; and (3) of fixed magnitude. We performed an experiment in which participants were instructed to oscillate their fingers in either an in-phase or anti-phase pattern in time with a metronome and instructed them to either “hold-on” or “let-go” should they feel the pattern begin to change, yielding a 2 by 2 within-subjects design. We observed that as the metronome frequency was increased from 1.00 to 3.00 Hz, fluctuations in the relative phase between the fingers were significantly altered both by the starting coordinative pattern as well as the participant’s intention to “hold it on” or “let it go”. Specifically, the intention to hold on to the anti-phase pattern delayed the spontaneous transition to in-phase, an effect that was paired with increased fluctuations beyond the critical frequency. These observations were analyzed under the extended Haken–Kelso–Bunz (HKB) model which describes the non-linear stochastic dynamics of the order parameter (relative phase) as a gradient descent on a certain potential. Our analysis, in line with experimental results, suggests that intention transforms the HKB potential not only by stabilizing unstable coordination states but also (paradoxically) by increasing fluctuations around them. Such findings may offer new interpretative light on the relation between intention and fluctuations in the coordination dynamics of living things. Full article
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19 pages, 3039 KB  
Article
TraceLAB: A MATLAB Toolbox for Interindividual Synchrony Analysis of Facial Expression and Head Movement Data Acquired via Trace
by Felix Carter, Mike Richardson, Danaë Stanton Fraser and Iain D. Gilchrist
Entropy 2026, 28(5), 503; https://doi.org/10.3390/e28050503 - 29 Apr 2026
Viewed by 527
Abstract
Facial expressions transmit information about internal states, both during social interaction and in response to shared stimuli such as films. When individuals view the same content, synchrony in their expressions reflects shared information processing, and the degree to which their expressions correlate indicates [...] Read more.
Facial expressions transmit information about internal states, both during social interaction and in response to shared stimuli such as films. When individuals view the same content, synchrony in their expressions reflects shared information processing, and the degree to which their expressions correlate indicates how similarly their perceptual and affective systems are responding to the common input. This makes interindividual expression synchrony a potential marker of engagement and subjective experience. However, the acquisition and analysis of facial data pose both ethical and technical challenges to researchers. ‘Trace’ is a research media player implemented in PsychoPy’s online platform Pavlovia, which captures anonymised facial landmark coordinates through a webcam, without the ethical and technical constraints of capturing and storing video images of participants. Nonetheless, its usefulness is currently limited due to the lack of available preprocessing and analysis tools. This paper describes the functionality of TraceLAB, a MATLAB-based toolbox designed for the preprocessing of Trace data: specifically, the formatting, aligning, and filtering of data. In addition, TraceLAB implements some novel analysis techniques to allow researchers to quantify interindividual synchrony of expressions (through correlated component analysis) and head movements (through Surrogate Synchrony), which may be interpreted as measures of shared information processing. These techniques are demonstrated here on both simulated and real datasets. Full article
(This article belongs to the Special Issue Synchronization and Information Patterns in Human Dynamics)
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21 pages, 1781 KB  
Article
A Post-Quantum Authentication and Key Agreement Protocol Based on Lattice-Based KEM for Secure Network Environments
by Xiaoping Chen, Wangyu Wu, Guangmin Liang, Haonan Tan and Yicheng Yu
Entropy 2026, 28(5), 490; https://doi.org/10.3390/e28050490 - 24 Apr 2026
Cited by 1 | Viewed by 706
Abstract
In emerging environments such as cloud computing and the Internet of Things (IoT), secure authentication and key negotiation play a crucial role in protecting data transmitted over public networks. However, many existing authentication protocols are still designed based on classical public-key cryptography primitives, [...] Read more.
In emerging environments such as cloud computing and the Internet of Things (IoT), secure authentication and key negotiation play a crucial role in protecting data transmitted over public networks. However, many existing authentication protocols are still designed based on classical public-key cryptography primitives, and quantum computing may threaten their security. To address this challenge, we propose a post-quantum authentication and key agreement protocol that uses the lattice-based Kyber key encapsulation mechanism (KEM). Our proposed protocol integrates cryptographic authentication, smart card protection, and post-quantum key encapsulation mechanisms, enabling mutual authentication between users and servers and securely establishing session keys. The security of the protocol is formally analyzed in the Real-or-Random (ROR) model under the random oracle assumption and the IND-CCA security of the underlying KEM scheme. Furthermore, through informal security analysis, we have further demonstrated that the protocol possesses important security properties, including anonymity, untraceability, perfect forward confidentiality, and resistance to known attacks. In addition, the computational cost and communication overhead of the proposed scheme are evaluated and compared with several representative authentication protocols. The results show that the proposed protocol can provide strong security while maintaining low computational cost and communication overhead. Full article
(This article belongs to the Special Issue Quantum Information Security)
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17 pages, 445 KB  
Article
On GRAND-Assisted Vector Random Linear Network Coding in Wireless Broadcasts
by Rina Su, Chengji Zhao, Qifu Sun and Linqi Song
Entropy 2026, 28(4), 450; https://doi.org/10.3390/e28040450 - 15 Apr 2026
Viewed by 850
Abstract
Recent works have combined random linear network coding (RLNC) with guessing random additive noise decoding (GRAND) to leverage RLNC packets to partially correct bit errors prior to RLNC decoding, so as to reduce the packet erasure rates in wireless broadcast networks. However, existing [...] Read more.
Recent works have combined random linear network coding (RLNC) with guessing random additive noise decoding (GRAND) to leverage RLNC packets to partially correct bit errors prior to RLNC decoding, so as to reduce the packet erasure rates in wireless broadcast networks. However, existing schemes are restricted to scalar RLNC over the finite field GF(2L). In this paper, we first formulate a general GRAND-assisted decoding framework for vector RLNC over the vector space GF(2)L, and further propose a design rule for vector RLNC schemes such that estimated error vectors can be efficiently obtained without incurring any additional computational overhead. Necessary and sufficient conditions for the correctness of every efficiently obtained estimated error vector are characterized. Two explicit vector RLNC schemes satisfying the proposed design rule are constructed. The first scheme is designed based on the matrix representation of GF(2L), and analytical results show that it achieves the same completion delay performance as the counterpart scalar RLNC scheme over GF(2L), while achieving up to a 37.3% reduction in coding computational complexity compared with the scalar one. The second scheme is designed based on sparse coding coefficient matrices. It further reduces computational complexity by up to 33.6% compared with the first scheme, at the cost of a slight degradation in completion delay performance. Full article
(This article belongs to the Special Issue Information and Coding Theory for Distributed Learning, Storage, Scheduling, and Security)
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12 pages, 892 KB  
Article
Ratio-Dependent Contrarian Activation in Opinion Dynamics
by Serge Galam
Entropy 2026, 28(4), 443; https://doi.org/10.3390/e28040443 - 14 Apr 2026
Cited by 2 | Viewed by 675
Abstract
I study the impact of mixed contrarians on the opinion dynamics of a heterogenous population with conformists using the Galam Majority Model. Activation of contrarians is a function of the majority/minority ratio in the local groups of discussion. Restricting the group size to [...] Read more.
I study the impact of mixed contrarians on the opinion dynamics of a heterogenous population with conformists using the Galam Majority Model. Activation of contrarians is a function of the majority/minority ratio in the local groups of discussion. Restricting the group size to 3, two types of contrarians are included in respective proportions c3,0 for configurations with ratio 3 to 0 and c2,1 for ratio 2 to 1. I then derive the explicit update equation and analytically obtain the fixed points, their stability, and the resulting full two-dimensional landscape of the dynamics of opinion. Setting c3,0=c2,1=c recovers the original results obtained with uniform contrarians. The findings allow for considering a wide spectrum of new disruptive strategies to secure either a majority/minority ending ensuring the opinion having the larger initial support to win, or a single attractor dynamics at fifty/fifty, which implies a random winner regardless of initial supports. Full article
(This article belongs to the Special Issue Entropy-Based Applications in Sociophysics, Third Edition)
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21 pages, 1178 KB  
Article
Soft-Community Kernel Rényi Spectrum for Semantic Uncertainty Estimation in Large Language Models
by Zongkai Li and Junliang Du
Entropy 2026, 28(4), 442; https://doi.org/10.3390/e28040442 - 14 Apr 2026
Viewed by 1186
Abstract
Uncertainty estimation is critical for deploying large language models (LLMs) in safety-sensitive and decision-critical applications. Recent approaches estimate semantic uncertainty by clustering multiple sampled responses into equivalence classes and measuring their diversity via entropy-based criteria. However, existing methods typically rely on greedy hard [...] Read more.
