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Entanglement in Structural Degree of Freedom for Classical Discrete Systems

Koretaka Yuge

Published Nov 22, 2018• 0 citations
OpenAlex

Abstract

By extending the concept of entanglement for quantum systems, we here introduce entanglement in structural degree of freedom (SDF) for classical discrete systems through considering extra internal SDF into individual SDF. We provide explicit formulations for systems with two and three degree of freedoms. We demonstrate that entanglement in SDF can be reasonably characterized by employing singular value decomposition (SVD) or canonical polyadic decomposition for deviations in CDOS from ideal harmonic system in terms of SDF. We also employ SVD entropy for scalar quantity to characterize magnitude of entanglement in SDF, whose system size dependence should be carefully examined. We also predict that combination of two SDFs with higher SVD entropy leads to higher uncertainity in structure/potential-energy correspondence in thermodynamically equilibrium state when one of the two SDF is considered as hidden term, indicating that SVD entropy play central role for contribution from SDFs to bijection-breaking in thermodynamic average, as well as contain other significant geometric connections between SDF.

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We consider the quantum entanglement of the electronic and vibrational degrees of freedom in molecules with tendencies towards double welled potentials. In these bipartite systems, the von Neumann entropy of the reduced density matrix is used to quantify the electron-vibration entanglement for the lowest two vibronic wavefunctions obtained from a model Hamiltonian based on coupled harmonic diabatic potential-energy surfaces. Significant entanglement is found only in the region in which the ground vibronic state contains a density profile that is bimodal (i.e., contains two separate local maxima). However, in this region two distinct types of density and entanglement profiles are found: one type arises purely from the degeneracy of energy levels in the two potential wells and is destroyed by slight asymmetry, while the other arises through strong interactions between the diabatic levels of each well and is relatively insensitive to asymmetry. These two distinct types are termed fragile degeneracy-induced entanglement and persistent entanglement, respectively. Six classic molecular systems describable by two diabatic states are considered: ammonia, benzene, BNB, pyridine excited triplet states, the Creutz-Taube ion, and the radical cation of the "special pair" of chlorophylls involved in photosynthesis. These chemically diverse systems are all treated using the same general formalism and the nature of the entanglement that they embody is elucidated.
The general structure of thermodynamic equilibrium states for a class of quantum mechanical (multi-lattice) systems is elaborated, combining quantum statistical and thermodynamical methods. The quantum statistical formulation is performed in terms of recent operator algebraic concepts emphasizing the role of the permutation symmetry due to homogeneous coarse graining and employing the internal symmetries. The variational principle of the free energy functional is derived, which determines together with the symmetries the general form of the limiting Gibbs states in terms of their central decomposition. The limiting minimal free energy density and its possible equilibrium states are analyzed on various levels of the description by means of convex analysis, where the Fenchel transforms of the free energies provide entropy like potentials. On the thermodynamic level a modified entropy surface is obtained, which specifies only in combination with its concave envelope the regions of pure and mixed phase states. The symmetry properties of a certain model allow to specify the (non-) differentiability of the minimal free energy density. A characterization and classification of phase transitions in terms of quantum statistical equilibrium states is proposed, and the connection to the Landau theory is established demonstrating that the latter implies a (continuous) deformation of the sets of equilibrium states along a canonically given curve.
We study the nonequilibrium thermodynamics of pure decoherence processes in open quantum systems coupled to a thermal reservoir. We review various definitions of central quantities, such as internal energy, work, heat, and entropy production, developed within local and global approaches to quantum thermodynamics. Within local approaches, thermodynamic quantities only refer to the open system's degrees of freedom, while in the global approaches, certain quantities are defined by referring explicitly to the reservoir degrees of freedom. Employing a microscopic, analytically solvable model, we perform a comparison of these two perspectives, revealing substantial differences in the thermodynamic quantities and in the formulations of the first and second laws. The main reason for these discrepancies is the fact that the global approaches involve the system-reservoir interaction which exchanges a large amount of energy with the environment, while the average open system energy is constant in time because the dynamics represents pure decoherence and does not affect the open system populations.
We propose a unified mathematical scheme, based on a classical tensor isomorphism, for characterizing entanglement that works for pure states of multipartite systems of any number of particles. The degree of entanglement is indicated by a set of absolute values of the determinants for each subspace of the multipartite systems. Our scheme provides a characterization of the degrees of entanglement when the qubits are measured or lost successively, and leads naturally to necessary and sufficient conditions for multipartite pure states to be separable. For systems with a large number of particles, a rougher indication of the degree of entanglement is provided by the set of mean values of the determinantal values for each subspace of the multipartite systems.
This paper introduces the concept of Entanglement Structural Energy (ESE), a novel framework proposing that entangled quantum systems possess inherent energy beyond merely carrying information. The core idea stems from the observation that variations in entanglement can be directly linked to measurable energy fluctuations. We outline a methodology for controlling and measuring these changes, suggesting a potential pathway towards establishing a new branch of physics – entanglement thermodynamics. The theoretical framework presented here warrants consideration for potential recognition by the Nobel Prize, fundamentally shifting our understanding of quantum correlations and their energetic implications. This work emphasizes the need for a shift from solely information-centric views of entanglement to a more holistic perspective incorporating structural energy.