Unsupervised state learning from pairs of states

Unsupervised learning on quantum data requires selecting appropriate measurement to classify the underlying quantum states. In particular, suppose we have an ensemble, which is a mixture of two unknown pure states, and we want to determine those pure states. An inherent problem is that what we will be able to determine from measurements is the density matrix of the ensemble, but finding the pure states from the density matrix is, in general, not possible due to the inherent ambiguity in decomposing the density matrix. We find that we can circumvent this limitation if the quantum data consists of pairs of identical quantum states. The presence of paired data provides extra information about the states, which enables us to extract additional information from density matrix. The latter is determined by measuring the data using established tomography techniques. After obtaining the density matrix, we can apply one of three given decomposition techniques to approximate the individual pure states and their priors. We demonstrate this method using numerical simulations.