Uncertainty estimation is critical for deploying large language models (LLMs) in safety-sensitive and decision-critical applications. Recent approaches estimate semantic uncertainty by clustering multiple sampled responses into equivalence classes and measuring their diversity via entropy-based criteria. However, existing methods typically rely on greedy hard clustering and von Neumann entropy, which suffer from sensitivity to clustering order, noise in semantic equivalence judgments, and limited control over spectral contributions. In this work, we propose a principled information-theoretic framework for LLM semantic uncertainty estimation based on soft semantic communities and kernel Rényi entropy. Given multiple generations for a query, we construct a weighted semantic graph using pairwise semantic similarity scores and infer soft community assignments via weighted graph community detection. These soft assignments induce a positive semi-definite semantic kernel that captures the distribution of semantic modes without enforcing hard equivalence relations. Uncertainty is then quantified by the Rényi entropy of the kernel spectrum, yielding a tunable measure that interpolates between sensitivity to dominant semantic modes and long-tail semantic diversity. Compared to prior von Neumann entropy-based estimators, the proposed Rényi spectral uncertainty offers improved robustness to semantic noise, reduced dependence on clustering heuristics, and greater flexibility through its order parameter. Extensive experiments on question answering tasks demonstrate that our method provides more stable and discriminative uncertainty estimates, particularly under limited sampling budgets and noisy semantic judgments. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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25 pages, 712 KB  
Article
Decision-Making Under Model Misspecification: DRO with Robust Bayesian Ambiguity Sets
by Charita Dellaporta, Patrick O’Hara and Theodoros Damoulas
Entropy 2026, 28(4), 430; https://doi.org/10.3390/e28040430 - 11 Apr 2026
Viewed by 956
Abstract
Distributionally Robust Optimisation (DRO) protects risk-averse decision-makers by considering the worst-case risk within an ambiguity set of distributions based on the empirical distribution or a model. To further guard against finite, noisy data, model-based approaches admit Bayesian formulations that propagate uncertainty from the [...] Read more.
Distributionally Robust Optimisation (DRO) protects risk-averse decision-makers by considering the worst-case risk within an ambiguity set of distributions based on the empirical distribution or a model. To further guard against finite, noisy data, model-based approaches admit Bayesian formulations that propagate uncertainty from the posterior to the decision-making problem. However, when the model is misspecified, the decision-maker must stretch the ambiguity set to contain the data-generating process (DGP), leading to overly conservative decisions. We address this challenge by introducing DRO with Robust ayesian Ambiguity Sets (DRO-RoBAS) to model misspecification. These are Maximum Mean Discrepancy ambiguity sets centred at a robust posterior predictive distribution that incorporates beliefs about the DGP. We show that the resulting optimisation problem obtains a dual formulation in the Reproducing Kernel Hilbert Space and we give probabilistic guarantees on the tolerance level of the ambiguity set. Our method outperforms other Bayesian and empirical DRO approaches in out-of-sample performance on the Newsvendor and Portfolio problems with various cases of model misspecification. Full article
(This article belongs to the Special Issue Statistical Inference: Theory and Methods)
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35 pages, 580 KB  
Article
From Quantum Time to Manifestly Covariant QFT: On the Need for a Quantum-Action-Based Quantization
by Nahuel L. Diaz
Entropy 2026, 28(4), 425; https://doi.org/10.3390/e28040425 - 10 Apr 2026
Viewed by 650
Abstract
In quantum time (QT) schemes, time is promoted to a degree of freedom, allowing Lorentz covariance to be made explicit for single particles. We ask whether this can be lifted to QFT so that Lorentz covariance becomes manifest at the Hilbert-space level, rather [...] Read more.
In quantum time (QT) schemes, time is promoted to a degree of freedom, allowing Lorentz covariance to be made explicit for single particles. We ask whether this can be lifted to QFT so that Lorentz covariance becomes manifest at the Hilbert-space level, rather than being hidden as in the standard canonical formulation. We address this question by proposing a second-quantized approach in which the elementary particle is the QT particle itself, leading naturally to the notion of spacetime field algebras and of quantum action. We show, however, that a naive many-body construction runs into inconsistencies. To pinpoint their origin we introduce a classical counterpart of the second-quantized formalism, spacetime classical mechanics (SCM), and prove a no-go theorem: Dirac quantization of SCM collapses back to standard QFT and therefore hides covariance. We circumvent this problem by presenting a quantum-action-based quantization that yields a spacetime version of quantum mechanics (SQM), making covariance manifest for (interacting) QFTs. Finally, we show that this resolution is tied to a genuine spacetime generalization of the notion of a quantum state, required by causality and closely connected to recent “states over time” proposals and, in dS/CFT–motivated settings, to microscopic notions of timelike entanglement and emergent time. Full article
(This article belongs to the Special Issue Time in Quantum Mechanics)
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21 pages, 1058 KB  
Article
Sequential Change Detection with Local Differential Privacy
by Lixing Zhang, Xuran Liu, Ruizhi Zhang and Liyan Xie
Entropy 2026, 28(4), 402; https://doi.org/10.3390/e28040402 - 2 Apr 2026
Viewed by 937
Abstract
Sequential change detection is a fundamental problem in statistics and signal processing, with the CUSUM procedure widely used to achieve minimax detection delay under a prescribed false alarm rate when pre- and post-change distributions are fully known. However, in many practical settings, raw [...] Read more.
Sequential change detection is a fundamental problem in statistics and signal processing, with the CUSUM procedure widely used to achieve minimax detection delay under a prescribed false alarm rate when pre- and post-change distributions are fully known. However, in many practical settings, raw observations cannot be shared with a trusted central curator, and privacy must be enforced at the data source, which prevents the computation of exact CUSUM statistics. We therefore introduce a local differentially private (DP) variant called LDP-CUSUM, which first applies a local DP mechanism to transform the raw data into privatized observations and then applies a CUSUM procedure to detect the change. We derive closed-form bounds on the average run length to false alarm and on the worst-case average detection delay, explicitly characterizing the tradeoff among privacy level, false alarm rate, and detection efficiency. Numerical simulations and a real-data case study were conducted to demonstrate the detection efficiency of our proposed LDP-CUSUM across various scenarios. Full article
(This article belongs to the Special Issue Information-Theoretic Security and Privacy)
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16 pages, 526 KB  
Article
Symmetric and Antisymmetric Quantum States from Graph Structure and Orientation
by Matheus R. de Jesus, Eduardo O. C. Hoefel and Renato M. Angelo
Entropy 2026, 28(4), 386; https://doi.org/10.3390/e28040386 - 1 Apr 2026
Viewed by 720
Abstract
Graph states provide a powerful framework for describing multipartite entanglement in quantum information science. In their standard formulation, graph states are generated by controlled-Z interactions and naturally encode symmetric exchange properties. Here we establish a precise correspondence between graph topology and exchange [...] Read more.
Graph states provide a powerful framework for describing multipartite entanglement in quantum information science. In their standard formulation, graph states are generated by controlled-Z interactions and naturally encode symmetric exchange properties. Here we establish a precise correspondence between graph topology and exchange symmetry by proving that a graph state is fully symmetric under particle permutations if and only if the underlying graph is complete. We then introduce a generalized graph-based construction using a non-commutative two-qudit gate, denoted GR, which requires directed edges and an explicit vertex ordering. We show that complete directed graphs generate fully antisymmetric multipartite states when endowed with appropriate orientations. Together, these results provide a unified graph-theoretic description of bosonic and fermionic exchange symmetry based on graph completeness and edge orientation. Full article
(This article belongs to the Special Issue Graph Theory and Its Applications in Quantum Mechanics)
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23 pages, 401 KB  
Article
Econophysics and the Landauer Principle: A Redefinition of Economic Temperature
by Edward Bormashenko and Igor Shendrik
Entropy 2026, 28(4), 376; https://doi.org/10.3390/e28040376 - 26 Mar 2026
Cited by 1 | Viewed by 2126
Abstract
A redefinition of key notions in econophysics based on the Landauer principle is proposed. Marginal economic temperature is defined via the economic Landauer principle and is proportional to the minimal monetary cost associated with erasing or transmitting one bit of information in a [...] Read more.
A redefinition of key notions in econophysics based on the Landauer principle is proposed. Marginal economic temperature is defined via the economic Landauer principle and is proportional to the minimal monetary cost associated with erasing or transmitting one bit of information in a given economic system. The actual economic temperature is defined as the effective monetary cost per irreversibly processed bit in a given economic system. The introduced definitions are particularly relevant for high-frequency trading environments. The Clausius formulation of the Second Law is reformulated for economic systems as the statement that money cannot spontaneously flow from a colder economic subsystem to a hotter one. Economic analogues of Carnot and Szilard engines are analyzed. The Carathéodory formulation of the Second Law is reformulated as follows: in every neighborhood of any equilibrium economic state there exist states that cannot be reached by processes that do not expend money or information. An optimal-power Curzon–Ahlborn economic engine is discussed. Illustrative examples of the introduced economic temperature are provided. Finally, a redefinition of economic entropy based on information processing is proposed. First and Third Laws of economic thermodynamics are proposed. Full article
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16 pages, 527 KB  
Article
Evaluating Sample-Based Krylov Quantum Diagonalization for Heisenberg Models with Applications to Materials Science
by Neel Misciasci, Roman Firt, Jonathan E. Mueller, Triet Friedhoff, Chinonso Onah, Aaron Schulze and Sarah Mostame
Entropy 2026, 28(4), 367; https://doi.org/10.3390/e28040367 - 24 Mar 2026
Viewed by 1475
Abstract
We evaluate the Sample-based Krylov Quantum Diagonalization (SKQD) algorithm on one- and two-dimensional Heisenberg models, including strongly correlated regimes in which the ground state is dense. Using problem-informed initial states and magnetization sector sweeps, we investigate SKQD for problems with non-sparse ground states, [...] Read more.
