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Special Issue "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 December 2023 | Viewed by 3529

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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

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Keywords

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

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Open AccessFeature PaperArticle

A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics

and

Entropy 2023, 25(5), 788; https://doi.org/10.3390/e25050788 - 12 May 2023

Viewed by 2595

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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Open AccessArticle

Entropic Dynamics in a Theoretical Framework for Biosystems

Richard L. Summers

Entropy 2023, 25(3), 528; https://doi.org/10.3390/e25030528 - 18 Mar 2023

Viewed by 437

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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