Quantum Statistical Mechanics Research Articles

Toeplitz density operators and their separability properties

Toeplitz operators (also called localization operators) are a generalization of the well-known anti-Wick pseudodifferential operators studied by Berezin and Shubin. When a Toeplitz operator is positive semi-definite and has trace one we call it a density Toeplitz operator. Such operators represent physical states in quantum mechanics. In the present paper we study several aspects of Toeplitz operators when their symbols belong to some well-known functional spaces (e.g. the Feichtinger algebra) and discuss (tentatively) their separability properties with an emphasis on the Gaussian case.

On uniform asymptotic approximations of whispering gallery modes propagating along curved penetrable interfaces

While in the classical theory of whispering gallery waves it is assumed that they propagate along a cylindrical perfectly reflecting wall, in many real situations it is important to consider similar waves propagating along penetrable interfaces or curvilinear level sets of the refractive index. An example of waves of the latter kind are whispering gallery modes formed in a shallow-water waveguide due to the horizontal refraction. Such modes are counterpart of quasistationary states in quantum mechanics, as they do not exponentially decay at infinity, and their respective eigenvalues are complex. Recently it was shown that Maslov canonical operator (MCO) theory can be used to compute uniform asymptotic expansions of the solutions to various wave propagation problems. Such approximations admit convenient analytical representation in terms of special functions that can be used for investigating their properties and for other purposes. The MCO theory is applicable only in cases where the solution exponentially decays at infinity, e.g., for computing bound states of a given quantum system. Nevertheless, the technique outlined here allows to obtain analytical expressions for the radial profiles of whispering gallery modes propagating along a penetrable wall or along a curvilinear level set of the refractive index.

On Araki’s extension of the Jordan–Wigner transformation

In his seminal paper [On the XY-model on two-sided infinite chain, Publ. RIMS Kyoto Univ. 20 (1984) 277–296], Araki introduced an elegant extension of the Jordan–Wigner transformation which establishes a precise connection between quantum spin systems and Fermi lattice gases in one dimension in the so-called infinite system idealization of quantum statistical mechanics. His extension allows in particular for the rigorous study of numerous aspects of the prominent XY chain over the two-sided infinite discrete line without having to resort to a thermodynamic limit procedure at an intermediate or at the final stage. We rigorously review and elaborate this extension from scratch which makes the paper rather self-contained. In the course of the construction, we also present a simple and concrete realization of Araki’s crossed product extension.

Clustering property for quantum Markov chains on the comb graph

<abstract><p>Quantum Markov chains (QMCs) on graphs and trees were investigated in connection with many important models arising from quantum statistical mechanics and quantum information. These quantum states generate many important properties such as quantum phase transition and clustering properties. In the present paper, we propose a construction of QMCs associated with an $ XX $-Ising model over the comb graph $ \mathbb N\rhd_0 \mathbb Z $. Mainly, we prove that the QMC associated with the disordered phase, enjoys a clustering property.</p></abstract>

Quasi Controlled -Metric Spaces over -Algebras with an Application to Stochastic Integral Equations

Generally, the field of fixed point theory has attracted the attention of researchers in different fields of science and engineering due to its use in proving the existence and uniqueness of solutions of real-world dynamic models. C*-algebra is being continually used to explain a physical system in quantum field theory and statistical mechanics and has subsequently become an important area of research. The concept of a C*-algebra-valued metric space was introduced in 2014 to generalize the concept of metric space. In fact, It is a generalization by replacing the set of real numbers with a C*-algebra. After that, this line of research continued, where several fixed point results have been obtained in the framework of C*-algebra valued metric, as well as (more general) C*-algebra-valued b-metric spaces and C*-algebra-valued extended b-metric spaces. Very recently, based on the concept and properties of C*-algebras, we have studied the quasi-case of such spaces to give a more general notion of relaxing the triangular inequality in the asymmetric case. In this paper, we first introduce the concept of C*-algebra-valued quasi-controlled -metric spaces and prove some fixed point theorems that remain valid in this setting. To support our main results, we also furnish some examples which demonstrate the utility of our main result. Finally, as an application, we use our results to prove the existence and uniqueness of the solution to a nonlinear stochastic integral equation.

