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arXiv:1202.1376

On the entropy of decoherence matrix for quantum walks

Norio Konno, Etsuo Segawa

Published Feb 7, 2012• 1 citation
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Abstract

The decoherence matrix studied by Gudder and Sorkin (2011) can be considered as a map from the set of all the pairs of $n$-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.

DOI
10.48550/arxiv.1202.1376

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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.
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.
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.
In this paper, we consider a discrete-time quantum walk on theN-cycle governed by the condition that at every time step of the walk, the option persists, with probabilityp, 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.
We investigate quantum dynamical systems defined on a finite dimensional Hilbert space and subjected to an interaction with an environment. The rate of decoherence of initially pure states, measured by the increase of their von Neumann entropy, averaged over an ensemble of random pure states, is proved to be bounded from above by the partial entropy used to define the ALF dynamical entropy. The rate of decoherence induced by the sequence of the von Neumann projectors measurements is shown to be maximal, if the measurements are performed in a randomly chosen basis. The numerically observed linear increase of entropies is attributed to free-independence of the measured observable and the unitary dynamical map.