Entropy

16 pages, 424 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

On the Asymptotic Capacity of Information-Theoretic Privacy-Preserving Epidemiological Data Collection

by Jiale Cheng, Nan Liu and Wei Kang

Entropy 2023, 25(4), 625; https://doi.org/10.3390/e25040625 - 6 Apr 2023

Cited by 4 | Viewed by 1828

The paradigm-shifting developments of cryptography and information theory have focused on the privacy of data-sharing systems, such as epidemiological studies, where agencies are collecting far more personal data than they need, causing intrusions on patients’ privacy. To study the capability of the data [...] Read more.

The paradigm-shifting developments of cryptography and information theory have focused on the privacy of data-sharing systems, such as epidemiological studies, where agencies are collecting far more personal data than they need, causing intrusions on patients’ privacy. To study the capability of the data collection while protecting privacy from an information theory perspective, we formulate a new distributed multiparty computation problem called privacy-preserving epidemiological data collection. In our setting, a data collector requires a linear combination of K users’ data through a storage system consisting of N servers. Privacy needs to be protected when the users, servers, and data collector do not trust each other. For the users, any data are required to be protected from up to E colluding servers; for the servers, any more information than the desired linear combination cannot be leaked to the data collector; and for the data collector, any single server can not know anything about the coefficients of the linear combination. Our goal is to find the optimal collection rate, which is defined as the ratio of the size of the user’s message to the total size of downloads from N servers to the data collector. For achievability, we propose an asymptotic capacity-achieving scheme when

E < N - 1

, by applying the cross-subspace alignment method to our construction; for the converse, we proved an upper bound of the asymptotic rate for all achievable schemes when

E < N - 1

. Additionally, we show that a positive asymptotic capacity is not possible when

E \geq N - 1

. The results of the achievability and converse meet when the number of users goes to infinity, yielding the asymptotic capacity. Our work broadens current researches on data privacy in information theory and gives the best achievable asymptotic performance that any epidemiological data collector can obtain. Full article

(This article belongs to the Special Issue Advances in Information and Coding Theory)

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19 pages, 337 KiB

Open AccessEditor’s ChoiceArticle

Relating a System’s Hamiltonian to Its Entropy Production Using a Complex Time Approach

by Michael C. Parker and Chris Jeynes

Entropy 2023, 25(4), 629; https://doi.org/10.3390/e25040629 - 6 Apr 2023

Cited by 10 | Viewed by 2662

Abstract

We exploit the properties of complex time to obtain an analytical relationship based on considerations of causality between the two Noether-conserved quantities of a system: its Hamiltonian and its entropy production. In natural units, when complexified, the one is simply the Wick-rotated complex [...] Read more.

We exploit the properties of complex time to obtain an analytical relationship based on considerations of causality between the two Noether-conserved quantities of a system: its Hamiltonian and its entropy production. In natural units, when complexified, the one is simply the Wick-rotated complex conjugate of the other. A Hilbert transform relation is constructed in the formalism of quantitative geometrical thermodynamics, which enables system irreversibility to be handled analytically within a framework that unifies both the microscopic and macroscopic scales, and which also unifies the treatment of both reversibility and irreversibility as complementary parts of a single physical description. In particular, the thermodynamics of two unitary entities are considered: the alpha particle, which is absolutely stable (that is, trivially reversible with zero entropy production), and a black hole whose unconditional irreversibility is characterized by a non-zero entropy production, for which we show an alternate derivation, confirming our previous one. The thermodynamics of a canonical decaying harmonic oscillator are also considered. In this treatment, the complexification of time also enables a meaningful physical interpretation of both “imaginary time” and “imaginary energy”. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics III)

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35 pages, 1146 KiB

Open AccessEditor’s ChoiceArticle

Dimensionless Groups by Entropic Similarity: I — Diffusion, Chemical Reaction and Dispersion Processes

by Robert K. Niven

Entropy 2023, 25(4), 617; https://doi.org/10.3390/e25040617 - 5 Apr 2023

Cited by 4 | Viewed by 2120

Abstract

Since the time of Buckingham in 1914, dimensional analysis and similarity arguments based on dimensionless groups have served as powerful tools for the analysis of systems in all branches of science and engineering. Dimensionless groups are generally classified into those arising from geometric [...] Read more.

Since the time of Buckingham in 1914, dimensional analysis and similarity arguments based on dimensionless groups have served as powerful tools for the analysis of systems in all branches of science and engineering. Dimensionless groups are generally classified into those arising from geometric similarity, based on ratios of length scales; kinematic similarity, based on ratios of velocities or accelerations; and dynamic similarity, based on ratios of forces. We propose an additional category of dimensionless groups based on entropic similarity, defined by ratios of (i) entropy production terms; (ii) entropy flow rates or fluxes; or (iii) information flow rates or fluxes. Since all processes involving work against friction, dissipation, diffusion, dispersion, mixing, separation, chemical reaction, gain of information or other irreversible changes are driven by (or must overcome) the second law of thermodynamics, it is appropriate to analyze them directly in terms of competing entropy-producing and transporting phenomena and the dominant entropic regime, rather than indirectly in terms of forces. In this study, entropic groups are derived for a wide variety of diffusion, chemical reaction and dispersion processes relevant to fluid mechanics, chemical engineering and environmental engineering. It is shown that many dimensionless groups traditionally derived by kinematic or dynamic similarity (including the Reynolds number) can also be recovered by entropic similarity—with a different entropic interpretation—while many new dimensionless groups can also be identified. The analyses significantly expand the scope of dimensional analysis and similarity arguments for the resolution of new and existing problems in science and engineering. Full article

(This article belongs to the Section Multidisciplinary Applications)

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16 pages, 2958 KiB

Open AccessEditor’s ChoiceArticle

On the Different Abilities of Cross-Sample Entropy and K-Nearest-Neighbor Cross-Unpredictability in Assessing Dynamic Cardiorespiratory and Cerebrovascular Interactions

by Alberto Porta, Vlasta Bari, Francesca Gelpi, Beatrice Cairo, Beatrice De Maria, Davide Tonon, Gianluca Rossato and Luca Faes

Entropy 2023, 25(4), 599; https://doi.org/10.3390/e25040599 - 1 Apr 2023

Cited by 13 | Viewed by 1956

Abstract

Nonlinear markers of coupling strength are often utilized to typify cardiorespiratory and cerebrovascular regulations. The computation of these indices requires techniques describing nonlinear interactions between respiration (R) and heart period (HP) and between mean arterial pressure (MAP) and mean cerebral blood velocity (MCBv). [...] Read more.

Nonlinear markers of coupling strength are often utilized to typify cardiorespiratory and cerebrovascular regulations. The computation of these indices requires techniques describing nonlinear interactions between respiration (R) and heart period (HP) and between mean arterial pressure (MAP) and mean cerebral blood velocity (MCBv). We compared two model-free methods for the assessment of dynamic HP–R and MCBv–MAP interactions, namely the cross-sample entropy (CSampEn) and k-nearest-neighbor cross-unpredictability (KNNCUP). Comparison was carried out first over simulations generated by linear and nonlinear unidirectional causal, bidirectional linear causal, and lag-zero linear noncausal models, and then over experimental data acquired from 19 subjects at supine rest during spontaneous breathing and controlled respiration at 10, 15, and 20

{b r e a t h s \cdot m i n u t e}^{- 1}

as well as from 13 subjects at supine rest and during 60° head-up tilt. Linear markers were computed for comparison. We found that: (i) over simulations, CSampEn and KNNCUP exhibit different abilities in evaluating coupling strength; (ii) KNNCUP is more reliable than CSampEn when interactions occur according to a causal structure, while performances are similar in noncausal models; (iii) in healthy subjects, KNNCUP is more powerful in characterizing cardiorespiratory and cerebrovascular variability interactions than CSampEn and linear markers. We recommend KNNCUP for quantifying cardiorespiratory and cerebrovascular coupling. Full article

(This article belongs to the Special Issue Nonlinear Dynamics in Cardiovascular Signals)

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15 pages, 500 KiB

Open AccessEditor’s ChoiceArticle

Time Series of Counts under Censoring: A Bayesian Approach

by Isabel Silva, Maria Eduarda Silva, Isabel Pereira and Brendan McCabe

Entropy 2023, 25(4), 549; https://doi.org/10.3390/e25040549 - 23 Mar 2023

Cited by 1 | Viewed by 1935

Abstract

Censored data are frequently found in diverse fields including environmental monitoring, medicine, economics and social sciences. Censoring occurs when observations are available only for a restricted range, e.g., due to a detection limit. Ignoring censoring produces biased estimates and unreliable statistical inference. The [...] Read more.

Censored data are frequently found in diverse fields including environmental monitoring, medicine, economics and social sciences. Censoring occurs when observations are available only for a restricted range, e.g., due to a detection limit. Ignoring censoring produces biased estimates and unreliable statistical inference. The aim of this work is to contribute to the modelling of time series of counts under censoring using convolution closed infinitely divisible (CCID) models. The emphasis is on estimation and inference problems, using Bayesian approaches with Approximate Bayesian Computation (ABC) and Gibbs sampler with Data Augmentation (GDA) algorithms. Full article

(This article belongs to the Special Issue Discrete-Valued Time Series)

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39 pages, 8736 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

NaRnEA: An Information Theoretic Framework for Gene Set Analysis

by Aaron T. Griffin, Lukas J. Vlahos, Codruta Chiuzan and Andrea Califano

Entropy 2023, 25(3), 542; https://doi.org/10.3390/e25030542 - 21 Mar 2023

Cited by 3 | Viewed by 4804

Abstract

Gene sets are being increasingly leveraged to make high-level biological inferences from transcriptomic data; however, existing gene set analysis methods rely on overly conservative, heuristic approaches for quantifying the statistical significance of gene set enrichment. We created Nonparametric analytical-Rank-based Enrichment Analysis (NaRnEA) to [...] Read more.