We evaluate the Sample-based Krylov Quantum Diagonalization (SKQD) algorithm on one- and two-dimensional Heisenberg models, including strongly correlated regimes in which the ground state is dense. Using problem-informed initial states and magnetization sector sweeps, we investigate SKQD for problems with non-sparse ground states, where energy accuracy and sampling efficiency are theoretically anticipated to degrade. Our studies reveal that SKQD reproduces ground-state energies and field-dependent magnetization across a range of anisotropies. Benchmarks against DMRG and exact diagonalization show consistent qualitative agreement, with accuracy improving systematically in more anisotropic regimes. We further demonstrate SKQD on quantum hardware by implementing 18- and 30-qubit Heisenberg chains, obtaining magnetization curves that match theoretical expectations. Simulations on the IBM Nighthawk processor for 64-qubit two-dimensional square lattice systems further indicate that the method remains effective beyond one-dimensional geometries. Full article
(This article belongs to the Section Quantum Information)
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29 pages, 1964 KB  
Article
Unified Space–Time-Message Interference Alignment: An End-to-End Learning Approach
by Elaheh Sadeghabadi and Steven Blostein
Entropy 2026, 28(2), 249; https://doi.org/10.3390/e28020249 - 21 Feb 2026
Viewed by 559
Abstract
This paper investigates the performance of a multi-user multiple-input single-output (MU-MISO) broadcast channel under the practical constraints of imperfect, delayed, and quantized channel state information at the transmitter (CSIT). Conventional interference alignment (IA) strategies—classified into spatial (SIA), temporal (TIA), and message-domain (MIA) techniques— [...] Read more.
This paper investigates the performance of a multi-user multiple-input single-output (MU-MISO) broadcast channel under the practical constraints of imperfect, delayed, and quantized channel state information at the transmitter (CSIT). Conventional interference alignment (IA) strategies—classified into spatial (SIA), temporal (TIA), and message-domain (MIA) techniques— typically designed for specific, idealized CSI regimes and often rely on successive interference cancellation (SIC) at the receiver. However, the iterative structure of SIC is highly susceptible to error propagation, particularly under CSI uncertainty and high-order modulation. We propose Deep-STMIA, a novel end-to-end deep learning framework that jointly optimizes interference management across the space, time, and message domains. Using a neural network-based autoencoder architecture with structural message-domain regularization, Deep-STMIA learns to mitigate the catastrophic effects of error propagation and adapts to a continuum of CSIT conditions. Simulation results demonstrate that Deep-STMIA matches the performance of degrees-of-freedom (DoF) optimal benchmarks in extreme CSI regimes and significantly outperforms state-of-the-art baselines, such as rate-splitting multiple access (RSMA), in practical imperfect CSIT scenarios. Full article
(This article belongs to the Special Issue Wireless Communications: Signal Processing Perspectives, 2nd Edition)
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13 pages, 1844 KB  
Article
The Properties of Plasma Sheath Containing the Primary Electrons with a Cairns Distribution
by Yida Zhang and Jiulin Du
Entropy 2026, 28(2), 237; https://doi.org/10.3390/e28020237 - 18 Feb 2026
Viewed by 868
Abstract
We study the properties of a plasma sheath containing cold positive ions, secondary electrons, and primary electrons with a Cairns distribution (a non-thermal velocity distribution). We derive the generalized Bohm criterion and Bohm speed, the new floating potential at the wall, and the [...] Read more.
We study the properties of a plasma sheath containing cold positive ions, secondary electrons, and primary electrons with a Cairns distribution (a non-thermal velocity distribution). We derive the generalized Bohm criterion and Bohm speed, the new floating potential at the wall, and the new critical secondary electron emission coefficient. We show that these properties of the plasma sheath depend significantly on the α-parameter in the non-thermal α-distribution, and so they are generally different from those of the plasma sheath if the primary electrons were assumed to be a Maxwellian distribution. Full article
(This article belongs to the Special Issue Nonextensive Statistical Mechanics in Astrophysics)
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47 pages, 645 KB  
Review
A Survey of Lattice-Based Physical-Layer Security for Wireless Systems with p-Modular Lattice Constructions
by Hassan Khodaiemehr, Khadijeh Bagheri, Amin Mohajer, Chen Feng, Daniel Panario and Victor C. M. Leung
Entropy 2026, 28(2), 235; https://doi.org/10.3390/e28020235 - 18 Feb 2026
Viewed by 847
Abstract
Physical-layer security (PLS) provides an information-theoretic framework for securing wireless communications by exploiting channel and signal-structure asymmetries, thereby avoiding reliance on computational hardness assumptions. Within this setting, lattice codes and their algebraic constructions play a central role in achieving secrecy over Gaussian and [...] Read more.
Physical-layer security (PLS) provides an information-theoretic framework for securing wireless communications by exploiting channel and signal-structure asymmetries, thereby avoiding reliance on computational hardness assumptions. Within this setting, lattice codes and their algebraic constructions play a central role in achieving secrecy over Gaussian and fading wiretap channels. This article offers a comprehensive survey of lattice-based wiretap coding, covering foundational concepts in algebraic number theory, Construction A over number fields, and the structure of modular and unimodular lattice families. We review key secrecy metrics, including secrecy gain, flatness factor, and equivocation, and consolidate classical and recent results to provide a unified perspective that links wireless-channel models with their underlying algebraic lattice structures. In addition, we review a newly proposed family of p-modular lattices in Khodaiemehr, H., 2018 constructed from cyclotomic fields Q(ζp) for primes p≡1(mod4) via a generalized Construction A framework. We characterize their algebraic and geometric properties and establish a non-existence theorem showing that such constructions cannot be extended to prime-power cyclotomic fields Q(ζpn) with n>1. Finally, motivated by the fact that these p-modular lattices naturally yield mixed-signature structures for which classical theta series diverge, we integrate recent advances on indefinite theta series and modular completions. Drawing on Vignéras’ differential framework and generalized error functions, we outline how modularly completed indefinite theta series provide a principled analytic foundation for defining secrecy-relevant quantities in the indefinite setting. Overall, this work serves both as a survey of algebraic lattice techniques for PLS and as a source of new design insights for secure wireless communication systems. Full article
(This article belongs to the Special Issue Wireless Communications: Signal Processing Perspectives, 2nd Edition)
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20 pages, 1721 KB  
Article
RL-Based Parallel LDPC Decoding with Clustered Scheduling
by Yusuf Ozkan, Yauhen Yakimenka and Jörg Kliewer
Entropy 2026, 28(2), 215; https://doi.org/10.3390/e28020215 - 12 Feb 2026
Viewed by 1041
Abstract
We propose a reinforcement learning (RL)-based decoding framework for high-throughput parallel decoding of low-density parity-check (LDPC) codes using clustered scheduling. Parallel LDPC decoders must balance error-correction performance and decoding latency while avoiding memory conflicts. To address this trade-off, we construct clusters of check [...] Read more.
We propose a reinforcement learning (RL)-based decoding framework for high-throughput parallel decoding of low-density parity-check (LDPC) codes using clustered scheduling. Parallel LDPC decoders must balance error-correction performance and decoding latency while avoiding memory conflicts. To address this trade-off, we construct clusters of check nodes that satisfy a two-edge independence property, which enables conflict-free row-parallel belief propagation. An RL agent is trained offline to assign Q-values to clusters and to prioritize their update order during decoding. To overcome the exponential storage requirements of existing RL-based scheduling methods, we introduce the Q-Sum method, which approximates cluster-level Q-values as the sum of Q-values of individual check nodes, reducing storage complexity from exponential to linear in the number of check nodes. We further propose an On-the-Fly clustering strategy that enforces two-edge independence dynamically during decoding and provides additional flexibility when static clustering is not feasible. Simulation results for array-based LDPC codes over additive white Gaussian noise (AWGN) channels show that the proposed methods improve the latency-versus-performance trade-off of parallel LDPC decoders, achieving lower decoding latency and higher throughput while maintaining error rates comparable to state-of-the-art decoding methods. Full article
(This article belongs to the Special Issue Coding Theory and Its Applications)
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20 pages, 597 KB  
Article
BeamNet: Unsupervised Beamforming for ISAC Systems Under Imperfect CSI
by Helitha Nimnaka, Samiru Gayan, Ruhui Zhang, Hazer Inaltekin and H. Vincent Poor
Entropy 2026, 28(2), 175; https://doi.org/10.3390/e28020175 - 3 Feb 2026
Viewed by 1392
Abstract
Integrated sensing and communication (ISAC) is expected to be a key enabler for future wireless networks, improving spectral and hardware efficiency by jointly performing radar sensing and wireless communication within a unified framework. This paper proposes BeamNet, an unsupervised deep learning framework [...] Read more.
