Qubit-qudit entanglement transfer in defect centers with high-spin nuclei

TL;DR

The paper addresses how to accumulate entanglement between distant memory qudits stored in defect-center nuclei by repeatedly transferring entanglement from electron-spin communication qubits through the Ising hyperfine interaction. It introduces a universal, driving-free phase-set protocol that deterministically generates maximal entanglement for qudits with dimension $d=2^n$ and provides probabilistic schemes for other $d$, including a two-iteration method for $d=3$ with $1/2$ success. The approach is instantiated in a generic defect-center network model and elaborated for two-node and multipartite networks, with concrete guidance on photonic entanglement generation and phase-controlled transfer. The results offer a scalable pathway to high-dimensional quantum networking and MBQC using nuclei with large $I$, exemplified by the ${}^{73}$GeV center in diamond and related systems. The framework promises faster, memory-efficient quantum communication and richer resource states in near-term solid-state platforms.$

Abstract

We propose a scheme for accumulating entanglement between long-lived qudits provided by central nuclear spins of defect centers. Assuming a generic setting, the electron spin of each node acts as the communication qubit and may be entangled with other nodes, e.g., through a spin-photon interface. The generally available Ising component of the hyperfine interaction is shown to facilitate repeated entanglement transfer onto memory qudits of arbitrary dimension $d\le 2I+1$ with $I$ the nuclear spin quantum number. When $d$ is set to an integer power of two, maximal entanglement can be generated deterministically and without intermittent driving of nuclear spins. The scheme is applicable to several candidate systems, including the $^{73}$Ge germanium vacancy in diamond.

Figures (5)

• Figure 1: Scheme for iteratively entangling two memory qudits $n_a,n_b$. The dash-dotted box contains the photonic setup with optical arms $O_1$ and $O_2$ for entangling remote electron spin qubits $e_a,e_b$, producing the Bell state $|\Psi^+\rangle$. The dashed box corresponds to one iteration of entanglement transfer. The $\frac{\pi}{2}$-gates are $Y^{1/2}$ rotations. The nuclear initial state $\ket{+_d}$ is defined in \ref{['eq:psi0']}.
• Figure 2: First iteration of entanglement transfer (${d}=8$, $\ket{\phi_{ee}}=\ket{\Psi^{+}}$), showing entanglement and probability for individual measurement outcomes $(j_a,j_b)$ as well as the expected entanglement, as functions of the CPHASE angle.
• Figure 3: Generating maximal entanglement deterministically in three iterations for ${d}=8$. The expected entanglement without postselection is shown as a function of the accumulated CPHASE angle. Only the fastest phase combinations are shown (set \ref{['eq:scheme2']}, any ordering).
• Figure 4: Generating maximal entanglement in two iterations for ${d}=3$, using a postselection of $j_{a}^{(\nu)}=j_{b}^{(\nu)}$ in each iteration $\nu$. Different panels correspond to different entangled electron states as indicated. The electron qubit splitting parameter is chosen to be $\xi=20$.
• Figure 5: Expected entanglement without postselection for ${d}=3$, for three different types of electron resource states. The electron qubit splitting parameter is chosen to be $\xi=20$ for the solid lines, while the black thin dashed lines are for $\xi=0$.