bibkey: donkerslootsondenheimersperling2026walks
authors: Emil K. F. Donkersloot; René Sondenheimer; Jan Sperling
year: 2026
title: "Multipartite entanglement dynamics in quantum walks"
doi: 10.48550/arXiv.2603.24679
url: https://arxiv.org/abs/2603.24679v1
claim: "The paper computes the geometric measure of multipartite entanglement of single-walker quantum walks in linear optical networks, where one photon in M modes is a generalized W state, studies the Hadamard walk on a circle of P positions with localized initial coins, and, from numerics for all tested graph sizes and times, expects the symmetric coin (|-> + i|+>)/sqrt 2 to always yield the maximum entanglement for the partition into individual modes."
strata_touched:
  - D5/S3/Quantum/Walks/SymmetricCoinMaximalWalkEntanglement
license: citation-only
triage: anchor

Donkersloot, Sondenheimer and Sperling, multipartite entanglement of quantum walks

E. K. F. Donkersloot, R. Sondenheimer and J. Sperling, arXiv:2603.24679v1 [quant-ph] (25 March 2026), DOI 10.48550/arXiv.2603.24679 (the arXiv DataCite DOI); the arXiv record gives no journal reference. Quotations are from the v1 TeX source, with macros expanded and cross-references omitted.

The Hadamard walk on a circle (Section II A): the coin space is $\mathrm{span}\{|-⟩,|+⟩\}$, the coin is $\hat{H}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$, the step operator moves $|-,x⟩$ to $|-,x-1⟩$ and $|+,x⟩$ to $|+,x+1⟩$ with periodic positions $x\in \{0,\dots ,P-1\}$, and $|\Psi (n)⟩={\hat{T}}^n|{\psi }_0⟩$ with $\hat{T}=\hat{S}(\hat{H}\otimes 1)$. A single walker is one photon in the $M=2P$ optical modes.

Entanglement (Sections II B–C): $g_{\max }$ is the supremum of $|⟨\vec{\chi }|\psi ⟩{|}^2$ over normalized product states for the partition, $E_g=1-g_{\max }$, and single-photon states are the generalized W states $|\psi ⟩={\sum }_{i=1}^M{\lambda }_i|1_i⟩$.

The expectation (Section III C 2), for localized initial states $(\cos \frac{\theta }{2}|-⟩+e^{i\varphi }\sin \frac{\theta }{2}|+⟩)\otimes |0⟩$ and the partition into individual modes:

The choice $\theta =\pi /2,\varphi =\pi /2$ for the angles that define the initial coin corresponds to a symmetric walk as the evolutions of the two components of the initial state, $|-,0⟩$ and $i|+,0⟩$, do not interfere. Hence, we expect this walk to always yield the maximum entanglement. This is indeed the observed behavior for all graph sizes and time steps we have tested.

Verified locator

Source: https://arxiv.org/abs/2603.24679v1

• DOI: https://doi.org/10.48550/arXiv.2603.24679 (the arXiv-issued DataCite DOI; no journal version is listed).
• URL: https://arxiv.org/abs/2603.24679v1 (the e-print is a gzipped tar archive; member arxiv-v1.tex, 1099 lines, md5 d663ab8d3f72affb6922ca126d775cee):
  • the Hadamard walk on a circle, l. 139–166;
  • the optical modes of a single walker, l. 168–213;
  • entanglement, $g_{\max }$ and $E_g$, l. 223–279;
  • generalized W states, l. 286–295;
  • localized initial states, l. 423–425;
  • the expectation, l. 651–654.
• The author list and the version date are from the arXiv metadata.