Gene sets are being increasingly leveraged to make high-level biological inferences from transcriptomic data; however, existing gene set analysis methods rely on overly conservative, heuristic approaches for quantifying the statistical significance of gene set enrichment. We created Nonparametric analytical-Rank-based Enrichment Analysis (NaRnEA) to facilitate accurate and robust gene set analysis with an optimal null model derived using the information theoretic Principle of Maximum Entropy. By measuring the differential activity of ~2500 transcriptional regulatory proteins based on the differential expression of each protein’s transcriptional targets between primary tumors and normal tissue samples in three cohorts from The Cancer Genome Atlas (TCGA), we demonstrate that NaRnEA critically improves in two widely used gene set analysis methods: Gene Set Enrichment Analysis (GSEA) and analytical-Rank-based Enrichment Analysis (aREA). We show that the NaRnEA-inferred differential protein activity is significantly correlated with differential protein abundance inferred from independent, phenotype-matched mass spectrometry data in the Clinical Proteomic Tumor Analysis Consortium (CPTAC), confirming the statistical and biological accuracy of our approach. Additionally, our analysis crucially demonstrates that the sample-shuffling empirical null models leveraged by GSEA and aREA for gene set analysis are overly conservative, a shortcoming that is avoided by the newly developed Maximum Entropy analytical null model employed by NaRnEA. Full article

(This article belongs to the Special Issue Information Theory in Computational Biology)

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14 pages, 1668 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

More Stages Decrease Dissipation in Irreversible Step Processes

by Peter Salamon, Bjarne Andresen, James Nulton, Ty N. F. Roach and Forest Rohwer

Entropy 2023, 25(3), 539; https://doi.org/10.3390/e25030539 - 21 Mar 2023

Cited by 2 | Viewed by 1821

Abstract

The dissipation in an irreversible step process is reduced when the number of steps is increased in any refinement of the steps in the process. This is a consequence of the ladder theorem, which states that, for any irreversible process proceeding by a [...] Read more.

The dissipation in an irreversible step process is reduced when the number of steps is increased in any refinement of the steps in the process. This is a consequence of the ladder theorem, which states that, for any irreversible process proceeding by a sequence of relaxations, dividing any relaxation step into two will result in a new sequence that is more efficient than the original one. This results in a more-steps-the-better rule, even when the new sequence of steps is not reoptimized. This superiority of many steps is well established empirically in, e.g., insulation and separation applications. In particular, the fact that the division of any step into two steps improves the overall efficiency has interesting implications for biological evolution and emphasizes thermodynamic length as a central measure for dissipation. Full article

(This article belongs to the Section Thermodynamics)

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15 pages, 2994 KiB

Open AccessFeature PaperEditor’s ChoiceReview

Quantum Chaos and Level Dynamics

by Jakub Zakrzewski

Entropy 2023, 25(3), 491; https://doi.org/10.3390/e25030491 - 13 Mar 2023

Cited by 6 | Viewed by 2411

Abstract

We review the application of level dynamics to spectra of quantally chaotic systems. We show that the statistical mechanics approach gives us predictions about level statistics intermediate between integrable and chaotic dynamics. Then we discuss in detail different statistical measures involving level dynamics, [...] Read more.

We review the application of level dynamics to spectra of quantally chaotic systems. We show that the statistical mechanics approach gives us predictions about level statistics intermediate between integrable and chaotic dynamics. Then we discuss in detail different statistical measures involving level dynamics, such as level avoided-crossing distributions, level slope distributions, or level curvature distributions. We show both the aspects of universality in these distributions and their limitations. We concentrate in some detail on measures imported from the quantum information approach such as the fidelity susceptibility, and more generally, geometric tensor matrix elements. The possible open problems are suggested. Full article

(This article belongs to the Special Issue Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday)

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27 pages, 462 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Stochastic Expectation Maximization Algorithm for Linear Mixed-Effects Model with Interactions in the Presence of Incomplete Data

by Alandra Zakkour, Cyril Perret and Yousri Slaoui

Entropy 2023, 25(3), 473; https://doi.org/10.3390/e25030473 - 8 Mar 2023

Cited by 3 | Viewed by 2708

Abstract

The purpose of this paper is to propose a new algorithm based on stochastic expectation maximization (SEM) to deal with the problem of unobserved values when multiple interactions in a linear mixed-effects model (LMEM) are present. We test the effectiveness of the proposed [...] Read more.

The purpose of this paper is to propose a new algorithm based on stochastic expectation maximization (SEM) to deal with the problem of unobserved values when multiple interactions in a linear mixed-effects model (LMEM) are present. We test the effectiveness of the proposed algorithm with the stochastic approximation expectation maximization (SAEM) and Monte Carlo Markov chain (MCMC) algorithms. This comparison is implemented to highlight the importance of including the maximum effects that can affect the model. The applications are made on both simulated psychological and real data. The findings demonstrate that our proposed SEM algorithm is highly preferable to the other competitor algorithms. Full article

(This article belongs to the Special Issue Monte Carlo Simulation in Statistical Physics)

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24 pages, 6744 KiB

Open AccessEditor’s ChoiceArticle

Rate Distortion Theory for Descriptive Statistics

by Peter Harremoës

Entropy 2023, 25(3), 456; https://doi.org/10.3390/e25030456 - 5 Mar 2023

Cited by 3 | Viewed by 3108

Abstract

Rate distortion theory was developed for optimizing lossy compression of data, but it also has applications in statistics. In this paper, we illustrate how rate distortion theory can be used to analyze various datasets. The analysis involves testing, identification of outliers, choice of [...] Read more.

Rate distortion theory was developed for optimizing lossy compression of data, but it also has applications in statistics. In this paper, we illustrate how rate distortion theory can be used to analyze various datasets. The analysis involves testing, identification of outliers, choice of compression rate, calculation of optimal reconstruction points, and assigning “descriptive confidence regions” to the reconstruction points. We study four models or datasets of increasing complexity: clustering, Gaussian models, linear regression, and a dataset describing orientations of early Islamic mosques. These examples illustrate how rate distortion analysis may serve as a common framework for handling different statistical problems. Full article

(This article belongs to the Collection Feature Papers in Information Theory)

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10 pages, 4198 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Spectral Form Factor and Dynamical Localization

by Črt Lozej

Entropy 2023, 25(3), 451; https://doi.org/10.3390/e25030451 - 4 Mar 2023

Cited by 2 | Viewed by 2439

Abstract

Quantum dynamical localization occurs when quantum interference stops the diffusion of wave packets in momentum space. The expectation is that dynamical localization will occur when the typical transport time of the momentum diffusion is greater than the Heisenberg time. The transport time is [...] Read more.

Quantum dynamical localization occurs when quantum interference stops the diffusion of wave packets in momentum space. The expectation is that dynamical localization will occur when the typical transport time of the momentum diffusion is greater than the Heisenberg time. The transport time is typically computed from the corresponding classical dynamics. In this paper, we present an alternative approach based purely on the study of spectral fluctuations of the quantum system. The information about the transport times is encoded in the spectral form factor, which is the Fourier transform of the two-point spectral autocorrelation function. We compute large samples of the energy spectra (of the order of

10^{6}

levels) and spectral form factors of 22 stadium billiards with parameter values across the transition between the localized and extended eigenstate regimes. The transport time is obtained from the point when the spectral form factor transitions from the non-universal to the universal regime predicted by random matrix theory. We study the dependence of the transport time on the parameter value and show the level repulsion exponents, which are known to be a good measure of dynamical localization, depend linearly on the transport times obtained in this way. Full article

(This article belongs to the Special Issue Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday)

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12 pages, 484 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Two Features of the GINAR(1) Process and Their Impact on the Run-Length Performance of Geometric Control Charts

by Manuel Cabral Morais

Entropy 2023, 25(3), 444; https://doi.org/10.3390/e25030444 - 2 Mar 2023

Cited by 1 | Viewed by 1672

Abstract

The geometric first-order integer-valued autoregressive process (GINAR(1)) can be particularly useful to model relevant discrete-valued time series, namely in statistical process control. We resort to stochastic ordering to prove that the GINAR(1) process is a discrete-time Markov chain governed by a totally positive [...] Read more.

The geometric first-order integer-valued autoregressive process (GINAR(1)) can be particularly useful to model relevant discrete-valued time series, namely in statistical process control. We resort to stochastic ordering to prove that the GINAR(1) process is a discrete-time Markov chain governed by a totally positive order 2

({T P}_{2})

transition matrix.Stochastic ordering is also used to compare transition matrices referring to pairs of GINAR(1) processes with different values of the marginal mean. We assess and illustrate the implications of these two stochastic ordering results, namely on the properties of the run length of geometric charts for monitoring GINAR(1) counts. Full article

(This article belongs to the Special Issue Discrete-Valued Time Series)

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12 pages, 1348 KiB

Open AccessEditor’s ChoiceArticle

The Effect of On-Site Potentials on Supratransmission in One-Dimensional Hamiltonian Lattices

by Tassos Bountis and Jorge E. Macías-Díaz

Entropy 2023, 25(3), 423; https://doi.org/10.3390/e25030423 - 26 Feb 2023

Cited by 3 | Viewed by 1708

Abstract

We investigated a class of one-dimensional (1D) Hamiltonian N-particle lattices whose binary interactions are quadratic and/or quartic in the potential. We also included on-site potential terms, frequently considered in connection with localization phenomena, in this class. Applying a sinusoidal perturbation at one [...] Read more.