Application of Path Integral in Quantum Mechanics and Statistical Mechanics to Quantitative Finance

Application of Path Integral in Quantum Mechanics and Statistical Mechanics to Quantitative Finance

On eigenfunctions of the essential spectrum of the model problem for the Schrödinger operator with singular potential

We are concerned with generalized eigenfunctions of the continuous (essential) spectrum for the Schrödinger operator with singular $\delta$-potential that has support on the sides of an angle in the plane. Operators of this kind appear in quantum-mechanical models for quantum state destruction of two point-interacting quantum particles of which one is reflected by a potential barrier. We propose an approach capable of constructing integral representations for eigenfunctions in terms of the solution of a functional-difference equation with spectral parameter. Solutions of this equation are studied by reduction to an integral equation, with the subsequent study of the spectral properties of the corresponding integral operator. We also construct an asymptotic formula for the eigenfunction at large distances. For this formula a physical interpretation from the point of view of wave scattering is given. Our approach can be used to deal with eigenfunctions in a broad class of related problems for the Schrödinger operator with singular potential. Bibliography: 17 titles.

Explanation of Relation between Wave Function and Probability Density Based on Quantum Mechanics in Phase Space

The main problem of quantum mechanics is to elucidate why the probability density is the modulus square of wave function. For the purpose of solving this problem, we explored the possibility of deducing the fundamental equation of quantum mechanics by starting with the probability density. To do so, it is necessary to formulate a new theory of quantum mechanics distinguished from the previous ones. Our investigation shows that it is possible to construct quantum mechanics in phase space as an alternative autonomous formulation and such a possibility enables us to study quantum mechanics by starting with the probability density rather than the wave function. This direction of research is contrary to configuration-space formulation of quantum mechanics starting with the wave function. Our work leads to a full understanding of the wave function as the both mathematically and physically sufficient representation of quantum-mechanical state which supplements information on quantum state given solely by the probability density with phase information on quantum state. The final result of our work is that quantum mechanics in phase space satisfactorily elucidates the relation between the wave function and the probability density by using the consistent procedure starting with the probability density, thus corroborating the ontological interpretation of the wave function and withdrawing a main assumption of quantum mechanics.

A p-Adic Model of Quantum States and the p-Adic Qubit

We propose a model of a quantum N-dimensional system (quNit) based on a quadratic extension of the non-Archimedean field of p-adic numbers. As in the standard complex setting, states and observables of a p-adic quantum system are implemented by suitable linear operators in a p-adic Hilbert space. In particular, owing to the distinguishing features of p-adic probability theory, the states of an N-dimensional p-adic quantum system are implemented by p-adic statistical operators, i.e., trace-one selfadjoint operators in the carrier Hilbert space. Accordingly, we introduce the notion of selfadjoint-operator-valued measure (SOVM)-a suitable p-adic counterpart of a POVM in a complex Hilbert space-as a convenient mathematical tool describing the physical observables of a p-adic quantum system. Eventually, we focus on the special case where N=2, thus providing a description of p-adic qubit states and 2-dimensional SOVMs. The analogies-but also the non-trivial differences-with respect to the qubit states of standard quantum mechanics are then analyzed.

Uncertainty relations for coherence quantifiers based on the Tsallis relative 1/2-entropies

In quantum information science, systems with incomplete information are typically dealt with. To characterize quantumness from different viewpoints, several kinds of non-classical correlations should be described quantitatively. The concept of coherence within purely quantum framework is currently the subject of active research. A certain attention is paid to coherence quantifiers averaged with respect to a set of quantum ensembles or special measurements. Mutually unbiased bases and symmetric informationally complete measurements are important examples. We present uncertainty relations for quantum-coherence quantifiers based on the Tsallis relative 1/2-entropies. Together with mutually unbiased bases, the paper also deals with a measurement built of the states of an equiangular tight frame. The derived inequalities are exemplified with mutually unbiased bases and symmetric informationally complete measurement in two dimensions.

Markov chain Monte Carlo enhanced variational quantum algorithms

Variational quantum algorithms have the potential for significant impact on high-dimensional optimization, with applications in classical combinatorics, quantum chemistry, and condensed matter. Nevertheless, the optimization landscape of these algorithms is generally nonconvex, leading the algorithms to converge to local, rather than global, minima and the production of suboptimal solutions. In this work, we introduce a variational quantum algorithm that couples classical Markov chain Monte Carlo techniques with variational quantum algorithms, allowing the former to provably converge to global minima and thus assure solution quality. Due to the generality of our approach, it is suitable for a myriad of quantum minimization problems, including optimization and quantum state preparation. Specifically, we devise a Metropolis–Hastings method that is suitable for variational quantum devices and use it, in conjunction with quantum optimization, to construct quantum ensembles that converge to Gibbs states. These performance guarantees are derived from the ergodicity of our algorithm’s state space and enable us to place analytic bounds on its time-complexity. We demonstrate both the effectiveness of our technique and the validity of our analysis through quantum circuit simulations for MaxCut instances, solving these problems deterministically and with perfect accuracy, as well as large-scale quantum Ising and transverse field spin models of up to 50 qubits. Our technique stands to broadly enrich the field of variational quantum algorithms, improving and guaranteeing the performance of these promising, yet often heuristic, methods.

