Quantum walk on the line: entanglement and non-local initial conditions

Abstract

The conditional shift in the evolution operator of a quantum walk generates entanglement between the coin and position degrees of freedom. This entanglement can be quantified by the von Neumann entropy of the reduced density operator (entropy of entanglement). In the long time limit, it converges to a well defined value which depends on the initial state. Exact expressions for the asymptotic (long-time) entanglement are obtained for (i) localized initial conditions and (ii) initial conditions in the position subspace spanned by the +1 and -1 position eigenstates.

arXiv commentA few mistakes where corrected. One of them leads to a factor of 2 in eq. (49), the other results remain unchanged. In this version, several figures where replaced by color versions

References in corpus (9)

• Entanglement Measures and Purification Procedures
• Quantum random walks - an introductory overview
• A Quantum Random Walk Search Algorithm
• Implementing the quantum random walk
• Entanglement in coined quantum walks on regular graphs
• Decoherence in the quantum walk on the line
• Quantum random walk on the line as a markovian process
• Quantum Walks with Entangled Coins
• Measures of entanglement in quantum mechanics

Cited by in corpus (69)

• Quantum walks: a comprehensive review
• Realization of quantum walks with negligible decoherence in waveguide lattices
• Quantum walks and wavepacket dynamics on a lattice with twisted photons
• Photonic quantum walk in a single beam with twisted light
• Mimicking the probability distribution of a two-dimensional Grover walk with a single-qubit coin
• Dynamically Disordered Quantum Walk as a Maximal Entanglement Generator
• Quantum walk as a generalized measuring device
• Universal Nonlinear Disordered Wave Packet Subdiffusion: 12 Decades
• Entangling Power of Disordered Quantum Walks
• Decoherence in one-dimensional Quantum Walk
• Two-dimensional quantum walk under artificial magnetic field
• Quantum walks with sequential aperiodic jumps
• Experimental demonstration of quantum walks with initial superposition states
• Continuous-time quantum walks on one-dimension regular networks
• Nonlocality, quantum correlations, and violations of classical realism in the dynamics of two noninteracting quantum walkers
• Generation of hybrid maximally entangled states in a one-dimensional quantum walk
• Aperiodic space-inhomogeneous quantum walks: localization properties, energy spectra and enhancement of entanglement
• Genuine Parrondo's paradox in quantum walks with time-dependent coin operators
• Asymptotic entanglement in 1D quantum walks with a time-dependent coined
• The Limiting Distribution of Decoherent Quantum Random Walks
• Asymptotic entanglement in quantum walks from delocalized initial states
• Quantum walks: decoherence and coin-flipping games
• Coherent exciton transport and trapping on long-range interacting cycles
• Conditional Quantum Walk and Iterated Quantum Games
• Weak disorder enhancing the production of entanglement in quantum walks
• Connecting velocity and entanglement in quantum walks
• Disorder-free localization in quantum walks
• Maximal entanglement from quantum random walks
• Multiple transitions between normal and hyperballistic diffusion in quantum walks with time-dependent jumps
• Classical-like behavior in quantum walks with inhomogeneous, time-dependent coin operators
• Asymptotic properties of the Dirac quantum cellular automaton
• Thermodynamics of N-dimensional quantum walks
• Quantum walk on a cylinder
• Strongly trapped two-dimensional quantum walks
• Accelerated quantum walk, two-particle entanglement generation and localization
• Effects of non-local initial conditions in the Quantum Walk on the line
• Maximal coin-position entanglement generation in a quantum walk for the third step and beyond regardless of the initial state
• Enhancing entanglement with the generalized elephant quantum walk from localized and delocalized states
• Trojan Quantum Walks
• Scaling and crossover behaviour in a truncated long range quantum walk
• Quantum Walk-based Generation of Entanglement Between Two Walkers
• Maximal coin-walker entanglement in a ballistic quantum walk
• Negative correlations can play a positive role in disordered quantum walks
• Almost compact moving breathers with fine-tuned discrete time quantum walks
• Initial-state-dependent thermalization in open qubits
• Implementation of the quantum walk step operator in lateral quantum dots
• A quantum walk with a delocalized initial state: contribution from a coin-flip operator
• Tunneling effects in a one-dimensional quantum walk
• Quantum walks in two dimensions: controlling directional spreading with entangling coins and tunable disordered step operator
• Resonant quantum kicked rotor with two internal levels
• High-fidelity state transfer via quantum walks from delocalized states
• Entanglement entropy distinguishes PT-symmetry and topological phases in a class of non-unitary quantum walks
• History states of one-dimensional quantum walks
• Analytical expression for variance of homogeneous-position quantum walk with decoherent position
• Entanglement for discrete-time quantum walks on the line
• A simple coin for a $2d$ entangled walk
• Quantum walk for SU(1,1)
• Discrete time quantum walk of locally interacting walkers
• Measure-independent description of wave-particle duality via coherence
• Möbius Quantum Walk
• Limit theorems of a 3-state quantum walk and its application for discrete uniform measures
• Studies of braided non-Abelian anyons using anyonic tensor networks
• Optimization for the propagation of a multiparticle quantum walk in a one-dimensional lattice
• Quantum Magic in Discrete-Time Quantum Walk
• Directional quantum random walk induced by coherence
• Realization of the probability laws in the quantum central limit theorems by a quantum walk
• Continuous-Time Quantum Walk on Penrose Lattice
• Directionality and quantum backfire in continuous-time quantum walks from delocalized states: Exact results
• On the von Neumann entropy of certain quantum walks subject to decoherence