We investigated a class of one-dimensional (1D) Hamiltonian N-particle lattices whose binary interactions are quadratic and/or quartic in the potential. We also included on-site potential terms, frequently considered in connection with localization phenomena, in this class. Applying a sinusoidal perturbation at one end of the lattice and an absorbing boundary on the other, we studied the phenomenon of supratransmission and its dependence on two ranges of interactions,

0 < α < \infty

and

0 < β < \infty

, as the effect of the on-site potential terms of the Hamiltonian varied. In previous works, we studied the critical amplitude

A_{s} (α, Ω)

at which supratransmission occurs, for one range parameter

α

, and showed that there was a sharp threshold above which energy was transmitted in the form of large-amplitude nonlinear modes, as long as the driving frequency

Ω

lay in the forbidden band-gap of the system. In the absence of on-site potentials, it is known that

A_{s} (α, Ω)

increases monotonically the longer the range of interactions is (i.e., as

α ⟶ 0

). However, when on-site potential terms are taken into account,

A_{s} (α, Ω)

reaches a maximum at a low value of

α

that depends on

Ω

, below which supratransmission thresholds decrease sharply to lower values. In this work, we studied this phenomenon further, as the contribution of the on-site potential terms varied, and we explored in detail their effect on the supratransmission thresholds. Full article

(This article belongs to the Special Issue Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday)

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30 pages, 2586 KiB

Open AccessEditor’s ChoiceReview

Tsallis q-Statistics in Seismology

by Leonardo Di G. Sigalotti, Alejandro Ramírez-Rojas and Carlos A. Vargas

Entropy 2023, 25(3), 408; https://doi.org/10.3390/e25030408 - 23 Feb 2023

Cited by 17 | Viewed by 2960

Abstract

Non-extensive statistical mechanics (or q-statistics) is based on the so-called non-additive Tsallis entropy. Since its introduction by Tsallis, in 1988, as a generalization of the Boltzmann–Gibbs equilibrium statistical mechanics, it has steadily gained ground as a suitable theory for the description of [...] Read more.

Non-extensive statistical mechanics (or q-statistics) is based on the so-called non-additive Tsallis entropy. Since its introduction by Tsallis, in 1988, as a generalization of the Boltzmann–Gibbs equilibrium statistical mechanics, it has steadily gained ground as a suitable theory for the description of the statistical properties of non-equilibrium complex systems. Therefore, it has been applied to numerous phenomena, including real seismicity. In particular, Tsallis entropy is expected to provide a guiding principle to reveal novel aspects of complex dynamical systems with catastrophes, such as seismic events. The exploration of the existing connections between Tsallis formalism and real seismicity has been the focus of extensive research activity in the last two decades. In particular, Tsallis q-statistics has provided a unified framework for the description of the collective properties of earthquakes and faults. Despite this progress, our present knowledge of the physical processes leading to the initiation of a rupture, and its subsequent growth through a fault system, remains quite limited. The aim of this paper was to provide an overview of the non-extensive interpretation of seismicity, along with the contributions of the Tsallis formalism to the statistical description of seismic events. Full article

(This article belongs to the Section Entropy Reviews)

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15 pages, 418 KiB

Open AccessEditor’s ChoiceArticle

Design and Analysis of Joint Group Shuffled Scheduling Decoding Algorithm for Double LDPC Codes System

by Qiwang Chen, Yanzhao Ren, Lin Zhou, Chen Chen and Sanya Liu

Entropy 2023, 25(2), 357; https://doi.org/10.3390/e25020357 - 15 Feb 2023

Cited by 5 | Viewed by 2352

Abstract

In this paper, a joint group shuffled scheduling decoding (JGSSD) algorithm for a joint source-channel coding (JSCC) scheme based on double low-density parity-check (D-LDPC) codes is presented. The proposed algorithm considers the D-LDPC coding structure as a whole and applies shuffled scheduling to [...] Read more.

In this paper, a joint group shuffled scheduling decoding (JGSSD) algorithm for a joint source-channel coding (JSCC) scheme based on double low-density parity-check (D-LDPC) codes is presented. The proposed algorithm considers the D-LDPC coding structure as a whole and applies shuffled scheduling to each group; the grouping relies on the types or the length of the variable nodes (VNs). By comparison, the conventional shuffled scheduling decoding algorithm can be regarded as a special case of this proposed algorithm. A novel joint extrinsic information transfer (JEXIT) algorithm for the D-LDPC codes system with the JGSSD algorithm is proposed, by which the source and channel decoding are calculated with different grouping strategies to analyze the effects of the grouping strategy. Simulation results and comparisons verify the superiority of the JGSSD algorithm, which can adaptively trade off the decoding performance, complexity and latency. Full article

(This article belongs to the Special Issue Coding and Entropy)

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23 pages, 2911 KiB

Open AccessEditor’s ChoiceArticle

The Typical Set and Entropy in Stochastic Systems with Arbitrary Phase Space Growth

by Rudolf Hanel and Bernat Corominas-Murtra

Entropy 2023, 25(2), 350; https://doi.org/10.3390/e25020350 - 14 Feb 2023

Cited by 3 | Viewed by 2611

Abstract

The existence of the typical set is key for data compression strategies and for the emergence of robust statistical observables in macroscopic physical systems. Standard approaches derive its existence from a restricted set of dynamical constraints. However, given its central role underlying the [...] Read more.

The existence of the typical set is key for data compression strategies and for the emergence of robust statistical observables in macroscopic physical systems. Standard approaches derive its existence from a restricted set of dynamical constraints. However, given its central role underlying the emergence of stable, almost deterministic statistical patterns, a question arises whether typical sets exist in much more general scenarios. We demonstrate here that the typical set can be defined and characterized from general forms of entropy for a much wider class of stochastic processes than was previously thought. This includes processes showing arbitrary path dependence, long range correlations or dynamic sampling spaces, suggesting that typicality is a generic property of stochastic processes, regardless of their complexity. We argue that the potential emergence of robust properties in complex stochastic systems provided by the existence of typical sets has special relevance to biological systems. Full article

(This article belongs to the Special Issue The Statistical Foundations of Entropy II)

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12 pages, 574 KiB

Open AccessEditor’s ChoiceArticle

Toward Prediction of Financial Crashes with a D-Wave Quantum Annealer

by Yongcheng Ding, Javier Gonzalez-Conde, Lucas Lamata, José D. Martín-Guerrero, Enrique Lizaso, Samuel Mugel, Xi Chen, Román Orús, Enrique Solano and Mikel Sanz

Entropy 2023, 25(2), 323; https://doi.org/10.3390/e25020323 - 10 Feb 2023

Cited by 22 | Viewed by 4774

Abstract

The prediction of financial crashes in a complex financial network is known to be an NP-hard problem, which means that no known algorithm can efficiently find optimal solutions. We experimentally explore a novel approach to this problem by using a D-Wave quantum annealer, [...] Read more.

The prediction of financial crashes in a complex financial network is known to be an NP-hard problem, which means that no known algorithm can efficiently find optimal solutions. We experimentally explore a novel approach to this problem by using a D-Wave quantum annealer, benchmarking its performance for attaining a financial equilibrium. To be specific, the equilibrium condition of a nonlinear financial model is embedded into a higher-order unconstrained binary optimization (HUBO) problem, which is then transformed into a spin-

1 / 2

Hamiltonian with at most, two-qubit interactions. The problem is thus equivalent to finding the ground state of an interacting spin Hamiltonian, which can be approximated with a quantum annealer. The size of the simulation is mainly constrained by the necessity of a large number of physical qubits representing a logical qubit with the correct connectivity. Our experiment paves the way for the codification of this quantitative macroeconomics problem in quantum annealers. Full article

(This article belongs to the Special Issue Quantum Control and Quantum Computing)

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22 pages, 2941 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Characterizing the Impact of Communication on Cellular and Collective Behavior Using a Three-Dimensional Multiscale Cellular Model

by Moriah Echlin, Boris Aguilar and Ilya Shmulevich

Entropy 2023, 25(2), 319; https://doi.org/10.3390/e25020319 - 9 Feb 2023

Viewed by 2284

Abstract

Communication between cells enables the coordination that drives structural and functional complexity in biological systems. Both single and multicellular organisms have evolved diverse communication systems for a range of purposes, including synchronization of behavior, division of labor, and spatial organization. Synthetic systems are [...] Read more.

Communication between cells enables the coordination that drives structural and functional complexity in biological systems. Both single and multicellular organisms have evolved diverse communication systems for a range of purposes, including synchronization of behavior, division of labor, and spatial organization. Synthetic systems are also increasingly being engineered to utilize cell–cell communication. While research has elucidated the form and function of cell–cell communication in many biological systems, our knowledge is still limited by the confounding effects of other biological phenomena at play and the bias of the evolutionary background. In this work, our goal is to push forward the context-free understanding of what impact cell–cell communication can have on cellular and population behavior to more fully understand the extent to which cell–cell communication systems can be utilized, modified, and engineered. We use an in silico model of 3D multiscale cellular populations, with dynamic intracellular networks interacting via diffusible signals. We focus on two key communication parameters: the effective interaction distance at which cells are able to interact and the receptor activation threshold. We found that cell–cell communication can be divided into six different forms along the parameter axes, three asocial and three social. We also show that cellular behavior, tissue composition, and tissue diversity are all highly sensitive to both the general form and specific parameters of communication even when the cellular network has not been biased towards that behavior. Full article

(This article belongs to the Section Complexity)

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9 pages, 492 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Carnot Cycles in a Harmonically Confined Ultracold Gas across Bose–Einstein Condensation

by Ignacio Reyes-Ayala, Marcos Miotti, Michal Hemmerling, Romain Dubessy, Hélène Perrin, Victor Romero-Rochin and Vanderlei Salvador Bagnato

Entropy 2023, 25(2), 311; https://doi.org/10.3390/e25020311 - 8 Feb 2023

Cited by 5 | Viewed by 2528

Abstract

Carnot cycles of samples of harmonically confined ultracold

^{87}

Rb fluids, near and across Bose–Einstein condensation (BEC), are analyzed. This is achieved through the experimental determination of the corresponding equation of state in terms of the appropriate global thermodynamics for non-uniform confined fluids. [...] Read more.

