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

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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 10442

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Special Issue Editor

Dr. Carlo Cafaro

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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

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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

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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

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

Keywords

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

Open AccessArticle

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 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

(This article belongs to the Special Issue On the Role of Geometric and Entropic Arguments in Physics: From Classical Thermodynamics to Quantum Mechanics)

7 pages, 220 KiB

Open AccessArticle

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 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

R^{J}

for some J. While this suggests that the differentiable structure of

R^{J}

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

Q^{J}

R^{J}

remain empirically untestable. Full article

(This article belongs to the Special Issue On the Role of Geometric and Entropic Arguments in Physics: From Classical Thermodynamics to Quantum Mechanics)

12 pages, 274 KiB

Open AccessArticle

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 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

(This article belongs to the Special Issue On the Role of Geometric and Entropic Arguments in Physics: From Classical Thermodynamics to Quantum Mechanics)

18 pages, 2106 KiB

Open AccessFeature PaperArticle

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 5407

Abstract

In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator

ρ_{S}

. 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

ρ_{S}

. 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

ρ_{S}

. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from

ρ_{S}

. Using examples where the SOI are generated using representations of

S U (2)

S O (3)

, and

S O (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

ρ_{S}

. These two features are essential to typicality arguments regarding thermalization and Boltzmann’s original CG. Full article

(This article belongs to the Special Issue On the Role of Geometric and Entropic Arguments in Physics: From Classical Thermodynamics to Quantum Mechanics)

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13 pages, 459 KiB

Open AccessArticle

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 1182

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 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

(This article belongs to the Special Issue On the Role of Geometric and Entropic Arguments in Physics: From Classical Thermodynamics to Quantum Mechanics)

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Review

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15 pages, 308 KiB

Open AccessReview

The

f \leftrightarrow \tilde{f}

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.

F_{o p}

. This class of operator monotone function has a significant structure, which is worthy of study; indeed, any step in understanding

F_{o p}

, 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

f \leftrightarrow f

correspondence has been introduced, which associates a non-regular element of

F_{o p}

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

f \leftrightarrow \tilde{f}

correspondence. Full article

(This article belongs to the Special Issue On the Role of Geometric and Entropic Arguments in Physics: From Classical Thermodynamics to Quantum Mechanics)

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