Integrated sensing and communication (ISAC) is expected to be a key enabler for future wireless networks, improving spectral and hardware efficiency by jointly performing radar sensing and wireless communication within a unified framework. This paper proposes BeamNet, an unsupervised deep learning framework for transmit beamforming in dual-function radar-communication systems operating over general fading with imperfect channel state information (CSI). BeamNet maps noisy estimates of the communication and sensing channels to a transmit beamforming vector and is trained end-to-end by maximizing a weighted sum of the communication rate (CR) and sensing rate (SR), thereby learning the CR–SR Pareto frontier without beamforming labels or embedded optimization solvers. Using Rayleigh fading with perfect CSI, we first show that BeamNet reproduces the analytical Pareto-optimal beamforming solutions. We then use BeamNet to characterize, for Nakagami-m and Rician fading, the CR–SR trade-off across a range of fading parameters, and to assess robustness under distribution mismatch between training and test channels. Finally, under imperfect CSI, we demonstrate that BeamNet yields CR–SR trade-offs that are consistently sandwiched between the perfect-CSI and mismatched analytical baselines, outperforming the closed-form beamformer applied to imperfect CSI and recovering part of the performance loss caused by channel estimation errors. These results indicate that unsupervised learning offers a flexible and robust approach to ISAC beamforming in fading environments with imperfect channel knowledge. Full article
(This article belongs to the Special Issue Joint Sensing, Communication, and Computation)
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34 pages, 2540 KB  
Article
Way More than the Sum of Their Parts: From Statistical to Structural Mixtures
by James P. Crutchfield
Entropy 2026, 28(1), 111; https://doi.org/10.3390/e28010111 - 16 Jan 2026
Viewed by 757
Abstract
We show that mixtures comprising multicomponent systems typically are much more structurally complex than the sum of their parts; sometimes, infinitely more complex. We contrast this with the more familiar notion of statistical mixtures, demonstrating how statistical mixtures miss key aspects of emergent [...] Read more.
We show that mixtures comprising multicomponent systems typically are much more structurally complex than the sum of their parts; sometimes, infinitely more complex. We contrast this with the more familiar notion of statistical mixtures, demonstrating how statistical mixtures miss key aspects of emergent hierarchical organization. This leads us to identify a new kind of structural complexity inherent in multicomponent systems and to draw out broad consequences for system ergodicity. Full article
(This article belongs to the Section Statistical Physics)
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10 pages, 257 KB  
Article
Kolmogorovian Censorship, Predictive Incompleteness, and the Locality Loophole in Bell Experiments
by Philippe Grangier
Entropy 2026, 28(1), 80; https://doi.org/10.3390/e28010080 - 10 Jan 2026
Viewed by 900
Abstract
We revisit the status of quantum probabilities in light of Kolmogorovian Censorship (KC) and the Contexts, Systems, and Modalities (CSM) framework, and we discuss KC-based ideas with respect to superdeterminism, counterfactuality, and predictive incompleteness. After briefly recalling the technical content of KC and [...] Read more.
We revisit the status of quantum probabilities in light of Kolmogorovian Censorship (KC) and the Contexts, Systems, and Modalities (CSM) framework, and we discuss KC-based ideas with respect to superdeterminism, counterfactuality, and predictive incompleteness. After briefly recalling the technical content of KC and its scope, we show that KC correctly identifies that probabilities are classical within a fixed measurement context but does not by itself remove the conceptual tension that motivates nonlocal or conspiratorial explanations of Bell inequality violations. We argue that predictive incompleteness—the view that the quantum state is operationally incomplete until the measurement context is specified—provides a simple, minimal, and explanatory framework that preserves relativistic locality while matching experimental practice. Finally we clarify logical relations among these positions, highlight the assumptions behind them, and justify the move from Kolmogorov’s to Gleason’s framework for quantum probabilities. Full article
(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering IV)
19 pages, 335 KB  
Article
Refinements and Generalizations of the Shannon Lower Bound via Extensions of the Kraft Inequality
by Neri Merhav
Entropy 2026, 28(1), 76; https://doi.org/10.3390/e28010076 - 9 Jan 2026
Cited by 1 | Viewed by 946
Abstract
We derive a few extended versions of the Kraft inequality for lossy compression, which pave the way to the derivation of several refinements and extensions of the well-known Shannon lower bound in a variety of instances of rate-distortion coding. These refinements and extensions [...] Read more.
We derive a few extended versions of the Kraft inequality for lossy compression, which pave the way to the derivation of several refinements and extensions of the well-known Shannon lower bound in a variety of instances of rate-distortion coding. These refinements and extensions include sharper bounds for one-to-one codes and D-semifaithful codes, a Shannon lower bound for distortion measures based on sliding-window functions, and an individual-sequence counterpart of the Shannon lower bound. Full article
(This article belongs to the Special Issue Information Theory and Data Compression)
20 pages, 1577 KB  
Article
Unraveling the Network Signatures of Oncogenicity in Virus–Human Protein–Protein Interactions
by Francesco Zambelli, Vera Pancaldi and Manlio De Domenico
Entropy 2025, 27(12), 1248; https://doi.org/10.3390/e27121248 - 11 Dec 2025
Viewed by 986
Abstract
Background: Climate change, urbanization, and global mobility increase the risk of emerging infectious diseases with pandemic potential. There is a need for rapid methods that can assess their long-term effects on human health. In silico approaches are particularly suited to study processes that [...] Read more.
Background: Climate change, urbanization, and global mobility increase the risk of emerging infectious diseases with pandemic potential. There is a need for rapid methods that can assess their long-term effects on human health. In silico approaches are particularly suited to study processes that may manifest years later, under the assumption that perturbed biomolecular interactions underlie these outcomes. Here we focus on viral oncogenicity—the ability of viruses to increase cancer risk—which accounts for about 15% of global cancer cases. Methods: We characterize viruses through multilayer representations of protein–protein interaction (PPI) networks reconstructed from the human interactome. Statistical analyses of topological features, combined with interpretable machine learning models, are used to distinguish oncogenic from non-oncogenic viruses and to identify proteins with potential central role in these processes. Results: Our analysis reveals clear statistical differences between the network properties of oncogenic and non-oncogenic viruses. Furthermore, the machine learning approach enables classification of virus–host interaction networks and identification of relevant subsets of proteins associated with oncogenesis. Functional enrichment analysis highlights mechanisms related to viral oncogenicity, including chromatin structure and other processes linked to cancer development. Conclusions: This framework enables virus classification and highlights mechanisms underlying viral oncogenicity, providing a foundation for investigating long-term health effects of emerging pathogens. Full article
(This article belongs to the Special Issue Statistical and Physics-Based Interpretation of AI and Machine Learning Methods for Biomedical Data)
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19 pages, 2361 KB  
Article
Detrended Cross-Correlations and Their Random Matrix Limit: An Example from the Cryptocurrency Market
by Stanisław Drożdż, Paweł Jarosz, Jarosław Kwapień, Maria Skupień and Marcin Wątorek
Entropy 2025, 27(12), 1236; https://doi.org/10.3390/e27121236 - 6 Dec 2025
Cited by 2 | Viewed by 1435
Abstract
Correlations in complex systems are often obscured by nonstationarity, long-range memory, and heavy-tailed fluctuations, which limit the usefulness of traditional covariance-based analyses. To address these challenges, we construct scale- and fluctuation-dependent correlation matrices using the multifractal detrended cross-correlation coefficient ρr that selectively [...] Read more.
Correlations in complex systems are often obscured by nonstationarity, long-range memory, and heavy-tailed fluctuations, which limit the usefulness of traditional covariance-based analyses. To address these challenges, we construct scale- and fluctuation-dependent correlation matrices using the multifractal detrended cross-correlation coefficient ρr that selectively emphasizes fluctuations of different amplitudes. We examine the spectral properties of these detrended correlation matrices and compare them to the spectral properties of the matrices calculated in the same way from synthetic Gaussian and q-Gaussian signals. Our results show that detrending, heavy tails, and the fluctuation-order parameter r jointly produce spectra, which substantially depart from the random case even under the absence of cross-correlations in time series. Applying this framework to one-minute returns of 140 major cryptocurrencies from 2021 to 2024 reveals robust collective modes, including a dominant market factor and several sectoral components whose strength depends on the analyzed scale and fluctuation order. After filtering out the market mode, the empirical eigenvalue bulk aligns closely with the limit of random detrended cross-correlations, enabling clear identification of structurally significant outliers. Overall, the study provides a refined spectral baseline for detrended cross-correlations and offers a promising tool for distinguishing genuine interdependencies from noise in complex, nonstationary, heavy-tailed systems. Full article
(This article belongs to the Special Issue Entropy, Econophysics, and Complexity)
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104 pages, 2313 KB  
Review
Twist Fields in Many-Body Physics
by Benjamin Doyon
Entropy 2025, 27(12), 1230; https://doi.org/10.3390/e27121230 - 4 Dec 2025
Cited by 1 | Viewed by 1009
Abstract
The notion of twist fields has played a fundamental role in many-body physics. It is used to construct the so-called disorder parameter for the study of phase transitions in the classical Ising model of statistical mechanics, it is involved in the Jordan–Wigner transformation [...] Read more.