Carnot cycles of samples of harmonically confined ultracold

^{87}

Rb fluids, near and across Bose–Einstein condensation (BEC), are analyzed. This is achieved through the experimental determination of the corresponding equation of state in terms of the appropriate global thermodynamics for non-uniform confined fluids. We focus our attention on the efficiency of the Carnot engine when the cycle occurs for temperatures either above or below the critical temperature and when BEC is crossed during the cycle. The measurement of the cycle efficiency reveals a perfect agreement with the theoretical prediction

(1 - T_{L} / T_{H})

, with

T_{H}

and

T_{L}

serving as the temperatures of the hot and cold heat exchange reservoirs. Other cycles are also considered for comparison. Full article

(This article belongs to the Special Issue Ultracold Gases and Thermodynamics)

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19 pages, 339 KiB

Open AccessEditor’s ChoiceArticle

Uncertainty Relations in the Madelung Picture Including a Dissipative Environment

by Dieter Schuch and Moise Bonilla-Licea

Entropy 2023, 25(2), 312; https://doi.org/10.3390/e25020312 - 8 Feb 2023

Cited by 2 | Viewed by 1550

Abstract

In a recent paper, we have shown how in Madelung’s hydrodynamic formulation of quantum mechanics, the uncertainties are related to the phase and amplitude of the complex wave function. Now we include a dissipative environment via a nonlinear modified Schrödinger equation. The effect [...] Read more.

In a recent paper, we have shown how in Madelung’s hydrodynamic formulation of quantum mechanics, the uncertainties are related to the phase and amplitude of the complex wave function. Now we include a dissipative environment via a nonlinear modified Schrödinger equation. The effect of the environment is described by a complex logarithmic nonlinearity that vanishes on average. Nevertheless, there are various changes in the dynamics of the uncertainties originating from the nonlinear term. Again, this is illustrated explicitly using generalized coherent states as examples. With particular focus on the quantum mechanical contribution to the energy and the uncertainty product, connections can be made with the thermodynamic properties of the environment. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations III)

33 pages, 752 KiB

Open AccessEditor’s ChoiceArticle

Distributed Hypothesis Testing over a Noisy Channel: Error-Exponents Trade-Off

by Sreejith Sreekumar and Deniz Gündüz

Entropy 2023, 25(2), 304; https://doi.org/10.3390/e25020304 - 6 Feb 2023

Viewed by 3020

Abstract

A two-terminal distributed binary hypothesis testing problem over a noisy channel is studied. The two terminals, called the observer and the decision maker, each has access to n independent and identically distributed samples, denoted by

U

and

V

, respectively. The observer communicates [...] Read more.

A two-terminal distributed binary hypothesis testing problem over a noisy channel is studied. The two terminals, called the observer and the decision maker, each has access to n independent and identically distributed samples, denoted by

U

and

V

, respectively. The observer communicates to the decision maker over a discrete memoryless channel, and the decision maker performs a binary hypothesis test on the joint probability distribution of

(U, V)

based on

V

and the noisy information received from the observer. The trade-off between the exponents of the type I and type II error probabilities is investigated. Two inner bounds are obtained, one using a separation-based scheme that involves type-based compression and unequal error-protection channel coding, and the other using a joint scheme that incorporates type-based hybrid coding. The separation-based scheme is shown to recover the inner bound obtained by Han and Kobayashi for the special case of a rate-limited noiseless channel, and also the one obtained by the authors previously for a corner point of the trade-off. Finally, we show via an example that the joint scheme achieves a strictly tighter bound than the separation-based scheme for some points of the error-exponents trade-off. Full article

(This article belongs to the Special Issue Information Theory for Distributed Systems)

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19 pages, 1155 KiB

Open AccessEditor’s ChoiceArticle

Random Walks on Networks with Centrality-Based Stochastic Resetting

by Kiril Zelenkovski, Trifce Sandev, Ralf Metzler, Ljupco Kocarev and Lasko Basnarkov

Entropy 2023, 25(2), 293; https://doi.org/10.3390/e25020293 - 4 Feb 2023

Cited by 15 | Viewed by 3043

Abstract

We introduce a refined way to diffusely explore complex networks with stochastic resetting where the resetting site is derived from node centrality measures. This approach differs from previous ones, since it not only allows the random walker with a certain probability to jump [...] Read more.

We introduce a refined way to diffusely explore complex networks with stochastic resetting where the resetting site is derived from node centrality measures. This approach differs from previous ones, since it not only allows the random walker with a certain probability to jump from the current node to a deliberately chosen resetting node, rather it enables the walker to jump to the node that can reach all other nodes faster. Following this strategy, we consider the resetting site to be the geometric center, the node that minimizes the average travel time to all the other nodes. Using the established Markov chain theory, we calculate the Global Mean First Passage Time (GMFPT) to determine the search performance of the random walk with resetting for different resetting node candidates individually. Furthermore, we compare which nodes are better resetting node sites by comparing the GMFPT for each node. We study this approach for different topologies of generic and real-life networks. We show that, for directed networks extracted for real-life relationships, this centrality focused resetting can improve the search to a greater extent than for the generated undirected networks. This resetting to the center advocated here can minimize the average travel time to all other nodes in real networks as well. We also present a relationship between the longest shortest path (the diameter), the average node degree and the GMFPT when the starting node is the center. We show that, for undirected scale-free networks, stochastic resetting is effective only for networks that are extremely sparse with tree-like structures as they have larger diameters and smaller average node degrees. For directed networks, the resetting is beneficial even for networks that have loops. The numerical results are confirmed by analytic solutions. Our study demonstrates that the proposed random walk approach with resetting based on centrality measures reduces the memoryless search time for targets in the examined network topologies. Full article

(This article belongs to the Topic Complex Systems and Network Science)

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17 pages, 3668 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Sample, Fuzzy and Distribution Entropies of Heart Rate Variability: What Do They Tell Us on Cardiovascular Complexity?

by Paolo Castiglioni, Giampiero Merati, Gianfranco Parati and Andrea Faini

Entropy 2023, 25(2), 281; https://doi.org/10.3390/e25020281 - 2 Feb 2023

Cited by 13 | Viewed by 4195

Abstract

Distribution Entropy (DistEn) has been introduced as an alternative to Sample Entropy (SampEn) to assess the heart rate variability (HRV) on much shorter series without the arbitrary definition of distance thresholds. However, DistEn, considered a measure of cardiovascular complexity, differs substantially from SampEn [...] Read more.

Distribution Entropy (DistEn) has been introduced as an alternative to Sample Entropy (SampEn) to assess the heart rate variability (HRV) on much shorter series without the arbitrary definition of distance thresholds. However, DistEn, considered a measure of cardiovascular complexity, differs substantially from SampEn or Fuzzy Entropy (FuzzyEn), both measures of HRV randomness. This work aims to compare DistEn, SampEn, and FuzzyEn analyzing postural changes (expected to modify the HRV randomness through a sympatho/vagal shift without affecting the cardiovascular complexity) and low-level spinal cord injuries (SCI, whose impaired integrative regulation may alter the system complexity without affecting the HRV spectrum). We recorded RR intervals in able-bodied (AB) and SCI participants in supine and sitting postures, evaluating DistEn, SampEn, and FuzzyEn over 512 beats. The significance of “case” (AB vs. SCI) and “posture” (supine vs. sitting) was assessed by longitudinal analysis. Multiscale DistEn (mDE), SampEn (mSE), and FuzzyEn (mFE) compared postures and cases at each scale between 2 and 20 beats. Unlike SampEn and FuzzyEn, DistEn is affected by the spinal lesion but not by the postural sympatho/vagal shift. The multiscale approach shows differences between AB and SCI sitting participants at the largest mFE scales and between postures in AB participants at the shortest mSE scales. Thus, our results support the hypothesis that DistEn measures cardiovascular complexity while SampEn/FuzzyEn measure HRV randomness, highlighting that together these methods integrate the information each of them provides. Full article

(This article belongs to the Special Issue Assessing Complexity in Physiological Systems through Biomedical Signals Analysis II)

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9 pages, 1205 KiB

Open AccessEditor’s ChoiceArticle

Narrow Pore Crossing of Active Particles under Stochastic Resetting

by Weitao Zhang, Yunyun Li, Fabio Marchesoni, Vyacheslav R. Misko and Pulak K. Ghosh

Entropy 2023, 25(2), 271; https://doi.org/10.3390/e25020271 - 1 Feb 2023

Cited by 10 | Viewed by 2349

Abstract

We propose a two-dimensional model of biochemical activation process, whereby self-propelling particles of finite correlation times are injected at the center of a circular cavity with constant rate equal to the inverse of their lifetime; activation is triggered when one such particle hits [...] Read more.

We propose a two-dimensional model of biochemical activation process, whereby self-propelling particles of finite correlation times are injected at the center of a circular cavity with constant rate equal to the inverse of their lifetime; activation is triggered when one such particle hits a receptor on the cavity boundary, modeled as a narrow pore. We numerically investigated this process by computing the particle mean-first exit times through the cavity pore as a function of the correlation and injection time constants. Due to the breach of the circular symmetry associated with the positioning of the receptor, the exit times may depend on the orientation of the self-propelling velocity at injection. Stochastic resetting appears to favor activation for large particle correlation times, where most of the underlying diffusion process occurs at the cavity boundary. Full article

(This article belongs to the Section Statistical Physics)

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19 pages, 764 KiB

Open AccessEditor’s ChoiceArticle

Temporal, Structural, and Functional Heterogeneities Extend Criticality and Antifragility in Random Boolean Networks

by Amahury Jafet López-Díaz, Fernanda Sánchez-Puig and Carlos Gershenson

Entropy 2023, 25(2), 254; https://doi.org/10.3390/e25020254 - 31 Jan 2023

Cited by 12 | Viewed by 3957

Abstract

Most models of complex systems have been homogeneous, i.e., all elements have the same properties (spatial, temporal, structural, functional). However, most natural systems are heterogeneous: few elements are more relevant, larger, stronger, or faster than others. In homogeneous systems, criticality—a balance between change [...] Read more.

