<p>In this paper, we study the recurrence of Open Quantum Walks induced by finite-dimensional coins on the line (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation>) and on the grid (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {Z}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>). 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 <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Z</mi> </math></EquationSource> </InlineEquation>. Also, we present a general recurrence criterion for OQWs on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {Z}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> via the open quantum jump chain, obtained from a recurrence criterion for CTOQWs on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {Z}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>.</p>

Recurrence criteria for reducible homogeneous open quantum walks on the line and on the grid

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2026年4月29日

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Newton Loebens

摘要

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 $\mathbb {Z}^2$ $Z^{2}$ .