The notion of twist fields has played a fundamental role in many-body physics. It is used to construct the so-called disorder parameter for the study of phase transitions in the classical Ising model of statistical mechanics, it is involved in the Jordan–Wigner transformation in quantum chains and bosonisation in quantum field theory, and it is related to measures of entanglement in many-body quantum systems. I provide a pedagogical introduction to the notion of twist field and the concepts at its roots, and review some of its applications, focussing on the 1 + 1 dimension. This includes locality and extensivity, internal symmetries, semi-locality, the standard exponential form and HEGT fields, path-integral defects and Riemann surfaces, topological invariance, and twist families. Additional topics touched upon include renormalisation and form factors in relativistic quantum field theory, tau functions of integrable PDEs, thermodynamic and hydrodynamic principles, and branch-point twist fields for entanglement entropy. One-dimensional quantum systems such as chains (e.g., quantum Heisenberg model) and field theory (e.g., quantum sine-Gordon model) are the main focus, but I also explain how the notion applies to equilibrium statistical mechanics (e.g., classical Ising lattice model), and how some aspects can be adapted to one-dimensional classical dynamical systems (e.g., classical Toda chain). Full article
(This article belongs to the Special Issue Entanglement Entropy in Quantum Field Theory)
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20 pages, 4235 KB  
Article
Geometry-Based Bounds on the Capacity of Peak-Limited and Band-Limited Signals over the Additive White Gaussian Noise Channel at a High SNR
by Michael Peleg and Shlomo Shamai
Entropy 2025, 27(12), 1192; https://doi.org/10.3390/e27121192 - 24 Nov 2025
Cited by 1 | Viewed by 1024
Abstract
We present a new computable geometry-based upper bound on the capacity of peak-power-limited and band-limited signal over the Additive White Gaussian Noise Channel. The peak limit applies at continuous time. The bound is a function of the volume and shape of the transmitted [...] Read more.
We present a new computable geometry-based upper bound on the capacity of peak-power-limited and band-limited signal over the Additive White Gaussian Noise Channel. The peak limit applies at continuous time. The bound is a function of the volume and shape of the transmitted signal set, namely the body, in the space of Nyquist-rate samples, comprising all of the points the transmitted signal can reach. At a high SNR, the bound is tight, better than previously known upper bounds and, together with a known lower bound, provides the capacity at an asymptotically high SNR. We found, using a numerical evaluation, the high-SNR capacity of signals with the structure used in Cyclic Prefix assisted Frequency Domain Equalization (CP-FDE) and OFDM for sequence length of up to 100 Nyquist intervals, and we present a conjecture that this result is correct for any sequence length and does not depend on the CPA-FDE structure. This paper extends the methodology developed in previous works. The penalty in power efficiency at a high SNR due to the peak power constraint relative to an average power constraint is about 7.5 dB in the low-pass case and about 5.4 dB in the band-pass case. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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22 pages, 3086 KB  
Article
Nonclassicality and Coherent Error Detection via Pseudo-Entropy
by Assaf Katz, Shalom Bloch and Eliahu Cohen
Entropy 2025, 27(11), 1165; https://doi.org/10.3390/e27111165 - 17 Nov 2025
Viewed by 1055
Abstract
Pseudo-entropy is a complex-valued generalization of entanglement entropy defined on non-Hermitian transition operators and induced by post-selection. We present a simulation-based protocol for detecting nonclassicality and coherent errors in quantum circuits using this pseudo-entropy measure Sˇ, focusing on its imaginary part [...] Read more.
Pseudo-entropy is a complex-valued generalization of entanglement entropy defined on non-Hermitian transition operators and induced by post-selection. We present a simulation-based protocol for detecting nonclassicality and coherent errors in quantum circuits using this pseudo-entropy measure Sˇ, focusing on its imaginary part ℑSˇ as a diagnostic tool. Our method enables resource-efficient classification of phase-coherent errors, such as those from miscalibrated CNOT gates, even under realistic noise conditions. By quantifying the transition between classical-like and quantum-like behavior through threshold analysis, we provide theoretical benchmarks for error classification that can inform hardware calibration strategies. Numerical simulations demonstrate that 55% of the parameter space remains classified as classical-like (below classification thresholds) at hardware-calibrated sensitivity levels, with statistical significance confirmed through rigorous sensitivity analysis. Robustness to noise and comparison with standard entropy-based methods are demonstrated in a simulation. While hardware validation remains necessary, this work bridges theoretical concepts of nonclassicality with practical quantum error classification frameworks, providing a foundation for experimental quantum computing applications. Full article
(This article belongs to the Special Issue Classical and Quantum Networks: Theory, Modeling and Optimization, 2nd Edition)
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19 pages, 321 KB  
Article
Entropy Production and Irreversibility in the Linearized Stochastic Amari Neural Model
by Dario Lucente, Giacomo Gradenigo and Luca Salasnich
Entropy 2025, 27(11), 1104; https://doi.org/10.3390/e27111104 - 25 Oct 2025
Viewed by 1813
Abstract
One among the most intriguing results coming from the application of statistical mechanics to the study of the brain is the understanding that it, as a dynamical system, is inherently out of equilibrium. In the realm of non-equilibrium statistical mechanics and stochastic processes, [...] Read more.
One among the most intriguing results coming from the application of statistical mechanics to the study of the brain is the understanding that it, as a dynamical system, is inherently out of equilibrium. In the realm of non-equilibrium statistical mechanics and stochastic processes, the standard observable computed to determine whether a system is at equilibrium or not is the entropy produced along the dynamics. For this reason, we present here a detailed calculation of the entropy production in the Amari model, a coarse-grained model of the brain neural network, consisting of an integro-differential equation for the neural activity field, when stochasticity is added to the original dynamics. Since the way to add stochasticity is always to some extent arbitrary, particularly for coarse-grained models, there is no general prescription to do so. We precisely investigate the interplay between noise properties and the original model features, discussing in which cases the stationary state is in thermal equilibrium and which cases it is out of equilibrium, providing explicit and simple formulae. Following the derivation for the particular case considered, we also show how the entropy production rate is related to the variation in time of the Shannon entropy of the system. Full article
(This article belongs to the Section Non-equilibrium Phenomena)
16 pages, 1206 KB  
Article
Contrast Analysis on Spin Transport of Multi-Periodic Exotic States in the XXZ Chain
by Shixian Jiang, Jianpeng Liu and Yongqiang Li
Entropy 2025, 27(10), 1070; https://doi.org/10.3390/e27101070 - 15 Oct 2025
Viewed by 1179
Abstract
Quantum spin transport in integrable systems reveals a rich nonequilibrium phenomena that challenges the conventional hydrodynamic framework. Recent advances in ultracold atom experiments with state preparation and single-site addressing have enabled the understanding of this anomalous behavior. Particularly, the full universality characterization of [...] Read more.
Quantum spin transport in integrable systems reveals a rich nonequilibrium phenomena that challenges the conventional hydrodynamic framework. Recent advances in ultracold atom experiments with state preparation and single-site addressing have enabled the understanding of this anomalous behavior. Particularly, the full universality characterization of exotic initial states, as well as their measurement representation, remain unknown. By employing tensor network and contrast methods, we systematically investigate spin transport in the quantum XXZ spin chain and extract dynamical scaling exponents emerging from two paradigmatic and experimentally attainable initial states, i.e., multi-periodic domain-wall (MPDW) and spin-helix (SH) states. Our results using different values of anisotropic parameters Δ∈[0,1.2] demonstrate the evident impeded transport and the difference between the two states with increasing Δ values. Large-scale and consistent simulations confirm the contrast method as a viable scaling extraction approach for exotic states with periodicity within experimentally accessible timescales. Our work establishes a foundation for studying initial memory and the corresponding relations of emergent transport behavior in nonequilibrium quantum systems, opening avenues for the identification of their unique universality classes. Full article
(This article belongs to the Special Issue Emergent Phenomena in Quantum Many-Body Systems)
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11 pages, 275 KB  
Article
Relativistic Limits on the Discretization and Temporal Resolution of a Quantum Clock
by Tommaso Favalli
Entropy 2025, 27(10), 1068; https://doi.org/10.3390/e27101068 - 14 Oct 2025
Cited by 2 | Viewed by 889
Abstract
We provide a brief discussion regarding relativistic limits on the discretization and temporal resolution of time values in a quantum clock. Our clock is characterized by a time observable chosen to be the complement of a bounded and discrete Hamiltonian that can have [...] Read more.
