Complete Bell state measurement of diamond nuclear spins under a complete spatial symmetry at zero magnetic field

The symmetry of the space where a spin qubit resides plays an essential role in the manipulation of quantum entanglement, which governs the performance of quantum information systems. Application of a magnetic field, which is usually necessary for spin manipulation and readout, inevitably breaks the spatial symmetry to induce competition among quantization axes between internal and external fields, thus limiting the purity of the entanglement. If we could manipulate and readout entanglement under a zero magnetic field, we would be able to avoid the competition among quantization axes to achieve ideally high fidelity. We here demonstrate the complete Bell state measurement, which is a core element of quantum processing, of two carbon nuclear spins in the vicinity of a diamond nitrogen-vacancy center. The demonstration was made possible by holonomic entanglement manipulations based on the geometric phase with a polarized microwave under a zero magnetic field, where the quantization axis is uniquely defined by the hyperfine field. The demonstrated scheme allows high-fidelity entanglement processing even when magnetic fields cannot be applied to the integration of superconducting and spin qubits, thereby paving the way for building fault-tolerant distributed quantum computers and quantum repeater networks.