Unsupervised state learning from pairs of states

A fundamental challenge in unsupervised quantum learning is determining unknown pure states from an ensemble. When given a sequence of qubits, each guaranteed to be in one of two unknown pure states, standard measurements only reveal the ensemble's density matrix. However, density matrices generally admit multiple decompositions into pure states, making unique identification impossible without additional information. We demonstrate that supplying paired quantum data where each pair consists of two identical copies of the same unknown state which resolves this ambiguity. This pairing provides crucial quantum information that enables unique determination of the unknown states and their probabilities of occurrence. We numerically simulate measurements on these qubit pairs and show that the unknown states and their respective probabilities of occurrence can be inferred with high accuracy. Finally, we propose a practical experiment using spontaneous parametric down-conversion with polarization-based product measurements, demonstrating that this approach is realizable with existing quantum optics technology and that statistical analysis of the plausible parameter region confirms strong evidence for accurate state determination.