We study the temporal growth of the von Neumann entropy for dissipative quantum walks on networks. By using a phenomenological quantum master equation, the quantum stochastic walk (QSW), we are able to parametrically scan the crossover from purely coherent quantum walks to purely diffusive random walks. In the latter limit the entropy shows a logarithmic growth, which is proportional to the information dimension of the random walk on the network. Here we present results for the von Neumann entropy based on the reduced density operator of the QSW. It shows a similar logarithmic growth for a wide range of parameter values and networks. As a consequence, we propose the logarithmic growth rate of the von Neumann entropy to be a natural extension of the information dimension to dissipative quantum systems. We corroborate our results by comparing to numerically exact simulations.
We study the temporal growth of the von Neumann entropy for dissipative quantum walks on networks. By using a phenomenological quantum master equation, the quantum stochastic walk (QSW), we are able to parametrically scan the crossover from purely coherent quantum walks to purely diffusive random walks. In the latter limit the entropy shows a logarithmic growth, which is proportional to the information dimension of the random walk on the network. Here we present results for the von Neumann entropy based on the reduced density operator of the QSW. It shows a similar logarithmic growth for a wide range of parameter values and networks. As a consequence, we propose the logarithmic growth rate of the von Neumann entropy to be a natural extension of the information dimension to dissipative quantum systems. We corroborate our results by comparing to numerically exact simulations.
Abstract. The coin-position entanglement generated by the evolution operator of a discrete–time quantum walk is quantified, using the von Neumann entropy of the reduced density operator (entropy of entanglement). In the case of a single walker, the entropy of entanglement converges, in the long time limit, to a well defined value which depends on the initial state. Exact expressions are obtained for local and non-local initial conditions. We also discuss the asymptotic bi-partite entanglement generated by non-separable coin operations for two coherent quantum walkers. In this case, the entropy of entanglement is observed to increase logarithmically with time. 1.
This study investigates the crossover dynamics of a one-dimensional discrete-time quantum walk (DTQW) in noisy environments using the density matrix formalism and Kraus operator representation of quantum operations. Decoherence is modeled by bit-flip, dephasing, and bit-phase noise channels, while dissipation is represented by generalized amplitude damping channels. The analysis of crossover dynamics involves examining the probability distribution, scaling exponent of variance, coherence measures, and entropy metrics such as Shannon and von Neumann entropy. The DTQW produces a symmetrical probability distribution of displacement under decoherence, with distinct features in scaling exponents of mean squared displacement or variance. Under dissipation, DTQW produces a non-symmetric probability distribution and spreads faster than classical walk under extreme decoherence. These suggest a broad class of crossover transport behavior, which can be classified as pure quantum walk, quantum-like walk, semi-classical-like walk, and classical-like walk. Remarkably, maximum Shannon entropy is achieved at lower degrees of decoherence. This study also analyzes the effect of temporal disorder in the coin operator modeled using fractional Gaussian noise (fGn). The crossover dynamic occurs in the form of weak localization, followed by enhanced entanglement and information backflow. Notably, the correlation of fGn is transferred to the coin's degree of entanglement and transpires in the two-point correlation of the von Neumann entropy fluctuation. These findings can be useful for implementing DTQW-based protocols in quantum error correction, quantum cryptography, and quantum memory on the noisy intermediate-scale quantum (NISQ) platform.
