Designing three-way entangled and nonlocal two-way entangled single particle states via alternate quantum walks

TL;DR

The paper addresses generating genuine 3-way SPE among x, y and coin, and optimal nonlocal 2-way SPE between x and y, by evolving a single-particle state through a generalized 2D AQW with a resource-saving single-qubit coin. It introduces a formal framework and optimizes coin parameters and initial phases to maximize the $\pi$-tangle and negativity, respectively, while verifying CKW-type monogamy and LU invariance. It provides explicit optimal coins and time steps, demonstrates maximal entanglement values, and argues for photonic feasibility with PBS and Jones-plates. The results offer a scalable pathway for high-dimensional SPE in quantum information processing and hybrid quantum networks, with practical photonic realizations and accompanying open-source code.

Abstract

Entanglement with single-particle states is advantageous in quantum technology because of their ability to encode and process information more securely than their multi-particle analogs. Threeway and nonlocal two-way entangled single-particle states are desirable in this context. Herein, we generate genuine three-way entanglement from an initially separable state involving three degrees of freedom of a quantum particle, which evolves via a 2D alternate quantum walk employing a resource-saving single-qubit coin. We achieve maximum possible values for the three-way entanglement quantified by the π-tangle between the three degrees of freedom. We also generate optimal nonlocal two-way entanglement, quantified by the negativity between the nonlocal position degrees of freedom of the particle. This prepared architecture using quantum walks can be experimentally realized with a photon.

Figures (11)

• Figure 1: (a) 2D AQW describing a quantum particle (green diamond) shifting either along $x$ or $y$-direction after a coin operation via $C_x$ or $C_y$ operator. (b) 2D AQW protocol for generating both genuine 3-way and nonlocal 2-way entanglement for a single-particle state, starting from an arbitrary separable initial state given in Eq. (\ref{['eq1']}).
• Figure 2: Three-way entanglement $\pi_{av}$ vs time steps($t$) for spatial evolution sequences: (a) $M1_xM1_y...$ for arbitrary separable initial state (Eq. (\ref{['eq1']})) with $\phi=\pi,\frac{\pi}{2}$; (b) $M2_xM2_y...$ for an arbitrary separable initial state with $\phi=\frac{\pi}{8},\frac{\pi}{2}$. Note that at all time steps, $M1_xM1_y$ with $\phi=\pi$ and $M2_xM2_y$ with $\phi=\frac{\pi}{2}$ yield the best values for 3-way entanglement among the cases we have studied.
• Figure 3: (a) 3-way entanglement ($\pi_{xyc}$) between $x,y$ and coin DoF vs the initial state parameter $\theta$ for spatial evolution sequences: $M1_xM1_y$ for the phase variable (of the initial state) $\phi=\pi$, and $M2_xM2_y$ for $\phi=\frac{\pi}{2}$ at time step $t=15$; (b) Nonlocal 2-way entanglement ($N$) between $x,y$ DoF vs the initial state parameter $\theta$ for spatial evolution sequences: $G1_xG1_y$ for the phase variable (of the initial state) $\phi=\pi$, $G2_xG2_y$ for $\phi=\frac{\pi}{2}$ at time step $t=15$.
• Figure 4: Average nonlocal 2-way entanglement ($N_{av}$) between $x,y$ DoF vs time steps($t$) for spatial evolution sequences: (a) $G1_xG1_y...$ for the arbitrary separable initial state with $\phi=\pi,\frac{5\pi}{8}$, (b) $G2_xG2_y...$ for arbitrary separable initial state with $\phi=\frac{\pi}{8},\frac{\pi}{2}$. Note that $G1_xG1_y$ and $G2_xG2_y$ yield maximum 2-way entanglement respectively for $\phi=\pi$ and $\phi=\frac{\pi}{2}$.
• Figure 5: Single-photon based realization for generating genuine 3-way and nonlocal 2-way single-particle entangled states via 2D AQW at time steps up to $t=3$.