Three-flavoured neutrino oscillations and the Leggett–Garg inequality

Three-flavoured neutrino oscillations are investigated in the light of the Leggett–Garg inequality (LGI). The results obtained are: (a) The maximum violation of the LGI is 2.17036 for neutrino path length \(L_{1}=140.15 \) km and \(\Delta L=1255.7 \) km. (b) The presence of the mixing angle \(\theta _{13}\) enhances the maximum violation of LGI by \(4.6\%\). (c) The currently known mass hierarchy parameter \(\alpha = 0.0305\) increases the maximum violation of LGI by \(3.7\%\). (d) The presence of a CP-violating phase parameter enhances the maximum violation of LGI by \(0.24\%\), thus providing an alternative indicator of CP violation in three-flavoured neutrino oscillations. The outline of an experimental proposal is suggested whereby the findings of this investigation may be verified.     

Formulation of Leggett–Garg Inequalities in Terms of q-Entropies

As is well known, the macroscopic realism and the noninvasive measurability together lead to Leggett–Garg inequalities violated by quantum mechanics. We consider tests of the Leggett–Garg type with use of the q-entropies.For all q≥1, quantum mechanics predicts violations of an entire family of q-entropic inequalities of the Leggett–Garg type. Violations are exemplified with a quantum spin-s system. In general, entropic Leggett–Garg inequalities give only necessary conditions that some probabilistic model is compatible with the macrorealism in the broader sense. The presented q-entropic inequalities allow to widen a class of situations, in which an incompatibility with the macrorealism can be tested. In the considered example, both the strength and range of violations are somehow improved by varying q.We also examine q-entropic inequalities of the Leggett–Garg type in the case of detection inefficiencies, when the no-click event may occur in each observation. With the use of the q-entropic inequalities, the required amount of efficiency may be reduced.     

Tsallis relative α entropy of coherence dynamics in Grover′s search algorithm

Quantum coherence plays a central role in Grover’s search algorithm. We study the Tsallis relative a entropy of coherence dynamics of the evolved state in Grover’s search algorithm. We prove that the Tsallis relative a entropy of coherence decreases with the increase of the success probability, and derive the complementarity relations between the coherence and the success probability. We show that the operator coherence of the first H■ relies on the size of the database N, the success probabilit...     

Quantum uncertainty relations of Tsallis relative α entropy coherence based on MUBs

In this paper,we discuss quantum uncertainty relations of Tsallis relative α entropy coherence for a single qubit system based on three mutually unbiased bases.For α∈[1/2,1)U(1,2],the upper and lower bounds of sums of coherence are obtained.However,the above results cannot be verified directly for any α∈(0,1/2).Hence,we only consider the special case of α=1/n+1,where n is a positive integer,and we obtain the upper and lower bounds.By comparing the upper and lower bounds,we find that the upper bound is equal to the lower bound for the special α=1/2,and the differences between the upper and the lower bounds will increase as α increases.Furthermore,we discuss the tendency of the sum of coherence,and find that it has the same tendency with respect to the different θ or φ,which is opposite to the uncertainty relations based on the Rényi entropy and Tsallis entropy.     

Implication of Tsallis entropy in the Thomas–Fermi model for self-gravitating fermions

The Thomas–Fermi approach for self-gravitating fermions is revisited within the theoretical framework of the q$q$-statistics  . Starting from the q$q$-deformation of the Fermi–Dirac distribution function, a generalized Thomas–Fermi equation is derived. It is shown that the Tsallis entropy   preserves a scaling property of this equation. The q$q$-statistical   approach to Jeans’ instability in a system of self-gravitating fermions is also addressed. The dependence of the Jeans’ wavenumber (or the Jeans length) on the parameter q$q$ is traced. It is found that the q$q$-statistics makes the Fermionic system unstable at scales shorter than the standard Jeans length.     

The Weiss model and the Landau–Khalatnikov model for the switching of ferroelectrics

It is shown that the molecular field theory by P. Weiss formally leads to the switching kinetics of ferroelectrics, which is described by the well-known Landau–Khalatnikov equation. The switching has a critical character, taking place only at E_a>E_c (E_a: external field, E_c: coercive field). The results are checked by computer simulations.     

