Quantum Probability

In classical probability theory, we define the conditional probability of an outcome given a precondition to be the probability that the precndition is met and the outcome arises divided by the probability that the precondition is met. When the outcome may arise via various mutually exclusive intermediate circumstances, the probability of the outcome is a sum of terms, each of which multiplies a probability of an intermediate circumstance by the probability of the outcome given that intermediate. In quantum mechanics, the same pattern emerges, but with complex amplitudes replacing probabilities; and the actual probability of the outcome being the squared magnitude of its amplitude. My aim here is to explore how these two descriptions correspond, where they say distinct things and where they say the same thing in different ways.

In particular, in classical probability, one distinguishes between correlated circumstances and independent (or uncorrelated) ones; in the case of independent circumstances, the probability that all occur is simply the product of their separate probabilities; indeed, the degree of correlation between circumstances is expressed in terms of how far their joint probabilities depart from what they would be if independent. At some level, the quantum model ought to be encoding the correlations, so it's important to understand how it does so and whether what it's saying is in fact equivalent to the classical model or, if not, what the differences are.

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