Most models of complex systems have been homogeneous, i.e., all elements have the same properties (spatial, temporal, structural, functional). However, most natural systems are heterogeneous: few elements are more relevant, larger, stronger, or faster than others. In homogeneous systems, criticality—a balance between change and stability, order and chaos—is usually found for a very narrow region in the parameter space, close to a phase transition. Using random Boolean networks—a general model of discrete dynamical systems—we show that heterogeneity—in time, structure, and function—can broaden additively the parameter region where criticality is found. Moreover, parameter regions where antifragility is found are also increased with heterogeneity. However, maximum antifragility is found for particular parameters in homogeneous networks. Our work suggests that the “optimal” balance between homogeneity and heterogeneity is non-trivial, context-dependent, and in some cases, dynamic. Full article

(This article belongs to the Special Issue The Principle of Dynamical Criticality)

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25 pages, 21310 KiB

Open AccessEditor’s ChoiceArticle

Turn-Taking Mechanisms in Imitative Interaction: Robotic Social Interaction Based on the Free Energy Principle

by Nadine Wirkuttis, Wataru Ohata and Jun Tani

Entropy 2023, 25(2), 263; https://doi.org/10.3390/e25020263 - 31 Jan 2023

Cited by 8 | Viewed by 2836

Abstract

This study explains how the leader-follower relationship and turn-taking could develop in a dyadic imitative interaction by conducting robotic simulation experiments based on the free energy principle. Our prior study showed that introducing a parameter during the model training phase can determine leader [...] Read more.

This study explains how the leader-follower relationship and turn-taking could develop in a dyadic imitative interaction by conducting robotic simulation experiments based on the free energy principle. Our prior study showed that introducing a parameter during the model training phase can determine leader and follower roles for subsequent imitative interactions. The parameter is defined as

w

, the so-called meta-prior, and is a weighting factor used to regulate the complexity term versus the accuracy term when minimizing the free energy. This can be read as sensory attenuation, in which the robot’s prior beliefs about action are less sensitive to sensory evidence. The current extended study examines the possibility that the leader-follower relationship shifts depending on changes in

w

during the interaction phase. We identified a phase space structure with three distinct types of behavioral coordination using comprehensive simulation experiments with sweeps of

w

of both robots during the interaction. Ignoring behavior in which the robots follow their own intention was observed in the region in which both

w

s were set to large values. One robot leading, followed by the other robot was observed when one

w

was set larger and the other was set smaller. Spontaneous, random turn-taking between the leader and the follower was observed when both

w

s were set at smaller or intermediate values. Finally, we examined a case of slowly oscillating

w

in anti-phase between the two agents during the interaction. The simulation experiment resulted in turn-taking in which the leader-follower relationship switched during determined sequences, accompanied by periodic shifts of

w

s. An analysis using transfer entropy found that the direction of information flow between the two agents also shifted along with turn-taking. Herein, we discuss qualitative differences between random/spontaneous turn-taking and agreed-upon sequential turn-taking by reviewing both synthetic and empirical studies. Full article

(This article belongs to the Special Issue Brain Theory from Artificial Life)

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13 pages, 1425 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Quantum Bounds on the Generalized Lyapunov Exponents

by Silvia Pappalardi and Jorge Kurchan

Entropy 2023, 25(2), 246; https://doi.org/10.3390/e25020246 - 30 Jan 2023

Cited by 14 | Viewed by 2910

Abstract

We discuss the generalized quantum Lyapunov exponents

L_{q}

, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of [...] Read more.

We discuss the generalized quantum Lyapunov exponents

L_{q}

, defined from the growth rate of the powers of the square commutator. They may be related to an appropriately defined thermodynamic limit of the spectrum of the commutator, which plays the role of a large deviation function, obtained from the exponents

L_{q}

via a Legendre transform. We show that such exponents obey a generalized bound to chaos due to the fluctuation–dissipation theorem, as already discussed in the literature. The bounds for larger q are actually stronger, placing a limit on the large deviations of chaotic properties. Our findings at infinite temperature are exemplified by a numerical study of the kicked top, a paradigmatic model of quantum chaos. Full article

(This article belongs to the Special Issue Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday)

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25 pages, 434 KiB

Open AccessEditor’s ChoiceArticle

A Dynamic Programming Algorithm for Finding an Optimal Sequence of Informative Measurements

by Peter N. Loxley and Ka-Wai Cheung

Entropy 2023, 25(2), 251; https://doi.org/10.3390/e25020251 - 30 Jan 2023

Cited by 2 | Viewed by 3123

Abstract

An informative measurement is the most efficient way to gain information about an unknown state. We present a first-principles derivation of a general-purpose dynamic programming algorithm that returns an optimal sequence of informative measurements by sequentially maximizing the entropy of possible measurement outcomes. [...] Read more.

An informative measurement is the most efficient way to gain information about an unknown state. We present a first-principles derivation of a general-purpose dynamic programming algorithm that returns an optimal sequence of informative measurements by sequentially maximizing the entropy of possible measurement outcomes. This algorithm can be used by an autonomous agent or robot to decide where best to measure next, planning a path corresponding to an optimal sequence of informative measurements. The algorithm is applicable to states and controls that are either continuous or discrete, and agent dynamics that is either stochastic or deterministic; including Markov decision processes and Gaussian processes. Recent results from the fields of approximate dynamic programming and reinforcement learning, including on-line approximations such as rollout and Monte Carlo tree search, allow the measurement task to be solved in real time. The resulting solutions include non-myopic paths and measurement sequences that can generally outperform, sometimes substantially, commonly used greedy approaches. This is demonstrated for a global search task, where on-line planning for a sequence of local searches is found to reduce the number of measurements in the search by approximately half. A variant of the algorithm is derived for Gaussian processes for active sensing. Full article

(This article belongs to the Section Multidisciplinary Applications)

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16 pages, 1935 KiB

Open AccessEditor’s ChoiceArticle

Measurement-Based Quantum Thermal Machines with Feedback Control

by Bibek Bhandari, Robert Czupryniak, Paolo Andrea Erdman and Andrew N. Jordan

Entropy 2023, 25(2), 204; https://doi.org/10.3390/e25020204 - 20 Jan 2023

Cited by 12 | Viewed by 4095

Abstract

We investigated coupled-qubit-based thermal machines powered by quantum measurements and feedback. We considered two different versions of the machine: (1) a quantum Maxwell’s demon, where the coupled-qubit system is connected to a detachable single shared bath, and (2) a measurement-assisted refrigerator, where the [...] Read more.

We investigated coupled-qubit-based thermal machines powered by quantum measurements and feedback. We considered two different versions of the machine: (1) a quantum Maxwell’s demon, where the coupled-qubit system is connected to a detachable single shared bath, and (2) a measurement-assisted refrigerator, where the coupled-qubit system is in contact with a hot and cold bath. In the quantum Maxwell’s demon case, we discuss both discrete and continuous measurements. We found that the power output from a single qubit-based device can be improved by coupling it to the second qubit. We further found that the simultaneous measurement of both qubits can produce higher net heat extraction compared to two setups operated in parallel where only single-qubit measurements are performed. In the refrigerator case, we used continuous measurement and unitary operations to power the coupled-qubit-based refrigerator. We found that the cooling power of a refrigerator operated with swap operations can be enhanced by performing suitable measurements. Full article

(This article belongs to the Special Issue Thermodynamics in Quantum and Mesoscopic Systems)

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12 pages, 784 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Generalized Survival Probability

by David A. Zarate-Herrada, Lea F. Santos and E. Jonathan Torres-Herrera

Entropy 2023, 25(2), 205; https://doi.org/10.3390/e25020205 - 20 Jan 2023

Cited by 3 | Viewed by 2339

Abstract

Survival probability measures the probability that a system taken out of equilibrium has not yet transitioned from its initial state. Inspired by the generalized entropies used to analyze nonergodic states, we introduce a generalized version of the survival probability and discuss how it [...] Read more.

Survival probability measures the probability that a system taken out of equilibrium has not yet transitioned from its initial state. Inspired by the generalized entropies used to analyze nonergodic states, we introduce a generalized version of the survival probability and discuss how it can assist in studies of the structure of eigenstates and ergodicity. Full article

(This article belongs to the Special Issue Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday)

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22 pages, 2693 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Scarring in Rough Rectangular Billiards

by Felix M. Izrailev, German A. Luna-Acosta and J. A. Mendez-Bermudez

Entropy 2023, 25(2), 189; https://doi.org/10.3390/e25020189 - 18 Jan 2023

Viewed by 1972

Abstract

We study the mechanism of scarring of eigenstates in rectangular billiards with slightly corrugated surfaces and show that it is very different from that known in Sinai and Bunimovich billiards. We demonstrate that there are two sets of scar states. One set is [...] Read more.

We study the mechanism of scarring of eigenstates in rectangular billiards with slightly corrugated surfaces and show that it is very different from that known in Sinai and Bunimovich billiards. We demonstrate that there are two sets of scar states. One set is related to the bouncing ball trajectories in the configuration space of the corresponding classical billiard. A second set of scar-like states emerges in the momentum space, which originated from the plane-wave states of the unperturbed flat billiard. In the case of billiards with one rough surface, the numerical data demonstrate the repulsion of eigenstates from this surface. When two horizontal rough surfaces are considered, the repulsion effect is either enhanced or canceled depending on whether the rough profiles are symmetric or antisymmetric. The effect of repulsion is quite strong and influences the structure of all eigenstates, indicating that the symmetric properties of the rough profiles are important for the problem of scattering of electromagnetic (or electron) waves through quasi-one-dimensional waveguides. Our approach is based on the reduction of the model of one particle in the billiard with corrugated surfaces to a model of two artificial particles in the billiard with flat surfaces, however, with an effective interaction between these particles. As a result, the analysis is conducted in terms of a two-particle basis, and the roughness of the billiard boundaries is absorbed by a quite complicated potential. Full article

(This article belongs to the Special Issue Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday)

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12 pages, 329 KiB

Open AccessEditor’s ChoiceArticle

Pauli’s Electron in Ehrenfest and Bohm Theories, a Comparative Study

by Asher Yahalom

Entropy 2023, 25(2), 190; https://doi.org/10.3390/e25020190 - 18 Jan 2023

Cited by 2 | Viewed by 2021

Abstract

Electrons moving at slow speeds much lower than the speed of light are described by a wave function which is a solution of Pauli’s equation. This is a low-velocity limit of the relativistic Dirac equation. Here we compare two approaches, one of which [...] Read more.

