We study decoherence in the quantum walk on the xy-plane. We generalize the method of decoherent coin quantum walk, introduced by [T. A. Brun, et.al, Phys.Rev.A 67 (2003) 032304], which could be applicable to all sorts of decoherence in two dimensional quantum walks, irrespective of the unitary transformation governing the walk. As an application we study decoherence in the presence of broken line noise in which the quantum walk is governed by the two-dimensional Hadamard operator.
We study a one-parameter family of discrete-time quantum walk models on Z and Z 2 associated with the Hadamard walk. Weak convergence in the long-time limit of all moments of the walker's pseudovelocity on Z and Z 2 is proved. Symmetrization on Z and Z 2 is theoretically investigated, leading to the resolution of the Konno-Namiki-Soshi conjecture in the special case of symmetrization of the unbiased Hadamard walk on Z. A necessary condition for the existence of a phenomenon known as localization is given.
Partially inspired by [Erdal Karapinar, Ravi Agarwal and Hassen Aydi, Interpolative Reich-Rus-Ćirić type contractions on partial metric spaces, Mathematics 6 (2018), 256] and [V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlinear Anal. Forum 9(1) (2004), 43-53], we introduce a concept of interpolative Berinde weak contraction, and obtain some existence theorems for mappings satisfying such a contractive definition, and some of its extensions.
In this paper we consider the model with decoherence operators introduced by [Brun,T.A, et.al, Phys.Rev.A 67 (2003) 032304] which has recently been considered in the two-dimensional setting by [Ampadu,C., Brun-Type Formalism for Decoherence in Two Dimensional Quantum Walks, Communication in Theoretical Physics To Appear, arXiv:1104.2061] to obtain the limit of the decoherent quantum walk.
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