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 Limit theorems for the discrete-time quantum walk on a graph with joined half lines

2012 ◽  
Vol 12 (3&4) ◽  
pp. 314-333
Author(s):  
Kota Chisaki ◽  
Norio Konno ◽  
Etsuo Segawa
Keyword(s):  
Asymptotic Behavior ◽  
Discrete Time ◽  
Limit Theorems ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Half Line ◽  
Initial State ◽  
Time Quantum ◽  
Weak Convergence Theorem ◽  
The One

We consider a discrete-time quantum walk W_{t,\kappa} at time t on a graph with joined half lines J_\kappa, which is composed of \kappa half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W_{t,\kappa} can be reduced to the walk on a half line even if the initial state is asymmetric. For W_{t,\kappa}, we obtain two types of limit theorems. The first one is an asymptotic behavior of W_{t,\kappa} which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W_{t,\kappa}. On each half line, W_{t,\kappa} converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.

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.

The fog on: Generalized teleportation by means of discrete-time quantum walks on N-lines and N-cycles

2019 ◽  
Vol 33 (23) ◽  
pp. 1950270 ◽  
Author(s):  
Duc Manh Nguyen ◽  
Sunghwan Kim
Keyword(s):  
Discrete Time ◽  
Quantum Teleportation ◽  
Quantum Walk ◽  
Quantum Walks ◽  
One Dimensional ◽  
Time Quantum ◽  
Infinite Line ◽  
The One

The recent paper entitled “Generalized teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles” by Yang et al. [Mod. Phys. Lett. B 33(6) (2019) 1950069] proposed the quantum teleportation by means of discrete-time quantum walks on [Formula: see text]-lines and [Formula: see text]-cycles. However, further investigation shows that the quantum walk over the one-dimensional infinite line can be based over the [Formula: see text]-cycles and cannot be based on [Formula: see text]-lines. The proofs of our claims on quantum walks based on finite lines are also provided in detail.

scholarly journals CONTINUOUS-TIME QUANTUM WALKS ON TREES IN QUANTUM PROBABILITY THEORY

2006 ◽  
Vol 09 (02) ◽  
pp. 287-297 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Probability Theory ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Probability ◽  
Quantum Walks ◽  
Homogeneous Tree ◽  
One Dimensional ◽  
Time Quantum ◽  
New Type ◽  
The One

A quantum central limit theorem for a continuous-time quantum walk on a homogeneous tree is derived from quantum probability theory. As a consequence, a new type of limit theorems for another continuous-time walk introduced by the walk is presented. The limit density is similar to that given by a continuous-time quantum walk on the one-dimensional lattice.

scholarly journals Probing the Eigenstates Thermalization Hypothesis with Many-Particle Quantum Walks on Lattices

2017 ◽  
Vol 24 (02) ◽  
pp. 1750007
Author(s):  
Dibwe Pierrot Musumbu ◽  
Maria Przybylska ◽  
Andrzej J. Maciejewski
Keyword(s):  
Thermal State ◽  
Particle System ◽  
Hamiltonian Dynamics ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Grid Graphs ◽  
Time Quantum ◽  
Counting Statistics ◽  
The Many ◽  
The One

We simulate the dynamics of many-particle system of bosons and fermions using discrete time quantum walks on lattices. We present a computational proof of the behaviour of the simulated systems similar to the one observed in Hamiltonian dynamics during quantum thermalization. We record the time evolution of the entropy and the temperature of a specific particle configuration during the entire dynamics and observe how they relax to a state which we call the quantum walk thermal state. This observation is made on two types of lattices while simulating different numbers of particles walking on two grid graphs with 25 vertices. In each case, we observe that the vertices counting statistics, the temperature of the indexed configuration and the dimension of the effective configuration Hilbert space relax simultaneously and remain relaxed for the rest of the many-particle quantum walk.

scholarly journals Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 504
Author(s):  
Ce Wang ◽  
Caishi Wang
Keyword(s):  
Discrete Time ◽  
Quantum Information Processing ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Integer Lattice ◽  
Initial State ◽  
Higher Dimensional ◽  
Time Quantum ◽  
Gauss Type ◽  
Limit Probability

As a discrete-time quantum walk model on the one-dimensional integer lattice Z , the quantum walk recently constructed by Wang and Ye [Caishi Wang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897–1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer d ≥ 2 , by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice Z d , which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.

scholarly journals Transport Efficiency of Continuous-Time Quantum Walks on Graphs

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 85
Author(s):  
Luca Razzoli ◽  
Matteo G. A. Paris ◽  
Paolo Bordone
Keyword(s):  
Transport Properties ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Transport Processes ◽  
Initial State ◽  
Transport Efficiency ◽  
Harvesting Systems ◽  
Time Quantum ◽  
Modeling Transport

Continuous-time quantum walk describes the propagation of a quantum particle (or an excitation) evolving continuously in time on a graph. As such, it provides a natural framework for modeling transport processes, e.g., in light-harvesting systems. In particular, the transport properties strongly depend on the initial state and specific features of the graph under investigation. In this paper, we address the role of graph topology, and investigate the transport properties of graphs with different regularity, symmetry, and connectivity. We neglect disorder and decoherence, and assume a single trap vertex that is accountable for the loss processes. In particular, for each graph, we analytically determine the subspace of states having maximum transport efficiency. Our results provide a set of benchmarks for environment-assisted quantum transport, and suggest that connectivity is a poor indicator for transport efficiency. Indeed, we observe some specific correlations between transport efficiency and connectivity for certain graphs, but, in general, they are uncorrelated.

scholarly journals A QUANTUM WALK WITH A DELOCALIZED INITIAL STATE: CONTRIBUTION FROM A COIN-FLIP OPERATOR

2013 ◽  
Vol 11 (05) ◽  
pp. 1350053 ◽  
Author(s):  
TAKUYA MACHIDA
Keyword(s):  
Shift Operator ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Fourier Series Expansion ◽  
Initial Condition ◽  
Initial State ◽  
Limit Distributions ◽  
Time Quantum ◽  
Long Time ◽  
Position Shift

A unit evolution step of discrete-time quantum walks (QWs) is determined by both a coin-flip operator and a position-shift operator. The behavior of quantum walkers after many steps delicately depends on the coin-flip operator and an initial condition of the walk. To get the behavior, a lot of long-time limit distributions for the QWs starting with a localized initial state have been derived. In this paper, we compute limit distributions of a 2-state QW with a delocalized initial state, not a localized initial state, and discuss how the walker depends on the coin-flip operator. The initial state induced from the Fourier series expansion, which is called the (α, β) delocalized initial state in this paper, provides different limit density functions from the ones of the quantum walk with a localized initial state.

Return probability of quantum walks with final-time dependence

2011 ◽  
Vol 11 (9&10) ◽  
pp. 761-773
Author(s):  
Yusuke Ide ◽  
Norio Konno ◽  
Takuya Machida ◽  
Etsuo Segawa
Keyword(s):  
Random Walk ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Time Dependent ◽  
One Dimension ◽  
Final Time ◽  
Counting Approach ◽  
Return Probability ◽  
Time Quantum ◽  
Path Counting

We analyze final-time dependent discrete-time quantum walks in one dimension. We compute asymptotics of the return probability of the quantum walk by a path counting approach. Moreover, we discuss a relation between the quantum walk and the corresponding final-time dependent classical random walk.

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