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On the Role of Geometric and Entropic Arguments in Physics: From Classical Thermodynamics to Quantum Mechanics

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Quantum Information".

Deadline for manuscript submissions: 31 August 2024 | Viewed by 10416

Special Issue Editor


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Department of Nanoscale Science and Engineering, University at Albany, 1400 Washington Avenue, Albany, NY 12222, USA
Interests: classical and quantum information physics; complexity; entropy; inference; information geometry
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Geometry plays a special role in the description and, to a certain extent, in the understanding of various physical phenomena. The concepts of length, area, and volume are ubiquitous in physics and their meaning can prove quite helpful in explaining physical phenomena from a more intuitive perspective. The notions of “longer” and “shorter” are extensively used in virtually all disciplines. Indeed, geometric formulations of classical and quantum evolutions along with geometric descriptions of classical and quantum mechanical aspects of thermal phenomena are becoming increasingly important in science. Concepts, such as thermodynamic length, area law, and statistical volumes are omnipresent in geometric thermodynamics, general relativity, and statistical physics, respectively.

The concept of entropy finds application in essentially any realm of science, from classical thermodynamics to quantum information science. The notions of “hotter” and “cooler” are widely used in many fields. Entropy can be used to provide measures of distinguishability of classical probability distributions, as well as pure and mixed quantum states. It can also be used to propose measures of complexity for classical motion, quantum evolution, and entropic motion on curved statistical manifolds underlying the entropic dynamics of physical systems for which only partial knowledge of relevant information can be obtained. Furthermore, entropy can also be used to express the degree of entanglement in a quantum state specifying a composite quantum system. For instance, concepts such as Shannon entropy, von Neumann entropy, and Umegaki relative entropy are ubiquitous in classical information science, quantum information theory, and information geometric formulations of mixed quantum state evolutions, respectively.

The aim of this Special Issue is to collect works exhibiting novel connections among geometry, thermodynamics, and quantum information theoretic concepts. Clearly, special attention to the role played by entropic arguments in such connections will be warmly welcomed.

Dr. Carlo Cafaro
Guest Editor

Manuscript Submission Information

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

Keywords

  • complexity
  • differential geometry
  • entanglement
  • entropy
  • phase transitions
  • probability theory
  • quantum computing
  • quantum information
  • quantum mechanics
  • thermodynamics
  • statistical physics

Published Papers (6 papers)

