Efficient Self‐Guided Estimation of Schmidt Number via Local Measurements

Recent advances in experimental quantum platforms have enabled the development of devices capable of generating high‐dimensional quantum states, facilitating their use in quantum information processing tasks. High‐dimensional entangled states offer significant advantages in areas such as quantum key distribution, quantum communication, and quantum metrology. While the Schmidt rank (SR) characterizes entanglement for pure bipartite states in high‐dimensional spaces, its extension to mixed states is the Schmidt number (SN). However, experimentally determining the SN of unknown mixed states remains a significant challenge. Here, we propose an experimentally feasible method to determine the SN of unknown states, constructing a self‐guided SN witness scheme. Leveraging techniques from quantum state verification (QSV), this method allows SN estimation using only local measurements, thereby avoiding the need for direct projections onto high‐dimensional maximally entangled states. The self‐guided aspect is implemented through a Stochastic Gradient Algorithm (SGA) that systematically explores the space of local unitary operations to maximize a fidelity related to Schmidt number witnesses. We numerically demonstrate that for four‐dimensional entangled states, our scheme accurately estimates the SN, typically converging within approximately 200 iterations. Our results also show that the method is robust to realistic imperfections, highlighting its potential for practical applications in quantum information processing.