Construction of a 4-uniform 12-qubit entangled state based on recurrence relations

The quantification of quantum entangled states is essential for developing entanglement resources, and this process has profound implications for advancing quantum computing and quantum communication. To address the challenge of quantifying such systems beyond 10 qubits, we present a method for constructing high-dimensional k -uniform states using a 9-qubit entangled state and recurrence relations. By incrementally increasing the number of qubits, we construct 10-, 11-, and 12-qubit entangled states and verify their uniformity using reduced density matrices. We demonstrate for the first time that the 12-qubit state is 4-uniform as all 4-qubit subsystems exhibit maximal mixedness. Compared to conventional designs, our recurrence-relation-based approach emphasizes intrinsic multiqubit correlations, offering a direct method for quantifying high-dimensional entanglement. The 4-uniform state offers distinct advantages for quantum error correction and fault-tolerant computation, laying a foundation for large-scale quantum information systems.