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Research

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20 pages, 362 KiB  
Article
Maximum Geometric Quantum Entropy
by Fabio Anza and James P. Crutchfield
Entropy 2024, 26(3), 225; https://doi.org/10.3390/e26030225 - 01 Mar 2024
Viewed by 908
Abstract
Any given density matrix can be represented as an infinite number of ensembles of pure states. This leads to the natural question of how to uniquely select one out of the many, apparently equally-suitable, possibilities. Following Jaynes’ information-theoretic perspective, this can be framed [...] Read more.
Any given density matrix can be represented as an infinite number of ensembles of pure states. This leads to the natural question of how to uniquely select one out of the many, apparently equally-suitable, possibilities. Following Jaynes’ information-theoretic perspective, this can be framed as an inference problem. We propose the Maximum Geometric Quantum Entropy Principle to exploit the notions of Quantum Information Dimension and Geometric Quantum Entropy. These allow us to quantify the entropy of fully arbitrary ensembles and select the one that maximizes it. After formulating the principle mathematically, we give the analytical solution to the maximization problem in a number of cases and discuss the physical mechanism behind the emergence of such maximum entropy ensembles. Full article
7 pages, 220 KiB  
Article
Does the Differential Structure of Space-Time Follow from Physical Principles?
by Rathindra Nath Sen
Entropy 2024, 26(3), 179; https://doi.org/10.3390/e26030179 - 20 Feb 2024
Viewed by 644
Abstract
This article examines Wigner’s view on the unreasonable effectiveness of mathematics in the natural sciences, which was based on Cantor’s claim that ‘mathematics is a free creation of the human mind’. It is contended that Cantor’s claim is not relevant to physics [...] Read more.
This article examines Wigner’s view on the unreasonable effectiveness of mathematics in the natural sciences, which was based on Cantor’s claim that ‘mathematics is a free creation of the human mind’. It is contended that Cantor’s claim is not relevant to physics because it was based on his power set construction, which does not preserve neighborhoods of geometrical points. It is pointed out that the physical notion of Einstein causality can be defined on a countably infinite point set M with no predefined mathematical structure on it, and this definition endows M with a Tychonoff topology. Under Shirota’s theorem, M can therefore be embedded as a closed subspace of RJ for some J. While this suggests that the differentiable structure of RJ may follow from the principle of causality, the argument is constrained by the fact that the completion processes (analyzed here in some detail) required for the passage from QJ to RJ remain empirically untestable. Full article
12 pages, 274 KiB  
Article
Entropic Density Functional Theory
by Ahmad Yousefi and Ariel Caticha
Entropy 2024, 26(1), 10; https://doi.org/10.3390/e26010010 - 21 Dec 2023
Viewed by 931
Abstract
A formulation of density functional theory (DFT) is constructed as an application of the method of maximum entropy for an inhomogeneous fluid in thermal equilibrium. The use of entropy as a systematic method to generate optimal approximations is extended from the classical to [...] Read more.
A formulation of density functional theory (DFT) is constructed as an application of the method of maximum entropy for an inhomogeneous fluid in thermal equilibrium. The use of entropy as a systematic method to generate optimal approximations is extended from the classical to the quantum domain. This process introduces a family of trial density operators that are parameterized by the particle density. The optimal density operator is that which maximizes the quantum entropy relative to the exact canonical density operator. This approach reproduces the variational principle of DFT and allows a simple proof of the Hohenberg–Kohn theorem at finite temperature. Finally, as an illustration, we discuss the Kohn–Sham approximation scheme at finite temperature. Full article
18 pages, 2106 KiB  
Article
A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics
by Shannon Ray, Paul M. Alsing, Carlo Cafaro and H S. Jacinto
Entropy 2023, 25(5), 788; https://doi.org/10.3390/e25050788 - 12 May 2023
Viewed by 5404
Abstract
In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator ρS. We also define a quantum coarse-graining (CG) to study the volume where macrostates are the manifolds of purifications, [...] Read more.
In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator ρS. We also define a quantum coarse-graining (CG) to study the volume where macrostates are the manifolds of purifications, which we call surfaces of ignorance (SOI), and microstates are the purifications of ρS. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from ρS. Using examples where the SOI are generated using representations of SU(2), SO(3), and SO(N), we show two features of the CG: (1) A system beginning in an atypical macrostate of smaller volume evolves to macrostates of greater volume until it reaches the equilibrium macrostate in a process in which the system and environment become strictly more entangled, and (2) the equilibrium macrostate takes up the vast majority of the coarse-grained space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to a maximum entanglement between the system and the environment. To demonstrate feature (1) for the examples considered, we show that the volume behaves like the von Neumann entropy in that it is zero for pure states, maximal for maximally mixed states, and is a concave function with respect to the purity of ρS. These two features are essential to typicality arguments regarding thermalization and Boltzmann’s original CG. Full article
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13 pages, 459 KiB  
Article
Entropic Dynamics in a Theoretical Framework for Biosystems
by Richard L. Summers
Entropy 2023, 25(3), 528; https://doi.org/10.3390/e25030528 - 18 Mar 2023
Cited by 1 | Viewed by 1181
Abstract
Central to an understanding of the physical nature of biosystems is an apprehension of their ability to control entropy dynamics in their environment. To achieve ongoing stability and survival, living systems must adaptively respond to incoming information signals concerning matter and energy perturbations [...] Read more.
Central to an understanding of the physical nature of biosystems is an apprehension of their ability to control entropy dynamics in their environment. To achieve ongoing stability and survival, living systems must adaptively respond to incoming information signals concerning matter and energy perturbations in their biological continuum (biocontinuum). Entropy dynamics for the living system are then determined by the natural drive for reconciliation of these information divergences in the context of the constraints formed by the geometry of the biocontinuum information space. The configuration of this information geometry is determined by the inherent biological structure, processes and adaptive controls that are necessary for the stable functioning of the organism. The trajectory of this adaptive reconciliation process can be described by an information-theoretic formulation of the living system’s procedure for actionable knowledge acquisition that incorporates the axiomatic inference of the Kullback principle of minimum information discrimination (a derivative of Jaynes’ principle of maximal entropy). Utilizing relative information for entropic inference provides for the incorporation of a background of the adaptive constraints in biosystems within the operations of Fisher biologic replicator dynamics. This mathematical expression for entropic dynamics within the biocontinuum may then serve as a theoretical framework for the general analysis of biological phenomena. Full article
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Review

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15 pages, 308 KiB  
Review
The ff˜ Correspondence and Its Applications in Quantum Information Geometry
by Paolo Gibilisco
Entropy 2024, 26(4), 286; https://doi.org/10.3390/e26040286 - 26 Mar 2024
Viewed by 472
Abstract
Due to the classifying theorems by Petz and Kubo–Ando, we know that there are bijective correspondences between Quantum Fisher Information(s), operator means, and the class of symmetric, normalized operator monotone functions on the positive half line; this last class is usually denoted as  [...] Read more.
Due to the classifying theorems by Petz and Kubo–Ando, we know that there are bijective correspondences between Quantum Fisher Information(s), operator means, and the class of symmetric, normalized operator monotone functions on the positive half line; this last class is usually denoted as Fop. This class of operator monotone function has a significant structure, which is worthy of study; indeed, any step in understanding Fop, besides being interesting per se, immediately translates into a property of the classes of operator means and therefore of Quantum Fisher Information(s). In recent years, the ff correspondence has been introduced, which associates a non-regular element of Fop to any regular element of the same set. In terms of operator means, this amounts to associating a mean with multiplicative character to a mean that has an additive character. In this paper, we survey a number of different settings where this technique has proven useful in Quantum Information Geometry. In Sections 1–4, all the needed background is provided. In Sections 5–14, we describe the main applications of the ff˜ correspondence. Full article
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