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arXiv:1109.1600

On the von Neumann and Shannon Entropies for Quantum Walks on Z^2

Clement Boateng Ampadu

Published Sep 7, 2011• 3 citations
arXivHTMLPDF

Abstract

We give asymptotic behaviors of the von Neumann entropy and the Shannon entropy of discrete-time quantum walks on Z^2
DOI
10.48550/arxiv.1109.1600

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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.
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The decoherence matrix studied by Gudder and Sorkin (2011) can be considered as a map from the set of all the pairs of nnn-length paths to complex numbers, which is induced by the discrete-time quantum walk. The decoherence matrix is one of the decoherence functionals which present their historical quantum measure theory. In this paper, we compute the von Neumann entropy of the decoherence matrix. To do so, we use the result that the eigensystem of the decoherence matrix can be expressed by a corresponding correlated random walk.