The role of conformal symmetry in the Jackiw–Pi model

The Jackiw–Pi model in 2+1$2+1$ dimensions is a non-relativistic conformal field theory of charged particles with point-like self-interaction. For specific values of the interaction strengths the classical theory possesses vortex and multi-vortex solutions, which are all degenerate in energy. We compute the full set of first-order perturbative quantum corrections. Only the coupling constant g²$g^2$ requires renormalization; the fields and electric charge e are not renormalized. It is shown that in general the conformal symmetries are broken by an anomalous contribution to the conservation law, proportional to the β-function. However, the β  -function vanishes upon restricting the coupling constants to values g²=±e²$g^2=\pm e^2$, which includes the case in which vortex solutions exist. Therefore the existence of vortices also guarantees the preservation of the conformal symmetries.     

Quantum coherence and ground-state phase transition in a four-chain Bose–Hubbard model

We study the quantum coherence and ground-state phase transition of a four-chain Bose–Hubbard model with the long-range interaction. In a special four-chain Bose–Hubbard model,i.e., each chain only has one optical potential, four types of the ground-state phases are discovered. The effects of the disorder, the on-site interaction and the long-range interaction on the quantum coherence are studied. For the system without the long-range interaction, the quantum coherence changes from one periodic oscillation to two periodic oscillations as the onsite interaction increases. By considering the long-range interaction, the quantum coherence goes back to one periodic oscillation again. The on-site interaction itself suppresses the quantum coherence, both the on-site interaction and long-range interaction together enhance the quantum coherence with the weak disorder. If the disorder strength is increased beyond a critical value,they start to suppress the quantum coherence. In a regular four-chain Bose–Hubbard model, i.e.,each chain has many optical potentials, the ground-state phase transitions are obtained by using the cluster Gutzwiller mean-field method. Exotic ground-state phases are found, i.e., superfluid phase, integer Mott insulator phase, supersolid phase and loophole insulator phase. The combination of the loophole insulator phase and the supersolid phase expands the lobes with the half-integer filling per site for the small ratio β = t_■/t_⊥.     

The Feynman–Wilson gas and the Lund model

We derive a partition function for the Lund fragmentation model and compare it with that of a classical gas. For a fixed rapidity “volume” this partition function corresponds to a multiplicity distribution which is very close to a binomial distribution. We compare our results with the multiplicity distributions obtained from the JETSET Monte Carlo for several scenarios. Firstly, for the fragmentation vertices of the Lund string. Secondly, for the final state particles both with and without decays. Received: 27 August 1998 / Published online: 15 October 1998     

Energy–entropy competition and the effectiveness of stochastic gradient descent in machine learning

ABSTRACT

Finding parameters that minimise a loss function is at the core of many machine learning methods. The Stochastic Gradient Descent (SGD) algorithm is widely used and delivers state-of-the-art results for many problems. Nonetheless, SGD typically cannot find the global minimum, thus its empirical effectiveness is hitherto mysterious. We derive a correspondence between parameter inference and free energy minimisation in statistical physics. The degree of undersampling plays the role of temperature. Analogous to the energy–entropy competition in statistical physics, wide but shallow minima can be optimal if the system is undersampled, as is typical in many applications. Moreover, we show that the stochasticity in the algorithm has a non-trivial correlation structure which systematically biases it towards wide minima. We illustrate our argument with two prototypical models: image classification using deep learning and a linear neural network where we can analytically reveal the relationship between entropy and out-of-sample error.     

The ground state of the Frenkel–Kontorova model

The continual approximation of the ground state of the discrete Frenkel–Kontorova model is tested using a symmetric algorithm of numerical simulation. A “kaleidoscope effect” is found, which means that the curves representing the dependences of the relative extension of an N-atom chain vary periodically with increasing N. Stairs of structural transitions for N ? 1 are analyzed by the channel selection method with the approximation N = ∞. Images of commensurable and incommensurable structures are constructed. The commensurable–incommensurable phase transitions are stepwise.     

The Ising version of the t–J model

The t–J model is analysed in the limit of strong anisotropy, where the transverse components of electron spin are neglected. We propose a slave-particle-type approach that is valid, in contradiction to many of the standard approaches, in the low-doping regime and becomes exact for a half-filled system. We describe an effective method that allows to numerically study the system with the no-double-occupancy constraint rigorously taken into account at each lattice site. Then, we use this approach to demonstrate the destruction of the antiferromagnetic order by increasing the doping and formation of Nagaoka polarons in the strong interaction regime.     

