I am studying how to apply neural networks to the problem of Quantum State Tomography and I got confused when it comes to decide if this is a supervised or unsupervised learning problem.

At first, I came across the usage of Restricted Boltzmann Machines in the domain of generative models, which are used for unsupervised learning, which seems right since the state tomography is a problem where I do not have access to the state in question and the only information available are measurement outcomes: thus, I am feeding my learning algorithm with unlabeled data, that is, in an unsupervised fashion.

But then I found this paper and also this one describing QST in the PAC-learning framework, which as far as I know is a framework applied to supervised problems only, but the paper seems ok since they were dealing with a training set of $(E, Tr(E\rho))$ where $E$ are measurement operators and $Tr(E\rho)$ are the trace of the measurement operator times the density matrix. The elements of the training set were drawn i.i.d from a probability distribution.

My problem here is: in QST the probability distribution and the density matrix are unknown. So how can I formulate QST as pac-learning and how can it be suited into the framework of a supervised learning algorithm? Or it depends on what kind of algorithm I am using?

RBMs are not the only approach. Nowadays, researchers are using deep neural networks and a lot of generative modeling, the most promissing being the usage of Conditional GANs.

What am I missing?