We provide a brief discussion regarding relativistic limits on the discretization and temporal resolution of time values in a quantum clock. Our clock is characterized by a time observable chosen to be the complement of a bounded and discrete Hamiltonian that can have an equally spaced or a generic spectrum. In the first case, the time observable can be described by a Hermitian operator, and we find a limit in the discretization for the time eigenvalues. Nevertheless, in both cases, the time observable can be described by a POVM, and, by increasing the number of time states, we show how the bound on the minimum time quantum can be reduced and identify the conditions under which the clock values can be treated as continuous. Finally, we find a limit for the temporal resolution of our time observable when the clock is used (together with light signals) in a relativistic framework for the measurement of spacetime distances. Full article
(This article belongs to the Special Issue Time in Quantum Mechanics)
11 pages, 285 KB  
Article
Local Invariance of Divergence-Based Quantum Information Measures
by Christopher Popp, Tobias C. Sutter and Beatrix C. Hiesmayr
Entropy 2025, 27(10), 1051; https://doi.org/10.3390/e27101051 - 10 Oct 2025
Cited by 2 | Viewed by 1121
Abstract
Quantum information quantities, such as mutual information and entropies, are essential for characterizing quantum systems and protocols in quantum information science. In this contribution, we identify types of information measures based on generalized divergences and prove their invariance under local isometric or unitary [...] Read more.
Quantum information quantities, such as mutual information and entropies, are essential for characterizing quantum systems and protocols in quantum information science. In this contribution, we identify types of information measures based on generalized divergences and prove their invariance under local isometric or unitary transformations. Leveraging the reversal channel for local isometries together with the data-processing inequality, we establish invariance for information quantities used in both asymptotic and one-shot regimes without relying on the specific functional form of the underlying divergence. These invariances can be applied to improve the computation of such information quantities or optimize protocols and their output states, whose performance is determined by some invariant measure. Our results improve the capability to characterize and compute many operationally relevant information measures with application across the field of quantum information processing. Full article
(This article belongs to the Section Quantum Information)
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12 pages, 1262 KB  
Article
Ordinal Spectrum: Mapping Ordinal Patterns into Frequency Domain
by Mario Chavez and Johann H. Martínez
Entropy 2025, 27(10), 1027; https://doi.org/10.3390/e27101027 - 30 Sep 2025
Viewed by 1447
Abstract
Classical spectral analysis characterizes linear systems effectively but often fails to reveal the nonlinear temporal structure of chaotic dynamics. We introduce the ordinal spectrum, a frequency-domain characterization derived from the ordinal-pattern representation of a time series. Applied to both synthetic and real-world [...] Read more.
Classical spectral analysis characterizes linear systems effectively but often fails to reveal the nonlinear temporal structure of chaotic dynamics. We introduce the ordinal spectrum, a frequency-domain characterization derived from the ordinal-pattern representation of a time series. Applied to both synthetic and real-world datasets—including periodic, stochastic, and chaotic signals from physical, biological, and astronomical sources—the ordinal spectrum identifies the temporal scales implied in a possible chaotic behavior. By providing an interpretable, data-driven view of symbolic dynamics in the frequency domain, this approach complements state–space reconstructions and enhances the detection of nonlinear temporal organization that classical spectra may obscure. Its ability to distinguish between qualitatively different dynamics make it a useful tool for exploring complex time series across diverse scientific domains. Full article
(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)
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33 pages, 3062 KB  
Article
Gradient-Free De Novo Learning
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
Cited by 2 | Viewed by 3568
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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15 pages, 17666 KB  
Article
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
Cited by 1 | Viewed by 1172
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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24 pages, 756 KB  
Article
Complex Time Approach to the Hamiltonian and the Entropy Production of the Damped Harmonic Oscillator
by Kyriaki-Evangelia Aslani
Entropy 2025, 27(8), 883; https://doi.org/10.3390/e27080883 - 21 Aug 2025
Cited by 1 | Viewed by 2418
Abstract
The present work applies and extends the previously developed Quantitative Geometrical Thermodynamics (QGT) formalism to the derivation of a Hamiltonian for the damped harmonic oscillator (DHO) across all damping regimes. By introducing complex time, with the real part encoding entropy production and the [...] Read more.
The present work applies and extends the previously developed Quantitative Geometrical Thermodynamics (QGT) formalism to the derivation of a Hamiltonian for the damped harmonic oscillator (DHO) across all damping regimes. By introducing complex time, with the real part encoding entropy production and the imaginary part governing reversible dynamics, QGT provides a unified geometric framework for irreversible thermodynamics, showing that the DHO Hamiltonian can be obtained directly from the (complex) entropy production in a simple exponential form that is generalized across all damping regimes. The derived Hamiltonian preserves a modified Poisson bracket structure and embeds thermodynamic irreversibility into the system’s evolution. Moreover, the resulting expression coincides in form with the well-known Caldirola–Kanai Hamiltonian, despite arising from fundamentally different principles, reinforcing the validity of the QGT approach. The results are also compared with the GENERIC framework, showing that QGT offers an elegant alternative to existing approaches that maintains consistency with symplectic geometry. Furthermore, the imaginary time component is interpreted as isomorphic to the antisymmetric Poisson matrix through the lens of geometric algebra. The formalism opens promising avenues for extending Hamiltonian mechanics to dissipative systems, with potential applications in nonlinear dynamics, quantum thermodynamics, and spacetime algebra. Full article
(This article belongs to the Special Issue Geometry in Thermodynamics, 4th Edition)
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23 pages, 8311 KB  
Article
Active Inference with Dynamic Planning and Information Gain in Continuous Space by Inferring Low-Dimensional Latent States
by Takazumi Matsumoto, Kentaro Fujii, Shingo Murata and Jun Tani
Entropy 2025, 27(8), 846; https://doi.org/10.3390/e27080846 - 9 Aug 2025
Cited by 2 | Viewed by 3889
Abstract
Active inference offers a unified framework in which agents can exhibit both goal-directed and epistemic behaviors. However, implementing policy search in high-dimensional continuous action spaces presents challenges in terms of scalability and stability. Our previously proposed model, T-GLean, addressed this issue by enabling [...] Read more.
Active inference offers a unified framework in which agents can exhibit both goal-directed and epistemic behaviors. However, implementing policy search in high-dimensional continuous action spaces presents challenges in terms of scalability and stability. Our previously proposed model, T-GLean, addressed this issue by enabling efficient goal-directed planning through low-dimensional latent space search, further reduced by conditioning on prior habituated behavior. However, the lack of an epistemic term in minimizing expected free energy limited the agent’s ability to engage in information-seeking behavior that can be critical for attaining preferred outcomes. In this study, we present EFE-GLean, an extended version of T-GLean that overcomes this limitation by integrating epistemic value into the planning process. EFE-GLean generates goal-directed policies by inferring low-dimensional future posterior trajectories while maximizing expected information gain. Simulation experiments using an extended T-maze task—implemented in both discrete and continuous domains—demonstrate that the agent can successfully achieve its goals by exploiting hidden environmental information. Furthermore, we show that the agent is capable of adapting to abrupt environmental changes by dynamically revising plans through simultaneous minimization of past variational free energy and future expected free energy. Finally, analytical evaluations detail the underlying mechanisms and computational properties of the model. Full article
(This article belongs to the Special Issue Active Inference in Cognitive Neuroscience)
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24 pages, 9086 KB  
Article
Linking Optimization Success and Stability of Finite-Time Thermodynamics Heat Engines
by Julian Gonzalez-Ayala, David Pérez-Gallego, Alejandro Medina, José M. Mateos Roco, Antonio Calvo Hernández, Santiago Velasco and Fernando Angulo-Brown
Entropy 2025, 27(8), 822; https://doi.org/10.3390/e27080822 - 2 Aug 2025
Viewed by 1253
Abstract
In celebration of 50 years of the endoreversible Carnot-like heat engine, this work aims to link the thermodynamic success of the irreversible Carnot-like heat engine with the stability dynamics of the engine. This region of success is defined by two extreme configurations in [...] Read more.
In celebration of 50 years of the endoreversible Carnot-like heat engine, this work aims to link the thermodynamic success of the irreversible Carnot-like heat engine with the stability dynamics of the engine. This region of success is defined by two extreme configurations in the interaction between heat reservoirs and the working fluid. The first corresponds to a fully reversible limit, and the second one is the fully dissipative limit; in between both limits, the heat exchange between reservoirs and working fluid produces irreversibilities and entropy generation. The distance between these two extremal configurations is minimized, independently of the chosen metric, in the state where the efficiency is half the Carnot efficiency. This boundary encloses the region where irreversibilities dominate or the reversible behavior dominates (region of success). A general stability dynamics is proposed based on the endoreversible nature of the model and the operation parameter in charge of defining the operation regime. For this purpose, the maximum ecological and maximum Omega regimes are considered. The results show that for single perturbations, the dynamics rapidly directs the system towards the success region, and under random perturbations producing stochastic trajectories, the system remains always in this region. The results are contrasted with the case in which no restitution dynamics exist. It is shown that stability allows the system to depart from the original steady state to other states that enhance the system’s performance, which could favor the evolution and specialization of systems in nature and in artificial devices. Full article
(This article belongs to the Special Issue The First Half Century of Finite-Time Thermodynamics)
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25 pages, 8472 KB  
Article
Harnessing the Power of Pre-Trained Models for Efficient Semantic Communication of Text and Images
by Emrecan Kutay and Aylin Yener
Entropy 2025, 27(8), 813; https://doi.org/10.3390/e27080813 - 29 Jul 2025
Cited by 4 | Viewed by 2364
Abstract
This paper investigates point-to-point multimodal digital semantic communications in a task-oriented setup, where messages are classified at the receiver. We employ a pre-trained transformer model to extract semantic information and propose three methods for generating semantic codewords. First, we propose semantic quantization that [...] Read more.