In this manuscript, we study discrete-time dynamics of systems that arise in physics and information theory, and the measure of disorder in these systems known as dy- namical entropy. The study of dynamics in classical systems is done from two distinct viewpoints: random walks and dynamical systems. Random walks are probabilistic in nature and are described by stochastic processes. On the other hand, dynami- cal systems are described algebraically and deterministic in nature. The measure of disorder from either viewpoint is known as dynamical entropy. Entropy is an essential notion in physics and information theory. Motivated by the study of disorder for the positions and velocities of gas molecules, the notion of entropy was first introduced mathematically by Boltzmann near the end of the 19th Century and gives rise to the second law of thermodynamics. Almost eighty years after Boltzmann, Shannon became the father of the new field of information theory when he produced his groundbreaking works where he used entropy as a measure of information transfer between two sources. In the last two years of the 1950’s, Kolmogorov and Sinai extended the notions of Boltzmann to a dynamical entropy. The Kolmogorov-Sinai dynamical entropy gives a measure for the disorder of a system of particles (e.g. gas molecules) averaged over time, quantifying the uncertainty in the dynamics of a system. The advent of quantum mechanics and its pervasiveness in nature has required the development of non-commutative generalizations of dynamics and dynamical entropy to the quantum regime. Many of each have been proposed. In particular, we recall the definitions of quantum random walks, dynamical systems and Markov chains. We motivate each generalization by relating to its classical counterpart. Quantum dynamical entropy (QDE) is a generalization of the Kolmogorov-Sinai dynamical entropy to quantum mechanics. There have been numerous definitions for QDE beginning with that of Connes, Narnhofer and Thirring in 1987. We focus on the semi- classical approach given by Słomczyński and Życzkowski in 1994 and the quantum Markov chain approach which started with Accardi, Ohya and Watanabe in 1997. Linearity of a dynamical entropy means that the dynamical entropy of the n-fold composition of a dynamical map with itself is equal to n times the dynamical entropy of the map for every positive integer n. We show that the quantum dynamical entropy introduced by Słomczyński and Życzkowski is nonlinear in the time interval between successive measurements of a quantum dynamical system. This is in contrast to Kolmogorov-Sinai dynamical entropy for classical dynamical systems, which is linear in time. We also compute the exact values of quantum dynamical entropy for the Hadamard walk with varying Lüders-von Neumann instruments and partitions. In 1948, Shannon proved the Source Coding Theorem which gives upper and lower bounds on the minimal expected codeword length in terms of the entropy of a ran- dom variable. This theorem can be leveraged to give the minimal expected average codeword length for a string of symbols in terms of entropy rate, which can be in- terpreted as the dynamical entropy of a stochastic process. In 1994, Schumacher defined indeterminate-length quantum codes and proved a Quantum Source Coding Theorem. We introduce the notion of stochastic ensembles of pure states and give a novel representation in terms of quantum Markov chains. Moreover, this representa- tion allows us to extend the Quantum Source Coding Theorem, giving the minimal expected average codeword length of an indeterminate-length quantum code in terms of quantum dynamical entropy. This manuscript includes joint work with Dr. George Androulakis (Mathematics, University of South Carolina).
The universal dynamical phenomenon of von Neumann entropy plays an important role in understanding the nature of open many-body systems, as it uncovers general collective properties driven by correlation and quantum dynamics. Here, we reveal a universal time evolution behavior of bipartite von Neumann entropy in open quantum systems with dissipation into the surrounding environments. The coupling with the environment leads to a rapid growth of the entropy, resulting in an exotic short-time evolution law, S ( t ) ∼ − t ln t . Such a fast growth behavior is proved to be a general phenomenon for generic many-body systems in dissipative environments, as long as the systems exhibit well-defined fermionic quasiparticles. Interestingly, even for strongly interacting Luttinger liquids, which display vanishing fermionic but emergent bosonic quasiparticles, the same dynamical phenomenon is still found to persist. These results demonstrate a universal Liouvillian dynamical feature that has no counterpart in isolated Hamiltonian systems.
Based on studies of four specific networks, we conjecture a general relation between the walk dimensions dw of discrete-time random walks and quantum walks with the (self-inverse) Grover coin. In each case, we find that dw of the quantum walk takes on exactly half the value found for the classical random walk on the same geometry. Since walks on homogeneous lattices satisfy this relation trivially, our results for heterogeneous networks suggest that such a relation holds irrespective of whether translational invariance is maintained or not. To develop our results, we extend the renormalization-group analysis (RG) of the stochastic master equation to one with a unitary propagator. As in the classical case, the solution $\ensuremath{\rho}(x,t)$ in space and time of this quantum-walk equation exhibits a scaling collapse for a variable xdw/t in the weak limit, which defines dw and illuminates fundamental aspects of the walk dynamics, e.g., its mean-square displacement. We confirm the collapse for $\ensuremath{\rho}(x,t)$ in each case with extensive numerical simulation. The exact values for dw themselves demonstrate that RG is a powerful complementary approach to study the asymptotics of quantum walks that weak-limit theorems have not been able to access, such as for systems lacking translational symmetries beyond simple trees.
We have investigated the time evolution of a free particle in interaction with a phonon thermal bath, using the tight-binding approach. A dissipative quantum walk can be defined and many important non-equilibrium decoherence properties can be investigated analytically. The non-equilibrium statistics of a pure initial state have been studied. Our theoretical results indicate that the evolving wave-packet shows the suppression of Anderson’s boundaries (ballistic peaks) by the presence of dissipation. Many important relaxation properties can be studied quantitatively, such as von Neumann’s entropy and quantum purity. In addition, we have studied Wigner’s function. The timedependent behavior of the quantum entanglement between a free particle—in the lattice—and the phonon bath has been characterized analytically. This result strongly suggests the non-trivial time dependence of the off-diagonal elements of the reduced density matrix of the system. We have established a connection between the quantum decoherence and the dissipative parameter arising from interaction with the phonon bath. The time-dependent behavior of quantum correlations has also been pointed out, showing continuous transition from quantum random walk to classical random walk, when dissipation increases.