Electrons moving at slow speeds much lower than the speed of light are described by a wave function which is a solution of Pauli’s equation. This is a low-velocity limit of the relativistic Dirac equation. Here we compare two approaches, one of which is the more conservative Copenhagen’s interpretation denying a trajectory of the electron but allowing a trajectory to the electron expectation value through the Ehrenfest theorem. The said expectation value is of course calculated using a solution of Pauli’s equation. A less orthodox approach is championed by Bohm, and attributes a velocity field to the electron also derived from the Pauli wave function. It is thus interesting to compare the trajectory followed by the electron according to Bohm and its expectation value according to Ehrenfest. Both similarities and differences will be considered. Full article

(This article belongs to the Special Issue Advances in Relativistic Statistical Mechanics II)

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18 pages, 2322 KiB

Open AccessEditor’s ChoiceArticle

Reconstructing Sparse Multiplex Networks with Application to Covert Networks

by Jin-Zhu Yu, Mincheng Wu, Gisela Bichler, Felipe Aros-Vera and Jianxi Gao

Entropy 2023, 25(1), 142; https://doi.org/10.3390/e25010142 - 10 Jan 2023

Cited by 1 | Viewed by 2673

Abstract

Network structure provides critical information for understanding the dynamic behavior of complex systems. However, the complete structure of real-world networks is often unavailable, thus it is crucially important to develop approaches to infer a more complete structure of networks. In this paper, we [...] Read more.

Network structure provides critical information for understanding the dynamic behavior of complex systems. However, the complete structure of real-world networks is often unavailable, thus it is crucially important to develop approaches to infer a more complete structure of networks. In this paper, we integrate the configuration model for generating random networks into an Expectation–Maximization–Aggregation (EMA) framework to reconstruct the complete structure of multiplex networks. We validate the proposed EMA framework against the Expectation–Maximization (EM) framework and random model on several real-world multiplex networks, including both covert and overt ones. It is found that the EMA framework generally achieves the best predictive accuracy compared to the EM framework and the random model. As the number of layers increases, the performance improvement of EMA over EM decreases. The inferred multiplex networks can be leveraged to inform the decision-making on monitoring covert networks as well as allocating limited resources for collecting additional information to improve reconstruction accuracy. For law enforcement agencies, the inferred complete network structure can be used to develop more effective strategies for covert network interdiction. Full article

(This article belongs to the Special Issue Dynamics of Complex Networks)

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14 pages, 809 KiB

Open AccessEditor’s ChoiceArticle

Geometrical Bounds on Irreversibility in Squeezed Thermal Bath

by Chen-Juan Zou, Yue Li, Jia-Kun Xu, Jia-Bin You, Ching Eng Png and Wan-Li Yang

Entropy 2023, 25(1), 128; https://doi.org/10.3390/e25010128 - 9 Jan 2023

Cited by 4 | Viewed by 2651

Abstract

Irreversible entropy production (IEP) plays an important role in quantum thermodynamic processes. Here, we investigate the geometrical bounds of IEP in nonequilibrium thermodynamics by exemplifying a system coupled to a squeezed thermal bath subject to dissipation and dephasing, respectively. We find that the [...] Read more.

Irreversible entropy production (IEP) plays an important role in quantum thermodynamic processes. Here, we investigate the geometrical bounds of IEP in nonequilibrium thermodynamics by exemplifying a system coupled to a squeezed thermal bath subject to dissipation and dephasing, respectively. We find that the geometrical bounds of the IEP always shift in a contrary way under dissipation and dephasing, where the lower and upper bounds turning to be tighter occur in the situation of dephasing and dissipation, respectively. However, either under dissipation or under dephasing, we may reduce both the critical time of the IEP itself and the critical time of the bounds for reaching an equilibrium by harvesting the benefits of squeezing effects in which the values of the IEP, quantifying the degree of thermodynamic irreversibility, also become smaller. Therefore, due to the nonequilibrium nature of the squeezed thermal bath, the system–bath interaction energy has a prominent impact on the IEP, leading to tightness of its bounds. Our results are not contradictory with the second law of thermodynamics by involving squeezing of the bath as an available resource, which can improve the performance of quantum thermodynamic devices. Full article

(This article belongs to the Special Issue Quantum Thermodynamics: Fundamentals and Applications)

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18 pages, 4010 KiB

Open AccessEditor’s ChoiceArticle

Processing Real-Life Recordings of Facial Expressions of Polish Sign Language Using Action Units

by Anna Irasiak, Jan Kozak, Adam Piasecki and Tomasz Stęclik

Entropy 2023, 25(1), 120; https://doi.org/10.3390/e25010120 - 6 Jan 2023

Cited by 2 | Viewed by 2453

Abstract

Automatic translation between the national language and sign language is a complex process similar to translation between two different foreign languages. A very important aspect is the precision of not only manual gestures but also facial expressions, which are extremely important in the [...] Read more.

Automatic translation between the national language and sign language is a complex process similar to translation between two different foreign languages. A very important aspect is the precision of not only manual gestures but also facial expressions, which are extremely important in the overall context of a sentence. In this article, we present the problem of including facial expressions in the automation of Polish-to-Polish Sign Language (PJM) translation—this is part of an ongoing project related to a comprehensive solution allowing for the animation of manual gestures, body movements and facial expressions. Our approach explores the possibility of using action unit (AU) recognition in the automatic annotation of recordings, which in the subsequent steps will be used to train machine learning models. This paper aims to evaluate entropy in real-life translation recordings and analyze the data associated with the detected action units. Our approach has been subjected to evaluation by experts related to Polish Sign Language, and the results obtained allow for the development of further work related to automatic translation into Polish Sign Language. Full article

(This article belongs to the Special Issue Entropy in Real-World Datasets and Its Impact on Machine Learning)

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40 pages, 613 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Observations on the Lovász θ-Function, Graph Capacity, Eigenvalues, and Strong Products †

by Igal Sason

Entropy 2023, 25(1), 104; https://doi.org/10.3390/e25010104 - 4 Jan 2023

Cited by 7 | Viewed by 3395

Abstract

This paper provides new observations on the Lovász

θ

-function of graphs. These include a simple closed-form expression of that function for all strongly regular graphs, together with upper and lower bounds on that function for all regular graphs. These bounds are expressed [...] Read more.

This paper provides new observations on the Lovász

θ

-function of graphs. These include a simple closed-form expression of that function for all strongly regular graphs, together with upper and lower bounds on that function for all regular graphs. These bounds are expressed in terms of the second-largest and smallest eigenvalues of the adjacency matrix of the regular graph, together with sufficient conditions for equalities (the upper bound is due to Lovász, followed by a new sufficient condition for its tightness). These results are shown to be useful in many ways, leading to the determination of the exact value of the Shannon capacity of various graphs, eigenvalue inequalities, and bounds on the clique and chromatic numbers of graphs. Since the Lovász

θ

-function factorizes for the strong product of graphs, the results are also particularly useful for parameters of strong products or strong powers of graphs. Bounds on the smallest and second-largest eigenvalues of strong products of regular graphs are consequently derived, expressed as functions of the Lovász

θ

-function (or the smallest eigenvalue) of each factor. The resulting lower bound on the second-largest eigenvalue of a k-fold strong power of a regular graph is compared to the Alon–Boppana bound; under a certain condition, the new bound is superior in its exponential growth rate (in k). Lower bounds on the chromatic number of strong products of graphs are expressed in terms of the order and the Lovász

θ

-function of each factor. The utility of these bounds is exemplified, leading in some cases to an exact determination of the chromatic numbers of strong products or strong powers of graphs. The present research paper is aimed to have tutorial value as well. Full article

(This article belongs to the Special Issue Extremal and Additive Combinatorial Aspects in Information Theory)

15 pages, 832 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Implications of Spectral Interlacing for Quantum Graphs

by Junjie Lu, Tobias Hofmann, Ulrich Kuhl and Hans-Jürgen Stöckmann

Entropy 2023, 25(1), 109; https://doi.org/10.3390/e25010109 - 4 Jan 2023

Cited by 3 | Viewed by 2111

Abstract

Quantum graphs are ideally suited to studying the spectral statistics of chaotic systems. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The latter ones correspond to totally disassembled graphs with a spectrum being the superposition of the [...] Read more.

Quantum graphs are ideally suited to studying the spectral statistics of chaotic systems. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The latter ones correspond to totally disassembled graphs with a spectrum being the superposition of the spectra of the individual bonds. According to the interlacing theorem, Neumann and Dirichlet eigenvalues on average alternate as a function of the wave number, with the consequence that the Neumann spectral statistics deviate from random matrix predictions. There is, e.g., a strict upper bound for the spacing of neighboring Neumann eigenvalues given by the number of bonds (in units of the mean level spacing). Here, we present analytic expressions for level spacing distribution and number variance for ensemble averaged spectra of Dirichlet graphs in dependence of the bond number, and compare them with numerical results. For a number of small Neumann graphs, numerical results for the same quantities are shown, and their deviations from random matrix predictions are discussed. Full article

(This article belongs to the Special Issue Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday)

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18 pages, 1344 KiB

Open AccessEditor’s ChoiceArticle

COVID-19 Effects on the Relationship between Cryptocurrencies: Can It Be Contagion? Insights from Econophysics Approaches

by Dora Almeida, Andreia Dionísio, Isabel Vieira and Paulo Ferreira

Entropy 2023, 25(1), 98; https://doi.org/10.3390/e25010098 - 3 Jan 2023

Cited by 9 | Viewed by 2771

Abstract

Cryptocurrencies are relatively new and innovative financial assets. They are a topic of interest to investors and academics due to their distinctive features. Whether financial or not, extraordinary events are one of the biggest challenges facing financial markets. The onset of the COVID-19 [...] Read more.

