# One-Dimensional Quantum Walks with a Position-Dependent Coin

## TL;DR

This paper explores how a position-dependent coin in a one-dimensional quantum walk influences the probability distribution, entropy, and entanglement, revealing new control mechanisms for quantum walk dynamics.

## Contribution

It introduces a position-dependent coin model that enables diverse probability distributions and entropy behaviors, advancing control over quantum walk evolution.

## Key findings

- Probability distributions vary with rotation angle
- Entropy is generally reduced with position-dependent coins
- Entanglement leads to increased localization

## Abstract

We investigate the evolution of a discrete-time one-dimensional quantum walk driven by a position-dependent coin. The rotation angle which depends upon the position of a quantum particle parameterizes the coin operator. For different values of the rotation angle, we observe that such a coin leads to a variety of probability distributions, e.g. localized, periodic, classical-like, semi-classical-like, and quantum-like. Further, we study the Shannon entropy associated with position space and coin space of a quantum particle and compare it with the case of the position-independent coin. We show that the entropy is smaller for most values of the rotation angle as compared to the case of the position-independent coin. We also study the effect of entanglement on the behavior of probability distribution and Shannon entropy of a quantum walk by considering two identical position-dependent entangled coins. We observe that in general, a quantum particle becomes more localized as compared to the case of the position-independent coin and hence the corresponding Shannon entropy is minimum. Our results show that position-dependent coin can be used as a controlling tool of quantum walks.