I would like to get an idea of what "quantum probability" means and how it differs from classical frequentist or Bayesian probability. Can anyone enlighten me in non-too-technical terms?
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In Bayesian probability there is some objective truth that can be discovered with higher and higher certainty if we learn more information and update our distribution. The distribution does not fully describe the system. Rather it helps us to guess what the system might look like.
In the Copenhagen interpretation of quantum mechanics there is no objective truth beyond the wavefunction, which is essentially the distribution. If you know the wavefunction exactly, you have fully described the system.
In ordinary Bayesian probability when you do a measurement you learn more about the system and change the distribution. But that doesn't change the reality of the system. In quantum mechanics if you know the wavefunction you already know everything. In fact, in quantum mechanics when you do a measurement and change the distribution, you are actually changing the physical state of the system rather then only uncovering information about the system's previous state.
For another more technical discussion of quantum mechanics as a kind of probability theory with a 2-norm, see this interesting article. https://arxiv.org/abs/quant-ph/0101012