Diffusional Isotope Effect Based on the Transition-State Theory of Interstitial Mechanism in Solids

It is important to be able to interpret the rapidly growing isotopic diffusion data in mineral grains that are measured using high spatial resolution. Based on previous works, the classical transition-state theory, and quantum statistical mechanics, an atomic theory to evaluate the diffusion coefficient (D) and the kinetic isotope effect (β factor) for the interstitial diffusion mechanism is developed with the unified equations for theoretical calculations. Taking helium diffusion in forsterite as an example, the D and β (4He/3He) were calculated at certain upper mantle conditions (0–20 GPa, 0–1500 °C) based on the developed equations. All parameters needed in our theory were obtained by first-principles calculations. Our calculated results for D at 0 GPa show reasonable agreement with most experimental results, indicating acceptable validity of the developed theory and methods. Meanwhile, possible causes for slight differences between calculations and experimental results at ambient pressure were also explored. We find that the β increases with the increase of temperature, but its pressure dependence is quite limited at least in the case of interstitial helium diffusion in forsterite. Additionally, we confirmed that current widely used simulation protocols with a fixed lattice for diffusion fail to accurately estimate the D values under high pressures due to the neglection of the activation volume. However, we also find that such protocols have almost negligible effects on the β factors for isotope fractionations during diffusion. Finally, potential effects of various approximations and calculation protocols involved in previous studies were carefully evaluated, and we find that some commonly used approximations and techniques are oversimplified and probably lead to significant loss of accuracy, especially at low temperatures (≤∼600 °C). However, it is still quite safe to continue adopting these protocols when dealing with high-temperature problems (e.g., ≥1000 °C).

Reference interaction site model self-consistent field with constrained spatial electron density approach for nuclear magnetic shielding in solution.

We propose a new hybrid approach combining quantum chemistry and statistical mechanics of liquids for calculating the nuclear magnetic resonance (NMR) chemical shifts of solvated molecules. Based on the reference interaction site model self-consistent field with constrained spatial electron density distribution (RISM-SCF-cSED) method, the electronic structure of molecules in solution is obtained, and the expression for the nuclear magnetic shielding tensor is derived as the second-order derivative of the Helmholtz energy of the solution system. We implemented a method for calculating chemical shifts and applied it to an adenine molecule in water, where hydrogen bonding plays a crucial role in electronic and solvation structures. We also performed the calculations of 17O chemical shifts, which showed remarkable solvent dependence. While converged results could not be sometimes obtained using the conventional method, in the present framework with RISM-SCF-cSED, an adequate representation of electron density is guaranteed, making it possible to obtain an NMR shielding constant stably. This introduction of cSED is key to extending the method's applicability to obtain the chemical shift of various chemical species. The present demonstration illustrates our approach's superiority in terms of numerical robustness and accuracy.

An Intensional Probability Theory: Investigating the Link between Classical and Quantum Probabilities

The link between classical and quantum theories is discussed in terms of extensional and intensional viewpoints. The paper aims to bring evidence that classical and quantum probabilities are related by intensionalization, which means that by abandoning sets from classical probability one should obtain quantum theory. Unlike the extensional concept of a set, the intensional probability is attributed to the quantum ensemble, which is contextually dependent. The contextuality offers a consistent realization of the measurement problem, which should require the existence of the time operator. The time continuum by Brouwer has satisfied such a requirement, which makes it fundamental to mathematical physics. The statistical model it provides has been proven tremendously useful in a variety of applications.

Quantifying the degree of disorder and associated phenomena in materials through zentropy: Illustrated with Invar Fe3Pt

Disorder exists in all materials at finite temperatures, implicating that the degree of disorder (fDoD) is a key parameter to tailor macroscopic functionalities of materials. Here, we propose to quantify fDoD as a function of temperature using properties such as configurational entropy predicted by zentropy − a theory to represent total entropy of a system via a nested formula through the integration of quantum mechanics and statistical mechanics. Taking Invar Fe3Pt as an example through first-principles based calculations from 0 K to finite temperatures, we demonstrate the capability of the present approach in predicting fDoD and associated phenomena in Fe3Pt such as Curie temperature and negative thermal expansion, showing a good agreement with experimental data.

