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

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.

💡 Research Summary

The paper proposes a general and experimentally feasible protocol for transferring and accumulating entanglement from remote electron‑spin qubits (the communication qubits) to high‑spin nuclear‑spin qudits (the memory qudits) that reside in solid‑state defect centers. The authors consider a generic two‑node architecture in which each node contains an electron spin (a two‑level system) coupled via the hyperfine interaction to a central nuclear spin of arbitrary quantum number (I). The nuclear spin thus provides a Hilbert space of dimension (d\le 2I+1).

The protocol proceeds in three steps per iteration: (i) an entangled Bell state (|\phi_{ee}\rangle) is generated between the two distant electron spins, for example by a spin‑photon interface using optical cavities; (ii) in each node a controlled gate (C(U_{0},U_{1})) is applied, where the electron’s logical state (|0\rangle) or (|1\rangle) selects one of two unitary operations (U_{0}) or (U_{1}) acting on the nuclear spin; (iii) both electron spins are measured in the X basis (or any basis orthogonal to Z). The measurement outcome projects the nuclear pair onto a new pure two‑qudit state.

Mathematically the effective non‑unitary map for a given outcome ((j_{a},j_{b})) is denoted (T_{j_{a}j_{b}}). The authors introduce a Schmidt‑decomposition based entanglement measure (von‑Neumann entropy of the reduced density matrix) and derive explicit conditions under which the entanglement of the nuclear pair increases by exactly the amount of entanglement present in the electron Bell pair, i.e. a deterministic complete entanglement transfer. The key condition is that the two nuclear unitaries satisfy
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