Could a Classical Probability Theory Describe Quantum Systems?

Quantum Mechanics (QM) is a quantum probability theory based on the density matrix. The possibility of applying classical probability theory, which is based on the probability distribution function(PDF), to describe quantum systems is investigated in this work. In a sense this is also the question about the possibility of a Hidden Variable Theory (HVT) of Quantum Mechanics. Unlike Bell's inequality, which need to be checked experimentally, here HVT is ruled out by theoretical consideration. The approach taken here is to construct explicitly the most general HVT, which agrees with all results from experiments on quantum systems (QS), and to check its validity and acceptability. Our list of experimental facts of quantum objects, which all quantum theories are required to respect, includes facts on repeat quantum measurement. We show that it plays an essential role at showing that it is very unlikely that a classical theory can successfully reproduce all QS facts, even for a single spin-1/2 object. We also examine and rule out Bell's HVT and Bohm's HVT based on the same consideration.