one random variable is exactly as it was before, but random variables no
longer have to commute. Remarkably, quantum probability is empirically true,
in the same sense that classical probability is true in science. We simply
do not usually see it, because most of our affairs lie within a commutative
realm of the non-commutative reality.
Quantum probability is also an unavoidable generalization: it cannot exist
within classical probability. One rigorous version of this conclusion
is that quantum probability violates Bell-type inequalities. However,
traditional Bell-type protocols, which are also called non-locality
demonstrations, are statistically inefficient. They consume many quantum
bits of data for each bit of persuasion. As an example problem in quantum
probability, I will describe some new protocols that are more efficient. I
will also discuss upper bounds on how efficient any such protocol can be,
and related questions.