Revisiting k-Means Clustering Method for Quantum State Discrimination: Training Dataset and Assignment

Clustering algorithms have received increasing attention as methods for determining unknown quantum states and their potential application in quantum information processing. In particular, quantum k-means is a promising approach to this task; however, its implementation presents practical challenges associated with state preparation, measurement, and the statistical estimation of quantum information. In this work, we investigate the adaptation and integration of the k-means method elements for the characterisation and discrimination of single-qubit mixed states. We analyse the physical limitations of this process and consider possible alternatives, including C-SWAP and multiple-teleportation-based measurements, to estimate the number and locations of the pure-state components or clusters of them on the Bloch sphere. Classical post-processing is further explored to recover cluster information from the resulting measurement distributions as a complementary method. Additionally, the optimal parameter values for centroid assignment are analysed and discussed. In fact, the assignment of incoming states is formulated as a quantum-state discrimination problem for which the square-root measurement provides a suitable strategy. Although simple, this strategy is useful in a limited number of clusters or pure states.The results highlight the physical and statistical limitations of applying k-means to quantum state determination, support hybrid quantum–classical approaches and suggest the potential for experimental implementation of the proposed framework.