Smooth bijective mapping between the world of quantum mechanics and that of statistics

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To formalize the title - I'm looking for a smooth, bijective mapping between the space of linear maps between the $\mathbb{C}^\infty$ unit sphere and itself, and the space of linear maps between $(0,1)^\infty$ and itself. I wouldn't even know where to begin looking for something like that, or what it'd be called in more idiomatic language, so vocabulary lessons would also be appreciated.