Open quantum walks

Stéphane Attal, Francesco Petruccione, Christophe Sabot +1 more

Stéphane Attal, Francesco Petruccione, Ilya Sinayskiy

Newton Loebens

In this paper, we study the recurrence of Open Quantum Walks induced by finite-dimensional coins on the line ($\mathbb{Z}$) and on the grid ($\mathbb{Z}^2$). Two versions are considered: discrete-time open quantum walks (OQW) and continuous-time open quantum walks (CTOQW). We present three distinct recurrence criteria for OQWs on $\mathbb{Z}$, each adapted to different types of coins. The first criterion applies to coins whose auxiliary map has a unique invariant state, resulting in the first recurrence criterion for Lazy OQWs. The second one applies to Lazy OQWs of dimension 2, where we provide a complete characterization of the recurrence for this low-dimensional case. The third one presents a general criterion for finite-dimensional coins in the non-lazy case, which generalizes many of the previously known results for OQWs on $\mathbb{Z}$. Also, we present a general recurrence criterion for OQWs on $\mathbb{Z}^2$ via the open quantum jump chain, obtained from a recurrence criterion for CTOQWs on $\mathbb{Z}^2$.

Newton Loebens

Abstract In this paper, we study the recurrence of Open Quantum Walks induced by finite-dimensional coins on the line ( $$\mathbb {Z}$$ Z ) and on the grid ( $$\mathbb {Z}^2$$ Z 2 ). Two versions are considered: discrete-time open quantum walks (OQW) and continuous-time open quantum walks (CTOQW). We present three distinct recurrence criteria for OQWs on $$\mathbb {Z}$$ Z , each adapted to different types of coins. The first criterion applies to coins whose auxiliary map has a unique invariant state, resulting in the first recurrence criterion for Lazy OQWs. The second applies to Lazy OQWs of dimension 2, where we provide a complete characterization of the recurrence for this low-dimensional case. The third one presents a general criterion for finite-dimensional coins in the non-lazy case, which generalizes several of the previously known results for OQWs on $$\mathbb {Z}$$ Z . Also, we present a general recurrence criterion for OQWs on $$\mathbb {Z}^2$$ Z 2 via the open quantum jump chain, obtained from a recurrence criterion for CTOQWs on $$\mathbb {Z}^2$$ Z 2 .

Franklin de Lima Marquezino, Renato Portugal, Carlile Lavor

Ilya Sinayskiy, Francesco Petruccione

The concept of open quantum walks (OQW), quantum walks exclusively driven by the interaction with the external environment, is reviewed. OQWs are formulated as discrete completely positive maps on graphs. The basic properties of OQWs are summarised and new examples of OQWs on and their simulation by means of quantum trajectories are presented.