scholarly journals ON THE VON NEUMANN AND SHANNON ENTROPIES FOR QUANTUM WALKS ON Z2

2012 ◽  
Vol 10 (02) ◽  
pp. 1250020 ◽  
Author(s):  
CLEMENT AMPADU
Keyword(s):  
Discrete Time ◽  
Shannon Entropy ◽  
Quantum Walks ◽  
Von Neumann Entropy ◽  
Asymptotic Behaviors ◽  
Von Neumann ◽  
Time Quantum

We give asymptotic behaviors of the von Neumann entropy and the Shannon entropy of discrete-time time quantum walks on Z2.

Entanglement for quantum walks on the line

2011 ◽  
Vol 11 (9&10) ◽  
pp. 855-866
Author(s):  
Yusuke Ide ◽  
Norio Konno ◽  
Takuya Machida
Keyword(s):  
Information Theory ◽  
Shannon Entropy ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Von Neumann Entropy ◽  
Initial State ◽  
Von Neumann ◽  
Time Quantum ◽  
The One ◽  
Path Counting

The discrete-time quantum walk is a quantum counterpart of the random walk. It is expected that the model plays important roles in the quantum field. In the quantum information theory, entanglement is a key resource. We use the von Neumann entropy to measure the entanglement between the coin and the particle's position of the quantum walks. Also we deal with the Shannon entropy which is an important quantity in the information theory. In this paper, we show limits of the von Neumann entropy and the Shannon entropy of the quantum walks on the one dimensional lattice starting from the origin defined by arbitrary coin and initial state. In order to derive these limits, we use the path counting method which is a combinatorial method for computing probability amplitude.

scholarly journals On the von Neumann entropy of certain quantum walks subject to decoherence

2010 ◽  
Vol 20 (6) ◽  
pp. 1099-1115 ◽  
Author(s):  
CHAOBIN LIU ◽  
NELSON PETULANTE
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Von Neumann Entropy ◽  
Total Density ◽  
Projective Measurement ◽  
Time Step ◽  
Von Neumann ◽  
Bipartite Quantum System ◽  
Maximum Value ◽  
Time Quantum

In this paper, we consider a discrete-time quantum walk on the N-cycle governed by the condition that at every time step of the walk, the option persists, with probability p, of exercising a projective measurement on the coin degree of freedom. For a bipartite quantum system of this kind, we prove that the von Neumann entropy of the total density operator converges to its maximum value. Thus, when influenced by decoherence, the mutual information between the two subsystems corresponding to the space of the coin and the space of the walker must eventually diminish to zero. Put plainly, any level of decoherence greater than zero forces the system to become completely ‘disentangled’ eventually.

scholarly journals Discrete-time quantum walks generated by aperiodic fractal sequence of space coin operators

2018 ◽  
Vol 29 (10) ◽  
pp. 1850098 ◽  
Author(s):  
R. F. S. Andrade ◽  
A. M. C. Souza
Keyword(s):  
Wave Function ◽  
Probability Distribution ◽  
Discrete Time ◽  
Numerical Study ◽  
Entanglement Entropy ◽  
Quantum Effects ◽  
Quantum Walks ◽  
Cantor Set ◽  
One Dimensional ◽  
Time Quantum

Properties of one-dimensional discrete-time quantum walks (DTQWs) are sensitive to the presence of inhomogeneities in the substrate, which can be generated by defining position-dependent coin operators. Deterministic aperiodic sequences of two or more symbols provide ideal environments where these properties can be explored in a controlled way. Based on an exhaustive numerical study, this work discusses a two-coin model resulting from the construction rules that lead to the usual fractal Cantor set. Although the fraction of the less frequent coin [Formula: see text] as the size of the chain is increased, it leaves peculiar properties in the walker dynamics. They are characterized by the wave function, from which results for the probability distribution and its variance, as well as the entanglement entropy, were obtained. A number of results for different choices of the two coins are presented. The entanglement entropy has shown to be very sensitive to uncovering subtle quantum effects present in the model.

scholarly journals Dynamics and energy spectra of aperiodic discrete-time quantum walks

2017 ◽  
Vol 96 (1) ◽  
Author(s):  
N. Lo Gullo ◽  
C. V. Ambarish ◽  
Th. Busch ◽  
L. Dell'Anna ◽  
C. M. Chandrashekar
Keyword(s):  
Discrete Time ◽  
Quantum Walks ◽  
Energy Spectra ◽  
Time Quantum

The non-uniform stationary measure for discrete-time quantum walks in one dimension

2015 ◽  
Vol 15 (11&12) ◽  
pp. 1060-1075
Author(s):  
Norio Konno ◽  
Masato Takei
Keyword(s):  
Eigenvalue Problem ◽  
Discrete Time ◽  
Quantum Walks ◽  
Stationary Measure ◽  
Unitary Matrix ◽  
Uniform Measure ◽  
Simple Argument ◽  
Stationary Measures ◽  
Time Quantum ◽  
The One

We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2 $\times$ 2 unitary matrix $U$. In our previous paper \cite{Konno2014}, we proved that any uniform measure becomes the stationary measure of the QW by solving the corresponding eigenvalue problem. This paper reports that non-uniform measures are also stationary measures of the QW except when $U$ is diagonal. For diagonal matrices, we show that any stationary measure is uniform. Moreover, we prove that any uniform measure becomes a stationary measure for more general QWs not by solving the eigenvalue problem but by a simple argument.

Photonic discrete-time quantum walks [Invited]

2020 ◽  
Vol 18 (5) ◽  
pp. 052701
Author(s):  
Gaoyan Zhu ◽  
Lei Xiao ◽  
Bingzi Huo ◽  
Peng Xue
Keyword(s):  
Discrete Time ◽  
Quantum Walks ◽  
Time Quantum
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