This paper investigates point-to-point multimodal digital semantic communications in a task-oriented setup, where messages are classified at the receiver. We employ a pre-trained transformer model to extract semantic information and propose three methods for generating semantic codewords. First, we propose semantic quantization that uses quantized embeddings of source realizations as a codebook. We investigate the fixed-length coding, considering the source semantic structure and end-to-end semantic distortion. We propose a neural network-based codeword assignment mechanism incorporating codeword transition probabilities to minimize the expected semantic distortion. Second, we present semantic compression that clusters embeddings, exploiting the inherent semantic redundancies to reduce the codebook size, i.e., further compression. Third, we introduce a semantic vector-quantized autoencoder (VQ-AE) that learns a codebook through training. In all cases, we follow this semantic source code with a standard channel code to transmit over the wireless channel. In addition to classification accuracy, we assess pre-communication overhead via a novel metric we term system time efficiency. Extensive experiments demonstrate that our proposed semantic source-coding approaches provide comparable accuracy and better system time efficiency compared to their learning-based counterparts. Full article
(This article belongs to the Special Issue Semantic Information Theory)
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23 pages, 758 KB  
Article
Low-Complexity Automorphism Ensemble Decoding of Reed-Muller Codes Using Path Pruning
by Kairui Tian, Rongke Liu and Zheng Lu
Entropy 2025, 27(8), 808; https://doi.org/10.3390/e27080808 - 28 Jul 2025
Viewed by 1481
Abstract
The newly developed automorphism ensemble decoder (AED) leverages the rich automorphisms of Reed–Muller (RM) codes to achieve near maximum likelihood (ML) performance at short code lengths. However, the performance gain of AED comes at the cost of high complexity, as the ensemble size [...] Read more.
The newly developed automorphism ensemble decoder (AED) leverages the rich automorphisms of Reed–Muller (RM) codes to achieve near maximum likelihood (ML) performance at short code lengths. However, the performance gain of AED comes at the cost of high complexity, as the ensemble size required for near ML decoding grows exponentially with the code length. In this work, we address this complexity issue by focusing on the factor graph permutation group (FGPG), a subgroup of the full automorphism group of RM codes, to generate permutations for AED. We propose a uniform partitioning of FGPG based on the affine bijection permutation matrices of automorphisms, where each subgroup of FGPG exhibits permutation invariance (PI) in a Plotkin construction-based information set partitioning for RM codes. Furthermore, from the perspective of polar codes, we exploit the PI property to prove a subcode estimate convergence (SEC) phenomenon in the AED that utilizes successive cancellation (SC) or SC list (SCL) constituent decoders. Observing that strong SEC correlates with low noise levels, where the full decoding capacity of AED is often unnecessary, we perform path pruning to reduce the decoding complexity without compromising the performance. Our proposed SEC-aided path pruning allows only a subset of constituent decoders to continue decoding when the intensity of SEC exceeds a preset threshold during decoding. Numerical results demonstrate that, for the FGPG-based AED of various short RM codes, the proposed SEC-aided path pruning technique incurs negligible performance degradation, while achieving a complexity reduction of up to 67.6%. Full article
(This article belongs to the Special Issue Next-Generation Channel Coding: Theory and Applications)
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15 pages, 5676 KB  
Article
Transverse Self-Propulsion Enhances the Aggregation of Active Dumbbells
by Pasquale Digregorio, Claudio Basilio Caporusso, Lucio Mauro Carenza, Giuseppe Gonnella, Daniela Moretti, Giuseppe Negro, Massimiliano Semeraro and Antonio Suma
Entropy 2025, 27(7), 692; https://doi.org/10.3390/e27070692 - 27 Jun 2025
Cited by 1 | Viewed by 1448
Abstract
We investigate a two-dimensional system of active Brownian dumbbells using molecular dynamics simulations. In this model, each dumbbell is driven by an active force oriented perpendicular to the axis connecting its two constituent beads. We characterize the resulting phase behavior and find that, [...] Read more.
We investigate a two-dimensional system of active Brownian dumbbells using molecular dynamics simulations. In this model, each dumbbell is driven by an active force oriented perpendicular to the axis connecting its two constituent beads. We characterize the resulting phase behavior and find that, across all values of activity, the system undergoes phase separation between dilute and dense phases. The dense phase exhibits hexatic order, and for large enough activity, we observe a marked increase in local polarization, with dumbbells predominantly oriented towards the interior of the clusters. Compared to the case of axially self-propelled dumbbells, we find that the binodal region is enlarged towards lower densities at all activities. This shift arises because dumbbells with transverse propulsion can more easily form stable cluster cores, serving as nucleation seeds, and show a highly suppressed escaping rate from the cluster boundary. Finally, we observe that clusters exhibit spontaneous rotation, with the modulus of the angular velocity scaling as ω∼rg−2, where rg is the cluster’s radius of gyration. This contrasts with axially propelled dumbbells, where the scaling follows ω∼rg−1. We develop a simplified analytical model to rationalize this scaling behavior. Full article
(This article belongs to the Section Non-equilibrium Phenomena)
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24 pages, 5959 KB  
Article
An Information Geometry-Based Track-Before-Detect Algorithm for Range-Azimuth Measurements in Radar Systems
by Jinguo Liu, Hao Wu, Zheng Yang, Xiaoqiang Hua and Yongqiang Cheng
Entropy 2025, 27(6), 637; https://doi.org/10.3390/e27060637 - 14 Jun 2025
Cited by 2 | Viewed by 2332
Abstract
The detection of weak moving targets in heterogeneous clutter backgrounds is a significant challenge in radar systems. In this paper, we propose a track-before-detect (TBD) method based on information geometry (IG) theory applied to range-azimuth measurements, which extends the IG detectors to multi-frame [...] Read more.
The detection of weak moving targets in heterogeneous clutter backgrounds is a significant challenge in radar systems. In this paper, we propose a track-before-detect (TBD) method based on information geometry (IG) theory applied to range-azimuth measurements, which extends the IG detectors to multi-frame detection through inter-frame information integration. The approach capitalizes on the distinctive benefits of the information geometry detection framework in scenarios with strong clutter, while enhancing the integration of information across multiple frames within the TBD approach. Specifically, target and clutter trajectories in multi-frame range-azimuth measurements are modeled on the Hermitian positive definite (HPD) and power spectrum (PS) manifolds. A scoring function based on information geometry, which uses Kullback–Leibler (KL) divergence as a geometric metric, is then devised to assess these motion trajectories. Moreover, this study devises a solution framework employing dynamic programming (DP) with constraints on state transitions, culminating in an integrated merit function. This algorithm identifies target trajectories by maximizing the integrated merit function. Experimental validation using real-recorded sea clutter datasets showcases the effectiveness of the proposed algorithm, yielding a minimum 3 dB enhancement in signal-to-clutter ratio (SCR) compared to traditional approaches. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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29 pages, 3108 KB  
Article
Soft Classification in a Composite Source Model
by Yuefeng Cao, Jiakun Liu and Wenyi Zhang
Entropy 2025, 27(6), 620; https://doi.org/10.3390/e27060620 - 11 Jun 2025
Cited by 1 | Viewed by 1236
Abstract
A composite source model consists of an intrinsic state and an extrinsic observation. The fundamental performance limit of reproducing the intrinsic state is characterized by the indirect rate–distortion function. In a remote classification application, a source encoder encodes the extrinsic observation (e.g., image) [...] Read more.
A composite source model consists of an intrinsic state and an extrinsic observation. The fundamental performance limit of reproducing the intrinsic state is characterized by the indirect rate–distortion function. In a remote classification application, a source encoder encodes the extrinsic observation (e.g., image) into bits, and a source decoder plays the role of a classifier that reproduces the intrinsic state (e.g., label of image). In this work, we characterize the general structure of the optimal transition probability distribution, achieving the indirect rate–distortion function. This optimal solution can be interpreted as a “soft classifier”, which generalizes the conventionally adopted “classify-then-compress” scheme. We then apply the soft classification to aid the lossy compression of the extrinsic observation of a composite source. This leads to a coding scheme that exploits the soft classifier to guide reproduction, outperforming existing coding schemes without classification or with hard classification. Full article
(This article belongs to the Special Issue Semantic Information Theory)
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15 pages, 419 KB  
Article
Ordinal Random Processes
by Christoph Bandt
Entropy 2025, 27(6), 610; https://doi.org/10.3390/e27060610 - 7 Jun 2025
Cited by 2 | Viewed by 1130
Abstract
Ordinal patterns have proven to be a valuable tool in many fields. Here, we address the need for theoretical models. A paradigmatic example shows that a model for frequencies of ordinal patterns can be determined without any numerical values. We specify the important [...] Read more.
