There is lots of mathematical evidence that every quantum state maps 1-to-1 on a physical state¹, or at least be uniquely determined by the physical state². The

There is lots of mathematical evidence that every quantum state maps 1-to-1 on a physical state¹, or at least be uniquely determined by the physical state². The main assumption both make is that there is some physical state. So it doesn't really matter if it's a map or not - it tells you something about the territory. [1] The completeness of quantum theory for predicting measurement outcomes, https://arxiv.org/abs/1208.4123 https://arxiv.org/abs/1208.4123 [2] On the reality of the quantum state, https://arxiv.org/abs/1111.3328v3 https://arxiv.org/abs/1111.3328v3