Quantum dynamics using path integral coarse-graining.

The vibrational spectra of condensed and gas-phase systems are influenced by thequantum-mechanical behavior of light nuclei. Full-dimensional simulations of approximate quantum dynamics are possible thanks to the imaginary time path-integral (PI) formulation of quantum statistical mechanics, albeit at a high computational cost which increases sharply with decreasing temperature. By leveraging advances in machine-learned coarse-graining, we develop a PI method with the reduced computational cost of a classical simulation. We also propose a simple temperature elevation scheme to significantly attenuate the artifacts of standard PI approaches as well as eliminate the unfavorable temperature scaling of the computational cost. We illustrate the approach, by calculating vibrational spectra using standard models of water molecules and bulk water, demonstrating significant computational savings and dramatically improved accuracy compared to more expensive reference approaches. Our simple, efficient, and accurate method has prospects for routine calculations of vibrational spectra for a wide range of molecular systems - with an explicit treatment of the quantum nature of nuclei.

Quantum Mechanics: Statistical Balance Prompts Caution in Assessing Conceptual Implications.

Throughout quantum mechanics there is statistical balance, in the collective response of an ensemble of systems to differing measurement types. Statistical balance is a core feature of quantum mechanics, underlying quantum mechanical states, and not yet explained. The concept of "statistical balance" is here explored, comparing its meaning since 2019 with its original meaning in 2001. Statistical balance now refers to a feature of contexts in which: (a) there is a prescribed probability other than 0 or 1 for the collective response of an ensemble to one measurement type; and (b) the collective response of the same ensemble to another measurement type demonstrates that no well-defined value can be attributed, for the property relevant to the original measurement type, to individual members of the ensemble. In some unexplained way, the outcomes of single runs of a measurement of the original type "balance" each other to give an overall result in line with the prescribed probability. Unexplained statistical balance prompts caution in assessing the conceptual implications of entanglement, measurement, uncertainty, and two-slit and Bell-type analyses. Physicists have a responsibility to the wider population to be conceptually precise about quantum mechanics, and to make clear that many possible conceptual implications are uncertain.

Gravitational fields and quantum mechanics

Massive particles in spatially nonlocal, quantum mechanical wave functions should have interesting gravitational fields. Should their gravitational fields be classical? What aspects of their gravitational fields are observable? We study these questions using quantum field theory to gravitationally probe the wave functions of such quantum mechanical states. We find that the observable signals seem to indicate that it is very difficult to phenomenologically observe the quantum mechanical, spatially nonlocal superposition.

Flood Susceptibility Mapping Using Novel Hybrid Approach of Neural Network with Genetic Quantum Ensembles

Flood susceptibility mapping is an important method for flood research. In this paper, we combine a backpropagation neural network (BPNN) with a genetic quantum algorithm (GQA) for the first time to develop flood susceptibility mapping. The area on the Chinese side of the Tumen River Basin was selected as the research object. A set of flood conditioning factors was selected based on relevant literature and an actual situation and then validated using the chi-square test and correlation analysis. Different weights were assigned using stepwise weight assessment ratio analysis. Finally, modeling and flood susceptibility mapping using GQA-BPNN. As a reference, the same work was performed with both the pure BPNN and optimized BPNN using a genetic algorithm (GA). The results show that the area under the curve, root mean squared error, Nash-Sutcliffe coefficient and percentage of bias are significantly better for the GQA-BPNN than for the BPNN and GA-BPNN and that the flood sensitivity maps constructed by the GQA-BPNN have more flood points in high flood sensitivity areas. Therefore, the GQA-BPNN method can be considered an effective method for flood susceptibility mapping.

Bose–Einstein condensation of two-dimensional spin-1 Cooper pairs in a uniform magnetic field

Bose–Einstein condensation of two-dimensional spin-1 Cooper pairs in a uniform magnetic field is investigated within the framework of quantum statistical mechanics. We derive an exact analytical solution of two-dimensional spin-1 Cooper pairs in a uniform magnetic field, which involves a q-digamma function in mathematics. In two dimensions, it is found that the critical temperature and condensate fraction are monotonically increasing functions of magnetic field B, Landé factor g, and number density σ. Our investigation in two dimensions provides an example of the magnetic-field-tuned insulator-superconductor quantum phase transition.