Ordinal patterns have proven to be a valuable tool in many fields. Here, we address the need for theoretical models. A paradigmatic example shows that a model for frequencies of ordinal patterns can be determined without any numerical values. We specify the important concept of stationary order and the fundamental problems to be solved in order to establish a genuine statistical methodology for ordinal time series. Full article
(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)
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8 pages, 379 KB  
Article
Scaling Laws in Language Families
by Maelyson Rolim Fonseca dos Santos and Marcelo Andrade de Filgueiras Gomes
Entropy 2025, 27(6), 588; https://doi.org/10.3390/e27060588 - 31 May 2025
Cited by 1 | Viewed by 1434
Abstract
This article investigates scaling laws within language families using data from over six thousand languages and analyzes emergent patterns observed in Zipf-like classification graphs. Both macroscopic (based on the number of languages by family) and microscopic (based on the number of speakers by [...] Read more.
This article investigates scaling laws within language families using data from over six thousand languages and analyzes emergent patterns observed in Zipf-like classification graphs. Both macroscopic (based on the number of languages by family) and microscopic (based on the number of speakers by language within a family) aspects of these classifications are examined. Particularly noteworthy is the discovery of a distinct division among the fourteen largest contemporary language families, excluding Afro-Asiatic and Nilo-Saharan languages. These families are found to be distributed across three language family quadruplets, each characterized by significantly different exponents in the Zipf graphs. This finding sheds light on the underlying structure and organization of major language families, revealing intriguing insights into the nature of linguistic diversity and distribution. Full article
(This article belongs to the Special Issue Complexity Characteristics of Natural Language)
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23 pages, 2439 KB  
Article
The Origin of Shared Emergent Properties in Discrete Systems
by Les Hatton and Greg Warr
Entropy 2025, 27(6), 561; https://doi.org/10.3390/e27060561 - 26 May 2025
Cited by 2 | Viewed by 1808
Abstract
Here, we propose that the shared emergent properties reproducibly observed in discrete systems can be explained by a theory that embeds the Conservation of Hartley–Shannon Information (CoHSI) in a statistical mechanics framework. Specific predictions of global properties that represent the most likely equilibrium [...] Read more.
Here, we propose that the shared emergent properties reproducibly observed in discrete systems can be explained by a theory that embeds the Conservation of Hartley–Shannon Information (CoHSI) in a statistical mechanics framework. Specific predictions of global properties that represent the most likely equilibrium state should be apparent in all qualifying systems, regardless of provenance. We demonstrate that these predictions of emergent global properties hold true in systems as disparate as collections of software written in the programming language C and collections of proteins. The implication is that the emergence of such shared properties is not driven by any specific local mechanism as the systems are so different. This raises the interesting prospect that important properties of biological systems (exemplified here by the length and multiplicity distributions of proteins) have little, if anything, to do with natural selection. Similarly, the size distribution of components and the frequency of tokens observed in computer software in C emerge as the most likely states, and are thus properties that are divorced from human agency, regardless of functionality. Full article
(This article belongs to the Section Entropy and Biology)
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14 pages, 262 KB  
Article
Universal Encryption of Individual Sequences Under Maximal Information Leakage
by Neri Merhav
Entropy 2025, 27(6), 551; https://doi.org/10.3390/e27060551 - 24 May 2025
Viewed by 877
Abstract
We consider the Shannon cipher system in the framework of individual sequences and finite-state encrypters under the metric of maximal information leakage. A lower bound and an asymptotically matching upper bound on the leakage are derived, which lead to the conclusion that asymptotically [...] Read more.
We consider the Shannon cipher system in the framework of individual sequences and finite-state encrypters under the metric of maximal information leakage. A lower bound and an asymptotically matching upper bound on the leakage are derived, which lead to the conclusion that asymptotically minimum leakage can be attained by Lempel–Ziv compression followed by one-time pad encryption of the compressed bitstream. Full article
(This article belongs to the Special Issue Information Theory and Data Compression)
10 pages, 8363 KB  
Article
Improved Reconstruction of Chaotic Signals from Ordinal Networks
by Antonio Politi and Leonardo Ricci
Entropy 2025, 27(5), 499; https://doi.org/10.3390/e27050499 - 6 May 2025
Viewed by 1105
Abstract
Permutation entropy is customarily implemented to quantify the intrinsic indeterminacy of complex time series, under the assumption that determinism manifests itself by lowering the (permutation) entropy of the resulting symbolic sequence. We expect this to be roughly true, but, in general, it is [...] Read more.
Permutation entropy is customarily implemented to quantify the intrinsic indeterminacy of complex time series, under the assumption that determinism manifests itself by lowering the (permutation) entropy of the resulting symbolic sequence. We expect this to be roughly true, but, in general, it is not clear to what extent a given ordinal pattern indeed provides a faithful reconstruction of the original signal. Here, we address this question by attempting the reconstruction of the original time series by invoking an ergodic Markov approximation of the symbolic dynamics, thereby inverting the encoding procedure. Using the Hénon map as a testbed, we show that a meaningful reconstruction can also be made in the presence of a small observational noise. Full article
(This article belongs to the Special Issue Ordinal Patterns-Based Tools and Their Applications)
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14 pages, 617 KB  
Article
Iterative Forecasting of Financial Time Series: The Greek Stock Market from 2019 to 2024
by Evangelos Bakalis and Francesco Zerbetto
Entropy 2025, 27(5), 497; https://doi.org/10.3390/e27050497 - 4 May 2025
Cited by 3 | Viewed by 3062
Abstract
Predicting the evolution of financial data, if at all possible, would be very beneficial in revealing the ways in which different aspects of a global environment can impact local economies. We employ an iterative stochastic differential equation that accurately forecasts an economic time [...] Read more.
Predicting the evolution of financial data, if at all possible, would be very beneficial in revealing the ways in which different aspects of a global environment can impact local economies. We employ an iterative stochastic differential equation that accurately forecasts an economic time series’s next value by analysing its past. The input financial data are assumed to be consistent with an α-stable Lévy motion. The computation of the scaling exponent and the value of α, which characterises the type of the α-stable Lévy motion, are crucial for the iterative scheme. These two indices can be determined at each iteration from the form of the structure function, for the computation of which we use the method of generalised moments. Their values are used for the creation of the corresponding α-stable Lévy noise, which acts as a seed for the stochastic component. Furthermore, the drift and diffusion terms are calculated at each iteration. The proposed model is general, allowing the kind of stochastic process to vary from one iterative step to another, and its applicability is not restricted to financial data. As a case study, we consider Greece’s stock market general index over a period of five years, from September 2019 to September 2024, after the completion of bailout programmes. Greece’s economy changed from a restricted to a free market over the chosen era, and its stock market trading increments are likely to be describable by an α-stable L’evy motion. We find that α=2 and the scaling exponent H varies over time for every iterative step we perform. The forecasting points follow the same trend, are in good agreement with the actual data, and for most of the forecasts, the percentage error is less than 2%. Full article
(This article belongs to the Special Issue Entropy-Based Applications in Sociophysics II)
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24 pages, 1419 KB  
Article
Measurement-Induced Symmetry Restoration and Quantum Mpemba Effect
by Giuseppe Di Giulio, Xhek Turkeshi and Sara Murciano
Entropy 2025, 27(4), 407; https://doi.org/10.3390/e27040407 - 10 Apr 2025
Cited by 19 | Viewed by 3587
Abstract
Monitoring a quantum system can profoundly alter its dynamical properties, leading to non-trivial emergent phenomena. In this work, we demonstrate that dynamical measurements strongly influence the evolution of symmetry in many-body quantum systems. Specifically, we demonstrate that monitored systems governed by non-Hermitian dynamics [...] Read more.
Monitoring a quantum system can profoundly alter its dynamical properties, leading to non-trivial emergent phenomena. In this work, we demonstrate that dynamical measurements strongly influence the evolution of symmetry in many-body quantum systems. Specifically, we demonstrate that monitored systems governed by non-Hermitian dynamics exhibit a quantum Mpemba effect, where systems with stronger initial asymmetry relax faster to a symmetric state. Crucially, this phenomenon is purely measurement-induced: in the absence of measurements, we find states where the corresponding unitary evolution does not display any Mpemba effect. Furthermore, we uncover a novel measurement-induced symmetry restoration mechanism: below a critical measurement rate, the symmetry remains broken, but beyond a threshold, it is fully restored in the thermodynamic limit—along with the emergence of the quantum Mpemba effect. Full article
(This article belongs to the Special Issue Entanglement Entropy in Quantum Field Theory)
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