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.
Written by Eddy.