Probabilistic joint remote implementation of quantum operations

Joint remote implementation of quantum operations (JRIO) is crucial for distributed quantum computation. We propose a probabilistic JRIO scheme using a three-photon nonmaximally hyperentangled Greenberger-Horne-Zeilinger (GHZ) state, which is simultaneously entangled in the polarization and spatial-mode degrees of freedom (DOFs), as the shared quantum resource. This JRIO protocol operates on two core principles: transferring the quantum-state information of the target qubit to the senders’ joint system via polarization-DOF entanglement and applying local operations, then executing quantum operations on the receiver’s system using entanglement in the spatial-mode DOF. The construction of the proposed scheme requires weak cross-Kerr nonlinearities, X homodyne measurements, simple linear optical elements, and common single-photon detectors. This work extends the framework of the previous JRIO protocol to the scenario of a nonmaximally hyperentangled quantum channel, enriching the theoretical family of distributed quantum operation schemes.