Cryptocurrencies are relatively new and innovative financial assets. They are a topic of interest to investors and academics due to their distinctive features. Whether financial or not, extraordinary events are one of the biggest challenges facing financial markets. The onset of the COVID-19 pandemic crisis, considered by some authors a “black swan”, is one of these events. In this study, we assess integration and contagion in the cryptocurrency market in the COVID-19 pandemic context, using two entropy-based measures: mutual information and transfer entropy. Both methodologies reveal that cryptocurrencies exhibit mixed levels of integration before and after the onset of the pandemic. Cryptocurrencies displaying higher integration before the event experienced a decline in such link after the world became aware of the first cases of pneumonia in Wuhan city. In what concerns contagion, mutual information provided evidence of its presence solely for the Huobi Token, and the transfer entropy analysis pointed out Tether and Huobi Token as its main source. As both analyses indicate no contagion from the pandemic turmoil to these financial assets, cryptocurrencies may be good investment options in case of real global shocks, such as the one provoked by the COVID-19 outbreak. Full article

(This article belongs to the Special Issue Cryptocurrency Behavior under Econophysics Approaches)

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10 pages, 660 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Entanglement Dynamics and Classical Complexity

by Jiaozi Wang, Barbara Dietz, Dario Rosa and Giuliano Benenti

Entropy 2023, 25(1), 97; https://doi.org/10.3390/e25010097 - 3 Jan 2023

Cited by 1 | Viewed by 2109

Abstract

We study the dynamical generation of entanglement for a two-body interacting system, starting from a separable coherent state. We show analytically that in the quasiclassical regime the entanglement growth rate can be simply computed by means of the underlying classical dynamics. Furthermore, this [...] Read more.

We study the dynamical generation of entanglement for a two-body interacting system, starting from a separable coherent state. We show analytically that in the quasiclassical regime the entanglement growth rate can be simply computed by means of the underlying classical dynamics. Furthermore, this rate is given by the Kolmogorov–Sinai entropy, which characterizes the dynamical complexity of classical motion. Our results, illustrated by numerical simulations on a model of coupled rotators, establish in the quasiclassical regime a link between the generation of entanglement, a purely quantum phenomenon, and classical complexity. Full article

(This article belongs to the Special Issue Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday)

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17 pages, 354 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Invariant-Parameterized Exact Evolution Operator for SU(2) Systems with Time-Dependent Hamiltonian

by Hiromichi Nakazato, Alessandro Sergi, Agostino Migliore and Antonino Messina

Entropy 2023, 25(1), 96; https://doi.org/10.3390/e25010096 - 3 Jan 2023

Cited by 6 | Viewed by 2246

Abstract

We report the step-by-step construction of the exact, closed and explicit expression for the evolution operator

U (t)

of a localized and isolated qubit in an arbitrary time-dependent field, which for concreteness we assume to be a magnetic field. Our approach [...] Read more.

We report the step-by-step construction of the exact, closed and explicit expression for the evolution operator

U (t)

of a localized and isolated qubit in an arbitrary time-dependent field, which for concreteness we assume to be a magnetic field. Our approach is based on the existence of two independent dynamical invariants that enter the expression of

S U (2)

by means of two strictly related time-dependent, real or complex, parameters. The usefulness of our approach is demonstrated by exactly solving the quantum dynamics of a qubit subject to a controllable time-dependent field that can be realized in the laboratory. We further discuss possible applications to any

S U (2)

model, as well as the applicability of our method to realistic physical scenarios with different symmetry properties. Full article

(This article belongs to the Special Issue Quantum Nonstationary Systems)

31 pages, 1844 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Extreme Eigenvalues and the Emerging Outlier in Rank-One Non-Hermitian Deformations of the Gaussian Unitary Ensemble

by Yan V. Fyodorov, Boris A. Khoruzhenko and Mihail Poplavskyi

Entropy 2023, 25(1), 74; https://doi.org/10.3390/e25010074 - 30 Dec 2022

Cited by 7 | Viewed by 2355

Abstract

Complex eigenvalues of random matrices

J = GUE + i γ diag (1, 0, \dots, 0)

provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is [...] Read more.

Complex eigenvalues of random matrices

J = GUE + i γ diag (1, 0, \dots, 0)

provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is known that in the limit of large matrix dimensions

N ≫ 1

the eigenvalue density of J undergoes an abrupt restructuring at

γ = 1

, the critical threshold beyond which a single eigenvalue outlier (“broad resonance”) appears. We provide a detailed description of this restructuring transition, including the scaling with N of the width of the critical region about the outlier threshold

γ = 1

and the associated scaling for the real parts (“resonance positions”) and imaginary parts (“resonance widths”) of the eigenvalues which are farthest away from the real axis. In the critical regime we determine the density of such extreme eigenvalues, and show how the outlier gradually separates itself from the rest of the extreme eigenvalues. Finally, we describe the fluctuations in the height of the eigenvalue outlier for large but finite N in terms of the associated large deviation function. Full article

(This article belongs to the Special Issue Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday)

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52 pages, 9913 KiB

Open AccessFeature PaperEditor’s ChoiceReview

Quantum Chaos in the Dynamics of Molecules

by Kazuo Takatsuka

Entropy 2023, 25(1), 63; https://doi.org/10.3390/e25010063 - 29 Dec 2022

Cited by 6 | Viewed by 4747

Abstract

Quantum chaos is reviewed from the viewpoint of “what is molecule?”, particularly placing emphasis on their dynamics. Molecules are composed of heavy nuclei and light electrons, and thereby the very basic molecular theory due to Born and Oppenheimer gives a view that quantum [...] Read more.

Quantum chaos is reviewed from the viewpoint of “what is molecule?”, particularly placing emphasis on their dynamics. Molecules are composed of heavy nuclei and light electrons, and thereby the very basic molecular theory due to Born and Oppenheimer gives a view that quantum electronic states provide potential functions working on nuclei, which in turn are often treated classically or semiclassically. Therefore, the classic study of chaos in molecular science began with those nuclear dynamics particularly about the vibrational energy randomization within a molecule. Statistical laws in probabilities and rates of chemical reactions even for small molecules of several atoms are among the chemical phenomena requiring the notion of chaos. Particularly the dynamics behind unimolecular decomposition are referred to as Intra-molecular Vibrational energy Redistribution (IVR). Semiclassical mechanics is also one of the main research fields of quantum chaos. We herein demonstrate chaos that appears only in semiclassical and full quantum dynamics. A fundamental phenomenon possibly giving birth to quantum chaos is “bifurcation and merging” of quantum wavepackets, rather than “stretching and folding” of the baker’s transformation and the horseshoe map as a geometrical foundation of classical chaos. Such wavepacket bifurcation and merging are indeed experimentally measurable as we showed before in the series of studies on real-time probing of nonadiabatic chemical reactions. After tracking these aspects of molecular chaos, we will explore quantum chaos found in nonadiabatic electron wavepacket dynamics, which emerges in the realm far beyond the Born-Oppenheimer paradigm. In this class of chaos, we propose a notion of Intra-molecular Nonadiabatic Electronic Energy Redistribution (INEER), which is a consequence of the chaotic fluxes of electrons and energy within a molecule. Full article

(This article belongs to the Special Issue Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday)

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25 pages, 544 KiB

Open AccessFeature PaperEditor’s ChoiceReview

Diffusion Coefficient of a Brownian Particle in Equilibrium and Nonequilibrium: Einstein Model and Beyond

by Jakub Spiechowicz, Ivan G. Marchenko, Peter Hänggi and Jerzy Łuczka

Entropy 2023, 25(1), 42; https://doi.org/10.3390/e25010042 - 26 Dec 2022

Cited by 34 | Viewed by 8055

Abstract

The diffusion of small particles is omnipresent in many processes occurring in nature. As such, it is widely studied and exerted in almost all branches of sciences. It constitutes such a broad and often rather complex subject of exploration that we opt here [...] Read more.

The diffusion of small particles is omnipresent in many processes occurring in nature. As such, it is widely studied and exerted in almost all branches of sciences. It constitutes such a broad and often rather complex subject of exploration that we opt here to narrow our survey to the case of the diffusion coefficient for a Brownian particle that can be modeled in the framework of Langevin dynamics. Our main focus centers on the temperature dependence of the diffusion coefficient for several fundamental models of diverse physical systems. Starting out with diffusion in equilibrium for which the Einstein theory holds, we consider a number of physical situations outside of free Brownian motion and end by surveying nonequilibrium diffusion for a time-periodically driven Brownian particle dwelling randomly in a periodic potential. For this latter situation the diffusion coefficient exhibits an intriguingly non-monotonic dependence on temperature. Full article

(This article belongs to the Collection Foundations of Statistical Mechanics)

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12 pages, 296 KiB

Open AccessEditor’s ChoiceArticle

Entanglement-Assisted Quantum Codes from Cyclic Codes

by Francisco Revson F. Pereira and Stefano Mancini

Entropy 2023, 25(1), 37; https://doi.org/10.3390/e25010037 - 24 Dec 2022

Cited by 13 | Viewed by 2484

Abstract

Entanglement-assisted quantum-error-correcting (EAQEC) codes are quantum codes which use entanglement as a resource. These codes can provide better error correction than the (entanglement unassisted) codes derived from the traditional stabilizer formalism. In this paper, we provide a general method to construct EAQEC codes [...] Read more.

