Quantum coherence is a fundamental feature of quantum mechanics and a key factor that distinguishes quantum mechanics from classical theories. From theoretical and practical perspectives, the characterization and quantification of coherence are crucial problems in quantum information science. Although quantum coherence has been recognized as a quantum resource and a systematic framework for its quantification has been developed, existing measurements generally depend on a pre-fixed reference basis. This dependence poses significant challenges in practical scenarios where the reference frame may be misaligned or the measurement devices may be uncharacterized.

To overcome these limitations, we detect quantum coherence within a semi-device-independent (SDI) framework. We introduce the concept of an “incoherent source,” defined as a collection of unknown quantum states that are jointly diagonalizable on an unspecified basis. By using the rank analysis theory of Gram matrices, we transform the problem of coherence detection into the evaluation of experimental correlation matrices. This approach eliminates the need for prior knowledge of the state’s density matrix or the alignment of measurement bases, requiring only the assumption of a bounded Hilbert space dimension (e.g. qubits).

We systematically construct two types of coherence witnesses: linear inequalities and nonlinear determinant-based criteria. For the minimal resource case involving two preparations and two measurements, n=2,\ m=2, we derive a linear witness W_1 and prove its tight upper bounds for classical and coherent systems. Moreover, we demonstrate that the determinant of the data matrix B (or B^\rm T B) serves as a sharp, nonlinear witness. A non-zero determinant unambiguously implies \rmrank(B) \geqslant 2, providing a robust and conclusive test for coherence.

Furthermore, we demonstrate that this framework not only has the ability to detect coherence, but also has remarkable discriminatory power. By increasing the number of preparations and measurements, the rank of the underlying state correlation matrix A, which is larger than or equal to 3, can be probed. We show that \rmrank(A) \geqslant 3 necessitates a complex quantum system, thus requiring the full complex structure of the Hilbert space. We construct specific linear witnesses (e.g. W_3) that can distinguish three hierarchical levels: classical, real-quantum, and complex-quantum, based solely on experimental data. We also analytically demonstrate that although linear witnesses W_2 in n=3,\ m=3 scenarios fail to isolate complex structures due to geometric overlaps, the nonlinear determinant witness provides a definitive “pinpoint” identification of complex-number quantum systems.

In summary, we establish a comprehensive SDI theory for witnessing quantum coherence and complex-number structure, without the need for state tomography, or trusted measurements, or a pre-defined basis. Our results provide a novel tool for proving quantum resources, which is of great significance for fundamental research on device-independent cryptography, randomness generation, classical--quantum boundaries, and the role of complex numbers in quantum mechanics.