One of the fundamental postulates of Quantum Mechanics is that states of a system are linear combinations of ground/observable states. Could someone point out some of the experiments leading to such a model? Is there a more general theory that deals with states that are nonlinear functions of the observable states with Quantum Mechanics being its first order approximation?
First of all, let me point out this post: Is the universe linear? If so, why? where Ron Maimon explains why it is difficult to imagine quantum mechanics as a nonlinear theory.
Now, for experiments. First let me point out that this is difficult. How would you directly test whether quantum mechanics is linear? You can construct linear superpositions (which has been done), but that doesn't tell you that all states are linear superpositions. As long as your experiment confirms quantum mechanics, you can argue that it also confirms linearity, as it is so very fundamental.
The easiest way to test linearity is to have alternative theories to test against. Such theories are very difficult to construct as pointed out above (and because they often lead to inconsistencies).
Somewhat prominent theories include the one by Bialynicki-Birula and Mycielski and the one by Weinberg. Both theories can be attacked from theoretical grounds but have also been attacked on an experimental basis: The first one was tested (for instance) in the experiments reported here: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.44.765 The second one was tested (for instance) in the experiments here: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.63.1031 There are also more indirect tests. Polchinski pointed out that Weinberg's theory would lead to the possibility of superluminal communication. Every test trying to find superluminal communication could therefore (if testing the Polchinski scenario) test Weinberg's theory.
Finally, let me point out the article here: https://arxiv.org/pdf/1002.4673.pdf