Entanglement-assisted quantum-error-correcting (EAQEC) codes are quantum codes which use entanglement as a resource. These codes can provide better error correction than the (entanglement unassisted) codes derived from the traditional stabilizer formalism. In this paper, we provide a general method to construct EAQEC codes from cyclic codes. Afterwards, the method is applied to Reed–Solomon codes, BCH codes, and general cyclic codes. We use the Euclidean and Hermitian construction of EAQEC codes. Three families have been created: two families of EAQEC codes are maximal distance separable (MDS), and one is almost MDS or almost near MDS. The comparison of the codes in this paper is mostly based on the quantum Singleton bound. Full article

(This article belongs to the Special Issue Advances in Quantum Computing)

21 pages, 5934 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Chaos and Thermalization in the Spin-Boson Dicke Model

by David Villaseñor, Saúl Pilatowsky-Cameo, Miguel A. Bastarrachea-Magnani, Sergio Lerma-Hernández, Lea F. Santos and Jorge G. Hirsch

Entropy 2023, 25(1), 8; https://doi.org/10.3390/e25010008 - 21 Dec 2022

Cited by 19 | Viewed by 3328

Abstract

We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of [...] Read more.

We present a detailed analysis of the connection between chaos and the onset of thermalization in the spin-boson Dicke model. This system has a well-defined classical limit with two degrees of freedom, and it presents both regular and chaotic regions. Our studies of the eigenstate expectation values and the distributions of the off-diagonal elements of the number of photons and the number of excited atoms validate the diagonal and off-diagonal eigenstate thermalization hypothesis (ETH) in the chaotic region, thus ensuring thermalization. The validity of the ETH reflects the chaotic structure of the eigenstates, which we corroborate using the von Neumann entanglement entropy and the Shannon entropy. Our results for the Shannon entropy also make evident the advantages of the so-called “efficient basis” over the widespread employed Fock basis when investigating the unbounded spectrum of the Dicke model. The efficient basis gives us access to a larger number of converged states than what can be reached with the Fock basis. Full article

(This article belongs to the Special Issue Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday)

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25 pages, 2133 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Canonical Density Matrices from Eigenstates of Mixed Systems

by Mahdi Kourehpaz, Stefan Donsa, Fabian Lackner, Joachim Burgdörfer and Iva Březinová

Entropy 2022, 24(12), 1740; https://doi.org/10.3390/e24121740 - 29 Nov 2022

Cited by 3 | Viewed by 7153

Abstract

One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (

N \to \infty

) limit of large quantum many-body systems, canonical density matrices [...] Read more.

One key issue of the foundation of statistical mechanics is the emergence of equilibrium ensembles in isolated and closed quantum systems. Recently, it was predicted that in the thermodynamic (

N \to \infty

) limit of large quantum many-body systems, canonical density matrices emerge for small subsystems from almost all pure states. This notion of canonical typicality is assumed to originate from the entanglement between subsystem and environment and the resulting intrinsic quantum complexity of the many-body state. For individual eigenstates, it has been shown that local observables show thermal properties provided the eigenstate thermalization hypothesis holds, which requires the system to be quantum-chaotic. In the present paper, we study the emergence of thermal states in the regime of a quantum analog of a mixed phase space. Specifically, we study the emergence of the canonical density matrix of an impurity upon reduction from isolated energy eigenstates of a large but finite quantum system the impurity is embedded in. Our system can be tuned by means of a single parameter from quantum integrability to quantum chaos and corresponds in between to a system with mixed quantum phase space. We show that the probability for finding a canonical density matrix when reducing the ensemble of energy eigenstates of the finite many-body system can be quantitatively controlled and tuned by the degree of quantum chaos present. For the transition from quantum integrability to quantum chaos, we find a continuous and universal (i.e., size-independent) relation between the fraction of canonical eigenstates and the degree of chaoticity as measured by the Brody parameter or the Shannon entropy. Full article

(This article belongs to the Special Issue Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday)

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13 pages, 356 KiB

Open AccessEditor’s ChoiceArticle

Entropy Optimization, Generalized Logarithms, and Duality Relations

by Angel R. Plastino, Constantino Tsallis, Roseli S. Wedemann and Hans J. Haubold

Entropy 2022, 24(12), 1723; https://doi.org/10.3390/e24121723 - 25 Nov 2022

Cited by 6 | Viewed by 2384

Abstract

Several generalizations or extensions of the Boltzmann–Gibbs thermostatistics, based on non-standard entropies, have been the focus of considerable research activity in recent years. Among these, the power-law, non-additive entropies [...] Read more.

Several generalizations or extensions of the Boltzmann–Gibbs thermostatistics, based on non-standard entropies, have been the focus of considerable research activity in recent years. Among these, the power-law, non-additive entropies

S_{q} \equiv k \frac{1 - \sum_{i} p_{i}^{q}}{q - 1} (q \in R; S_{1} = S_{B G} \equiv - k \sum_{i} p_{i} \ln p_{i})

have harvested the largest number of successful applications. The specific structural features of the

S_{q}

thermostatistics, therefore, are worthy of close scrutiny. In the present work, we analyze one of these features, according to which the q-logarithm function

\ln_{q} x \equiv \frac{x^{1 - q} - 1}{1 - q} (\ln_{1} x = \ln x)

associated with the

S_{q}

entropy is linked, via a duality relation, to the q-exponential function characterizing the maximum-entropy probability distributions. We enquire into which entropic functionals lead to this or similar structures, and investigate the corresponding duality relations. Full article

(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)

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23 pages, 5468 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Functional Connectome of the Human Brain with Total Correlation

by Qiang Li, Greg Ver Steeg, Shujian Yu and Jesus Malo

Entropy 2022, 24(12), 1725; https://doi.org/10.3390/e24121725 - 25 Nov 2022

Cited by 14 | Viewed by 4658

Abstract

Recent studies proposed the use of Total Correlation to describe functional connectivity among brain regions as a multivariate alternative to conventional pairwise measures such as correlation or mutual information. In this work, we build on this idea to infer a large-scale (whole-brain) connectivity [...] Read more.

Recent studies proposed the use of Total Correlation to describe functional connectivity among brain regions as a multivariate alternative to conventional pairwise measures such as correlation or mutual information. In this work, we build on this idea to infer a large-scale (whole-brain) connectivity network based on Total Correlation and show the possibility of using this kind of network as biomarkers of brain alterations. In particular, this work uses Correlation Explanation (CorEx) to estimate Total Correlation. First, we prove that CorEx estimates of Total Correlation and clustering results are trustable compared to ground truth values. Second, the inferred large-scale connectivity network extracted from the more extensive open fMRI datasets is consistent with existing neuroscience studies, but, interestingly, can estimate additional relations beyond pairwise regions. And finally, we show how the connectivity graphs based on Total Correlation can also be an effective tool to aid in the discovery of brain diseases. Full article

(This article belongs to the Special Issue Applications of Information Theory in Neuroscience)

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11 pages, 3187 KiB

Open AccessEditor’s ChoiceArticle

Quantum Coherence in Loopless Superconductive Networks

by Massimiliano Lucci, Valerio Campanari, Davide Cassi, Vittorio Merlo, Francesco Romeo, Gaetano Salina and Matteo Cirillo

Entropy 2022, 24(11), 1690; https://doi.org/10.3390/e24111690 - 18 Nov 2022

Cited by 3 | Viewed by 1942

Abstract

Measurements indicating that planar networks of superconductive islands connected by Josephson junctions display long-range quantum coherence are reported. The networks consist of superconducting islands connected by Josephson junctions and have a tree-like topological structure containing no loops. Enhancements of superconductive gaps over specific [...] Read more.

Measurements indicating that planar networks of superconductive islands connected by Josephson junctions display long-range quantum coherence are reported. The networks consist of superconducting islands connected by Josephson junctions and have a tree-like topological structure containing no loops. Enhancements of superconductive gaps over specific branches of the networks and sharp increases in pair currents are the main signatures of the coherent states. In order to unambiguously attribute the observed effects to branches being embedded in the networks, comparisons with geometrically equivalent—but isolated—counterparts are reported. Tuning the Josephson coupling energy by an external magnetic field generates increases in the Josephson currents, along the above-mentioned specific branches, which follow a functional dependence typical of phase transitions. Results are presented for double comb and star geometry networks, and in both cases, the observed effects provide positive quantitative evidence of the predictions of existing theoretical models. Full article

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

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17 pages, 3674 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Universal Single-Mode Lasing in Fully Chaotic Billiard Lasers

by Mengyu You, Daisuke Sakakibara, Kota Makino, Yonosuke Morishita, Kazutoshi Matsumura, Yuta Kawashima, Manao Yoshikawa, Mahiro Tonosaki, Kazutaka Kanno, Atsushi Uchida, Satoshi Sunada, Susumu Shinohara and Takahisa Harayama

Entropy 2022, 24(11), 1648; https://doi.org/10.3390/e24111648 - 14 Nov 2022

Cited by 5 | Viewed by 2668

Abstract

By numerical simulations and experiments of fully chaotic billiard lasers, we show that single-mode lasing states are stable, whereas multi-mode lasing states are unstable when the size of the billiard is much larger than the wavelength and the external pumping power is sufficiently [...] Read more.

By numerical simulations and experiments of fully chaotic billiard lasers, we show that single-mode lasing states are stable, whereas multi-mode lasing states are unstable when the size of the billiard is much larger than the wavelength and the external pumping power is sufficiently large. On the other hand, for integrable billiard lasers, it is shown that multi-mode lasing states are stable, whereas single-mode lasing states are unstable. These phenomena arise from the combination of two different nonlinear effects of mode-interaction due to the active lasing medium and deformation of the billiard shape. Investigations of billiard lasers with various shapes revealed that single-mode lasing is a universal phenomenon for fully chaotic billiard lasers. Full article

(This article belongs to the Special Issue Quantum Chaos—Dedicated to Professor Giulio Casati on the Occasion of His 80th Birthday)

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