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A simple coin for a $2d$ entangled walk
by Ahmadullah Zahed, Kallol Sen
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Submission summary
| Ontological classification |
| Academic field: |
Physics |
| Specialties: |
- Mathematical Physics
- Quantum Physics
|
| Approaches: |
Theoretical, Computational |
Abstract
We analyze the effect of a simple coin operator, built out of Bell pairs, in a $2d$ Discrete Quantum Random Walk (DQRW) problem. The specific form of the coin enables us to find analytical and closed form solutions to the recursion relations of the DQRW. The coin induces entanglement between the spin and position degrees of freedom, which oscillates with time and reaches a constant value asymptotically. We probe the entangling properties of the coin operator further, by two different measures. First, by integrating over the space of initial tensor product states, we determine the {\it Entangling Power} of the coin operator. Secondly, we compute the {\it Generalized Relative R\'{e}nyi Entropy} between the corresponding density matrices for the entangled state and the initial pure unentangled state. Both the {\it Entangling Power} and {\it Generalized Relative R\'{e}nyi Entropy} behaves similar to the entanglement with time. Finally, in the continuum limit, the specific coin operator reduces the $2d$ DQRW into two $1d$ massive fermions coupled to synthetic gauge fields, where both the mass term and the gauge fields are built out of the coin parameters.
Author: Kallol Sen on 2023-05-25 [id 3683]
(in reply to Report 1 on 2023-05-24)The authors thank the reviewer for insightful comments on the work. The practical relevance of the exact solutions in 2d lies in the context of the algorithm. We have focussed on the question of finding a coin operator that induces entanglement in the 2d walk. A coin built from minimal set of bell pairs admits analytic solution while preserving the versatile dynamical aspects of the walk. This is precisely the context in which the analytical solutions become appealing. This serves as a building block for constructing more generalized walk algorithms in higher dimensions and for quantum searches. Also, with analytical solutions, essential dynamical features of the walk e.g. entanglement, entropy and entangling power are extrapolated for asymptotic expansions leading to interesting continuum limits connecting quantum field theories with real time simulations of their discrete versions.