Quench of non-Markovian coherence in the deep sub-Ohmic spin–boson model: A unitary equilibration scheme

The deep sub-Ohmic spin–boson model shows a longstanding non-Markovian coherence at low temperature. Motivating to quench this robust coherence, the thermal effect is unitarily incorporated into the time evolution of the model, which is calculated by the adaptive time-dependent density matrix renormalization group algorithm combined with the orthogonal polynomials theory. Via introducing a unitary heating operator to the bosonic bath, the bath is heated up so that a majority portion of the bosonic excited states is occupied. It is found in this situation the coherence of the spin is quickly quenched even in the coherent regime, in which the non-Markovian feature dominates. With this finding we come up with a novel way to implement the unitary equilibration, the essential term of the eigenstate-thermalization hypothesis, through a short-time evolution of the model.     

Beyond the Shannon–Khinchin formulation: The composability axiom and the universal-group entropy

The notion of entropy is ubiquitous both in natural and social sciences. In the last two decades, a considerable effort has been devoted to the study of new entropic forms, which generalize the standard Boltzmann–Gibbs (BG) entropy and could be applicable in thermodynamics, quantum mechanics and information theory. In Khinchin (1957), by extending previous ideas of Shannon (1948) and Shannon and Weaver (1949), Khinchin proposed a characterization of the BG entropy, based on four requirements, nowadays known as the Shannon–Khinchin (SK) axioms.     

The generalization of the Mermin–Wagner theorem and the possibility of long-range order in the isotropic discrete one-dimensional quantum Heisenberg model

The problem of existence of long-range order in the isotropic quantum Heisenberg model on the D=1 lattice is reconsidered in view of the possibility of sufficiently slow decaying exchange interaction with infinite effective radius. It is shown that the macrosopic arguments given by Landau and Lifshitz and then supported microscopically by Mermin and Wagner fail for this case so that the non-zero spontaneous magnetization may yet exist. This result was anticipated by Thouless on the grounds of phenomenological analysis, and we give its microscopic foundation, which amounts to the generalization of Mermin–Wagner theorem for the case of the infinite second moment of the exchange interaction. Two well known in lattice statistics models – i.e., Kac-I and Kac-II – illustrate our results.     

The uncertainty principle and Bekenstein–Hawking entropy corrections

There is much attention on the corrections to Bekenstein–Hawking entropy in area with a model-dependent coefficient. The corrections are generally composed of two parts: quantum corrections and thermal corrections. The generalized uncertainty principle (GUP), which will reduce to the conventional Heisenberg relation in situations of weak gravity, is one of the candidates to be utilized to obtain the quantum corrections to the Bekenstein–Hawking entropy. Recently the extended uncertainty principle (EUP) and generalized extended uncertainty principle (GEUP) are introduced to calculate entropy corrections with large length scales limit. In this paper, we obtain the quantum corrections to Bekenstein–Hawking entropy in four-dimensional Schwarzschild black holes based on the EUP and GEUP. Some attractive results are derived.     

Random-bond and random-anisotropy effects in the phase diagram of the Blume–Capel model

We report on Monte Carlo studies of the influence of quenched randomness on the phase diagram of the three-dimensional (3D) Blume–Capel model. The randomness is supposed to act either on the exchange coupling constants (bond randomness) or on the anisotropy distribution. With increasing disorder, first-order phase transitions are shown to change into second-order phase transitions. The trajectory of the tricritical point in the phase space as a function of disorder is presented. We have also calculated critical exponents at some points in the second-order phase region which show a change of universality class in agreement with the Harris criterion.     

Partial coherence in the core–halo picture of Bose–Einstein n-particle correlations

We study the influence of a possible coherent component in the boson source on the two-, three- and n-particle correlation functions in a generalized core–halo-type boson-emitting source. In particular, a simple formula is presented for the strength of the n-particle correlation functions for such systems. Graph rules are obtained to evaluate the correlation functions of arbitrarily high order. The importance of an experimental determination of the 4-th and 5-th order Bose–Einstein correlation function is emphasized. Received: 18 December 1998 / Published online: 20 May 1999