Entropy and Irreversibility in Classical and Quantum Mechanics

Review of the irreversibility problem in modern physics with new researches is given. Some characteristics of the Markov chains are specified and the important property of monotonicity of a probability is formulated. Using one thin inequality, the behavior of relative entropy in the classical case is considered. Further we pass to studying of the irreversibility phenomena in quantum problems. By new method is received the Lindblad’s equation and its physical essence is explained. Deep analogy between the classical Markov processes and development described by the Lindblad’s equation is conducted. Using method of comparison of the Lind-blad’s equation with the linear Langevin equation we receive a system of differential equations, which are more general, than the Caldeira-Leggett equation. Here we consider quantum systems without inverse influ-ence on a surrounding background with high temperature. Quantum diffusion of a single particle is consid-ered and possible ways of the permission of the Schr?dinger’s cat paradox and the role of an external world for the phenomena with quantum irreversibility are discussed. In spite of previous opinion we conclude that in the equilibrium environment is not necessary to postulate the processes with collapses of wave functions. Besides, we draw attention to the fact that the Heisenberg’s uncertainty relation does not always mean the restriction is usually the product of the average values of commuting variables. At last, some prospects in the problem of quantum irreversibility are discussed.

Two-body exceptional points in open dissipative systems

We study two-body non-Hermitian physics in the context of an open dissipative system depicted by the Lindblad master equation.Adopting a minimal lattice model of a handful of interacting fermions with single-particle dissipation,we show that the non-Hermitian effective Hamiltonian of the master equation gives rise to two-body scattering states with state-and interaction-dependent parity-time transition.The resulting two-body exceptional points can be extracted from the trace-preserving density-matrix dynamics of the same dissipative system with three atoms.Our results not only demonstrate the interplay of parity-time symmetry and interaction on the exact few-body level,but also serve as a minimal illustration on how key features of non-Hermitian few-body physics can be probed in an open dissipative many-body system.

Quantum exceptional points of non-Hermitian Hamiltonian and Liouvillian in dissipative quantum Rabi model

The open quantum system can be described by either a Lindblad master equation or a non-Hermitian Hamiltonian(NHH).However,these two descriptions usually have different exceptional points(EPs),associated with the degeneracies in the open quantum system.Here,considering a dissipative quantum Rabi model,we study the spectral features of EPs in these two descriptions and explore their connections.We find that,although the EPs in these two descriptions are usually different,the EPs of NHH will be consistent with the EPs of master equation in the weak coupling regime.Further,we find that the quantum Fisher information(QFI),which measures the statistical distance between quantum states,can be used as a signature for the appearance of EPs.Our study may give a theoretical guidance for exploring the properties of EPs in open quantum systems.

TensorFlow solver for quantum Page Rank in large-scale networks

Google Page Rank is a prevalent algorithm for ranking the significance of nodes or websites in a network,and a recent quantum counterpart for Page Rank algorithm has been raised to suggest a higher accuracy of ranking comparing to Google Page Rank.The quantum Page Rank algorithm is essentially based on quantum stochastic walks and can be expressed using Lindblad master equation,which,however,needs to solve the Kronecker products of an O(N^(4))dimension and requires severely large memory and time when the number of nodes N in a network increases above 150.Here,we present an efficient solver for quantum Page Rank by using the Runge-Kutta method to reduce the matrix dimension to O(N^(2))and employing Tensor Flow to conduct GPU parallel computing.We demonstrate its performance in solving quantum stochastic walks on Erdos-Rényi graphs using an RTX 2060 GPU.The test on the graph of 6000 nodes requires a memory of 5.5 GB and time of 223 s,and that on the graph of 1000 nodes requires 226 MB and 3.6 s.Compared with QSWalk,a currently prevalent Mathematica solver,our solver for the same graph of 1000 nodes reduces the required memory and time to only 0.2%and 0.05%.We apply the solver to quantum Page Rank for the USA major airline network with up to 922 nodes,and to quantum stochastic walk on a glued tree of 2186 nodes.This efficient solver for large-scale quantum Page Rank and quantum stochastic walks would greatly facilitate studies of quantum information in real-life applications.