26pTE-9 Asymptotic behaviors in Discrete Time Quantum Walks

Toshikazu Sunada, Tatsuya Tate

Yutaka Shikano

The discrete time quantum walk defined as a quantum-mechanical analogue of the discrete time random walk have recently been attracted from various and interdisciplinary fields. In this review, the weak limit theorem, that is, the asymptotic behavior, of the one-dimensional discrete time quantum walk is analytically shown. From the limit distribution of the discrete time quantum walk, the discrete time quantum walk can be taken as the quantum dynamical simulator of some physical systems.

Norio Konno

Adam S. Sajna, T. P. Polak, Antoni Wójcik +1 more

Open quantum walks often lead to a classical asymptotic behavior. Here we look for a simple open quantum walk whose asymptotic behavior can be nonclassical. We consider a discrete-time quantum walk on an $n$-cycle subject to a random coin-dependent phase shift at a single position. This finite system, whose evolution is described by only two Kraus operators, can exhibit all kinds of asymptotic behavior observable in quantum Markov chains: It either evolves towards a maximally mixed state, or partially mixed state, or tends to an oscillatory motion on an asymptotic orbit. We find that the asymptotic orbits do not have a product structure; therefore, the corresponding states can manifest entanglement between the position and the coin degrees of freedom, even if the system started in a product state.

X.-P. Xu

Clement Boateng Ampadu

We give asymptotic behaviors of the von Neumann entropy and the Shannon entropy of discrete-time quantum walks on Z^2